Progress in Physical Chemistry Volume 4: Ionic Motion in Materials with Disordered Structures - From Elementary Steps to Macroscopic Transport 9783486711448, 9783486705508

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Progress in Physical Chemistry Volume 4 Ionic Motion in Materials with Disordered Structures – From Elementary Steps to Macroscopic Transport by

Prof. Dr. Klaus Funke (Ed.) Series Editor: Helmut Baumgärtel

Oldenbourg Verlag München

Preface to the Series – like all natural sciences Physical Chemistry also is strongly affected by a development which leads to an inflation of information details which cannot be thoroughly covered in its depth by the regular journal world. Therefore the editors and the publisher of Zeitschrift für Physikalische Chemie have decided to provide a platform for scientists to present their research and results on a broader basis. Thus the book series “Progress in Physical Chemistry” has been created. The first volume of the series is devoted to “Different Aspects of Intermolecular Interaction”. It collects most important reviews on the topic published in Zeitschrift für Physikalische Chemie between 2004 and 2006. Volume 2 covers the results of the Collaborative Research Center (SFB) 277 of the German Research Foundation (DFG). Volume 3 covers the results of the Priority Programm (SPP) 1145 of the German Research Foundation (DFG). Volume 4 is the final report of the Collaborative Research Centre (SFB) 458 of the German Research Foundation (DFG). Helmut Baumgärtel (Series Editor)

Bibliografische Information der Deutschen Nationalbibliothek Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.d-nb.de abrufbar.

© 2011 Oldenbourg Wissenschaftsverlag GmbH Rosenheimer Straße 145, D-81671 München Telefon: (089) 45051-0 www.oldenbourg-verlag.de Das Werk einschließlich aller Abbildungen ist urheberrechtlich geschützt. Jede Verwertung außerhalb der Grenzen des Urheberrechtsgesetzes ist ohne Zustimmung des Verlages unzulässig und strafbar. Das gilt insbesondere für Vervielfältigungen, Übersetzungen, Mikroverfilmungen und die Einspeicherung und Bearbeitung in elektronischen Systemen. Lektorat: Birgit Zoglmeier Herstellung: Constanze Müller Einbandgestaltung: hauser lacour Gesamtherstellung: Druckhaus „Thomas Müntzer“ GmbH, Bad Langensalza Dieses Papier ist alterungsbeständig nach DIN/ISO 9706. ISBN 978-3-486-70550-8

Preface

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Synthesis and Characterisation of New Materials M. Burjanadze, Y. Karatas, N. Kaskhedikar, L. M. Kogel, S. Kloss, A.-C. Gentschev, M. M. Hiller, R. A. Müller, R. Stolina, P. Vettikuzha, H.-D. Wiemhöfer Salt-in-Polymer Electrolytes for Lithium Ion Batteries Based on Organo-Functionalized Polyphosphazenes and Polysiloxanes . . .

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R. Pöttgen, T. Dinges, H. Eckert, P. Sreeraj, H.-D. Wiemhöfer Lithium-Transition Metal-Tetrelides – Structure and Lithium Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

T. Nilges, M. Bawohl, O. Osters, S. Lange, J. Messel Silver(I)-(poly)chalcogenide Halides – Ion and Electron High Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Experimental Investigation of Disorder and Ionic Motion C. Brinkmann, S. Faske, B. Koch, M. Vogel NMR Multi-Time Correlation Functions of Ion Dynamics in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103

´ W. Imre, A. Bhide, C. Cramer M. Schönhoff, A. Mechanisms of Ion Conduction in Polyelectrolyte Multilayers and Complexes . . . . . . . . . . . . . . . . . . . . . . . . .

123

H. Eckert Short and Medium Range Order in Ion-Conducting Glasses Studied by Modern Solid State NMR Techniques . . . . . . . . . . . . .

159

H. Staesche, B. Roling Nonlinear DC and Dispersive Conductivity of Ion Conducting Glasses and Glass Ceramics . . . . . . . . . . . . . . . . . . . . . . .

223

K. Sunder, M. Grofmeier, R. Staskunaite, H. Bracht Dynamics of Network Formers and Modifiers in Mixed Cation Silicate Glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

245

N. A. Stolwijk, M. Wiencierz, J. Fögeling, J. Bastek, S. Obeidi The Use of Radiotracer Diffusion to Investigate Ionic Transport in Polymer Electrolytes: Examples, Effects, and Their Evaluation

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275

L. van Wüllen, T. Echelmeyer, N. Voigt, T. K.-J. Köster, G. Schiffmann Local Li Cation Coordination and Dynamics in Novel Solid Electrolytes . . . . . . . . . . . . . . . . . . . . . . . . . . .

303

M. Kunze, A. Schulz, H.-D. Wiemhöfer, H. Eckert, M. Schönhoff Transport Mechanisms of Ions in Graft-Copolymer Based Salt-in-Polymer Electrolytes . . . . . . . . . . . . . . . . . . .

339

G. Schmitz, R. Abouzari, F. Berkemeier, T. Gallasch, G. Greiwe, T. Stockhoff, F. Wunde Nanoanalysis and Ion Conductivity of Thin Film Battery Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . .

363

A. Schirmeisen, A. Taskiran, H. Bracht, B. Roling Ion Jump Dynamics in Nanoscopic Subvolumes Analyzed by Electrostatic Force Spectroscopy . . . . . . . . . . . . . . . .

399

Simulation and Development of Model Concepts S. Röthel, R. Friedrich, L. Lühning, A. Heuer Theoretical Description of Ion Conduction in Disordered Systems: From Linear to Nonlinear Response . . . . . . . . . . . . . . . .

423

K. Funke, R. D. Banhatti, D. M. Laughman, L. G. Badr, M. Mutke, ˇ A. Santi´ c, W. Wrobel, E. M. Fellberg, C. Biermann First and Second Universalities: Expeditions Towards and Beyond . .

459

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519

Keywords . . . . . . . . . . . . . . . . . . . . . . . . . . . .

520

Author Index

Preface

Volume 4 of “Progress in Physical Chemistry” comprises 15 invited minireview articles. Their common theme is the study of ionic motion in materials with disordered structures, from elementary steps to macroscopic transport. They thus address the most basic questions of SOLID STATE IONICS, a discipline which has its roots in the physics and chemistry of solids and has more recently become a rapidly developing branch of materials science and engineering. The contributions to this volume have been written by those who were project leaders and members of Sonderforschungsbereich (Collaborative Research Centre) 458 at Münster, Germany. SFB 458, entitled Ionenbewegung in Materialien mit ungeordneten Strukturen – vom Elementarschritt zum makroskopischen Transport, was funded by Deutsche Forschungsgemeinschaft, DFG, from January 2000 until December 2009. The contributions have also been published in a special topical issue of “Zeitschrift für Physikalische Chemie” (2010/10–12). In the field of SOLID STATE IONICS, interest is at present largely focused on applications of disordered ion-conducting materials in high-technology devices, including advanced solid-state battery systems, fuel cells and chemical sensors. However, regarding the ionic materials involved, a thorough understanding of their structures and ion dynamics is imperative for an improvement and optimisation of their desired transport properties. The endeavour to achieve this understanding has always been, and still is, a major scientific challenge, constituting an area of basic research in its own right. Unravelling the microscopic mechanisms behind the macroscopic transport phenomena was, indeed, the main objective of SFB 458. The present mini-review articles are meant to be concise reports on the progress made along these lines during the last few years. The thematic leitmotif of this volume is best captured in the SFB logo, as shown in Fig. 1. What at first sight seems to resemble the initials of Westfälische Wilhelms-Universität at Münster, is in effect a highly non-trivial, non-periodic dynamic energy landscape in which mobile ions hop from site to site. Here, each displacement of an ion immediately changes the shapes of the potentials felt by its neighbours, thereby influencing their further motion. Evidently, the combined appearance of disorder and interaction creates a most complicated many-particle problem.

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Preface

Fig. 1. The thematic leitmotif of this volume, captured in the logo of SFB 458.

In materials with high ionic conductivities, structural – and hence dynamic – disorder is the key property. Figure 2, which has been called an “evolving scheme of materials science”, can serve to highlight the remarkable implications of this property with regard to the movements of the ions. In the classification of Fig. 2, perfectly ordered crystals are placed at level one. At this level, there is no ionic transport, as the ions cannot leave their sites. Historically, the decisive step forward was made when site disorder was discovered and point-defect thermodynamics were developed. At this stage, which we call level two, ionic transport is accomplished by point defects moving randomly from site to site. In fact, modern materials science and engineering build on the concept of level two. Dramatic changes are encountered as we move on to ionic materials with disordered structures, i.e., from level two to level three. At this stage, it is impossible to define point defects in the sense of isolated structural elements. It is even no longer sensible to speak of defects, since the entire structure is disordered. As a consequence, ionic transport can surely no longer be described in terms of a random hopping of individual defects in a static energy landscape. Rather, one has to consider irregularly placed mobile ions that interact with each other and with their surrounding matrix. This scenario, as sketched in Fig. 1, is characterised by time-dependent single-particle potentials causing highly correlated ionic movements. The transition from level two to level three is found to be accompanied by prominent changes in the shapes of functions that are experimentally accessible. For instance, the ionic conductivity assumes a characteristic frequency dependence, and spin-lattice relaxation rates are no longer properly described by the model of Bloembergen, Purcell and Pound. While these features of the ion dynamics are common to level-three ion conductors, reflecting “universal” dynamic properties, there are others as well which specifically vary between materials, strongly depending on their structures and compositions. The multitude of level-three materials requires a guideline such as the one provided by the evolving scheme of Fig. 2, which contains sublevels ranging from 3a to 3d. The scheme starts out with systems that appear comparatively simple with regard to their structures and ion dynamics and then leads on to more complicated ones offering unprecedented possibilities for technical applications. It does, indeed, reflect the recent development of the field of SOLID

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Preface

3a. 2. 3b. 1. 3c.

3d.

Fig. 2. An evolving scheme of materials science. 1. Ideally ordered crystals, 2. Point disorder in crystals, 3a. Crystals with structural disorder, 3b. Ion-conducting glasses, 3c. Polymer electrolytes and ionic liquids, 3d. Thin-film and composite systems.

STATE IONICS as a whole, as well as the concomitant change of focus at SFB 458 during the past decade. In Fig. 2, ionic crystals with structurally disordered sub-lattices are placed at level 3a. This is where SOLID STATE IONICS began in its early stage. The transition to ion-conducting glasses, see level 3b, marks the loss of long range order, with questions arising with regard to possible positions and passageways of the mobile ions. Further complications are met at level 3c, where even the concept of a stable matrix has to be abandoned, since the systems considered are above the glass transition temperature (polymer electrolytes and ionic liquids). Finally, level 3d comprises those materials whose dynamic properties are largely determined by surfaces on meso or nano scales, i.e., thin films and composite systems. The present volume contains mini-reviews on all kinds of level-three materials. Some articles report on the synthesis and characterisation of new solid, polymer and composite electrolytes. In most projects, however, advanced experimental techniques have been employed in order to obtain new information on the structural environment and on the motion of the mobile ions, resolving their displacements over wide scales of space and time, from individual hops to macroscopic transport. Last but not least, the experimental data are interpreted and complemented by results obtained from numerical simulations and on the basis of model concepts. Within one decade, substantial progress has thus been made in developing a coherent view and a new understanding of ionic motion in materials with disordered structures.

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Preface

In June/July 2009, the 17th International Conference on Solid State Ionics, held in Toronto, provided a forum for all SFB project leaders and many other SFB members to present their most recent results. In February 2010, a final report colloquium, with oral reviews on all SFB projects, was held at Münster, being part of an International Bunsen Discussion Meeting. Both events have demonstrated the key role that SFB 458 has been playing within the international scientific community. At this point, the authors of this volume would like to thank the reviewers for their time and valuable comments. Special thanks go to the editorial staff of Zeitschrift für Physikalische Chemie for their kind help with the preparation of this volume. The establishment and funding of SFB 458 by Deutsche Forschungsgemeinschaft, the time and work invested by our reviewers, and the continuing support provided by DFG staff are most gratefully acknowledged by all members of our Collaborative Research Centre. Münster, September 2010 Klaus Funke



A 

SYNTHESISANDCHARACTERISATIONOFNEWMATERIALS ProjectsA2,A3andA5ofSFB458  

Salt-in-Polymer Electrolytes for Lithium Ion Batteries Based on Organo-Functionalized Polyphosphazenes and Polysiloxanes By Marina Burjanadze, Yunus Karatas#, Nitin Kaskhedikar, Lutz M. Kogel, Sebastian Kloss, Ann-Christin Gentschev, Martin M. Hiller, Romek A. Müller, Raphael Stolina, Preeya Vettikuzha, and Hans-Dieter Wiemhöfer∗ Institut für Anorganische und Analytische Chemie, Westfälische Wilhelms-Universität, Corrensstr. 28/30, 48149 Münster, Germany (Received August 23, 2010; accepted in revised form October 6, 2010)

Polymer Electrolyte / Polyphosphazene / Polysiloxane / Ionic Conductivity / Electrochemistry / Polymer Synthesis An overview is given on polymer electrolytes based on organo-functionalized polyphosphazenes and polysiloxanes. Chemical and electrochemical properties are discussed with respect to the synthesis, the choice of side groups and the goal of obtaining membranes and thin films that combine high ionic conductivity and mechanical stability. Electrochemical stability, concentration polarization and the role of transference numbers are discussed with respect to possible applications in lithium batteries. It is shown that the ionic conductivities of salt-in-polymer membranes without additives and plasticizers are limited to maximum conductivities around 10−4 S/cm. Nevertheless, a straightforward strategy based on additives can increase the conductivities to at least 10−3 S/cm and maybe further. In this context, the future role of polymers for safe, alternative electrolytes in lithium batteries will benefit from concepts based on polymeric gels, composites and hybrid materials. Presently developed polymer electrolytes with oligoether sidechains are electrochemically stable in the potential range 0–4.5 V (vs. Li/Li+ reference).

1. Introduction There are numerous reasons why polymer concepts are attractive for electrolytes and electrodes in lithium ion batteries. This was soon recognized when the first publications concentrating on polyethylene oxide (PEO) as the solvent had been released [1–4]. Not much later, the first publications appeared * Corresponding author. E-mail: [email protected]

# Present address: Ahi Evran Univ., Dept. Chem., 40200 Kirsehir, Turkey

Z. Phys. Chem. 224 (2010) 1439–1473 © by Oldenbourg Wissenschaftsverlag, München

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on lithium salt containing polyphosphazenes as electrolytes which showed distinctly higher ionic conductivities as compared to linear PEO [5–7]. It has already been recognized at early stage that the much higher segmental mobility of these inorganic polymers with the third row element phosphorous was responsible for the improved results. This holds also for the polysiloxanes which tend to exhibit high segmental mobility in connection with a low glass transition temperature. Accordingly, analogous experiments with polysiloxane based polymer electrolytes were published not much later [8,9]. A considerable number of general overviews are available describing the developments in the general field of polymer electrolytes since then [10–14]. The advantages of polymeric solvents for battery electrolytes are evident. Liquid electrolytes are volatile and impose safety problems in current lithium ion batteries. It would be attractive to replace them by incombustible salt-inpolymer materials with no leakage problem. Cross-linked polymer electrolytes result in a safe separation of electrode components, prevent short-circuit in H.-D. Wiemhöfer and minimize the dissolution of electrode components. Furthermore, ion or even mixed conducting polymers may be applied as additives or binders in future electrode structures where they help to stabilize the network of active particles and, at the same time, support ion and/or electron transport within the three-dimensional electrode structures [15–17]. From a chemical standpoint, polymer properties are fixed by the monomers and their functional groups which determine polarity, degree of inter- and intramolecular interaction, mechanical stability and segmental mobility. In addition, different polymers can be combined in blends or copolymers or used as part of composite or hybrid systems offering a whealth of options to influence the properties. Chemical cross-linking can be used to stabilize the mechanical properties. This overview focuses on the chemistry and electrochemistry of polyphosphazenes and polysiloxanes as host systems for dissolved salts. The emphasis lies on their use in ion conducting salt-in-polymer membranes, with the background of lithium ion batteries (although, apart from that, a series of proton conducting polyphosphazenes was synthesized and investigated, too, within this research [18–20]). This article compiles corresponding research results obtained within the SFB 458 which was devoted to the analysis of ion conduction and ion transport mechanisms in disordered materials. Recent trends and other work regarding ion conducting polyphosphazenes and polysiloxanes will be commented, too.

2. Polymer electrolytes with inorganic backbones First of all, we discuss the influence of chemistry on the conductive properties of simple salt-in-polymer systems based on polyphosphazenes and polysiloxanes. The emphasis lies on the choice of grafted side chains and of the lithium

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Scheme 1. Linear PEO and two polyether-grafted polymers with inorganic backbone are shown (graft copolymers are sometimes also termed comb polymers). The interaction with salt ions and also further properties depend on the choice of the side chains. The inorganic polymer backbones exhibit an exceptional segmental mobility. On the right above: illustration of the salt-inpolymer concept with a graft copolymer; on the right below: liquid OPS. Both, OPS and MEEP are viscous liquids due to an extremely low glass transition temperature (−54 and −83 ◦ C, respectively).

salts. Extremely low Tg values lead to poor mechanical properties. Therefore, approaches to achieve better mechanical stabilities without depressing the ionic conductivity are an issue lateron.

2.1 Basic chemical approach to salt-in-polymer materials Scheme 1 shows the basic monomeric units of these polymers with inorganic backbone as compared to the archetype of polymer electrolyte solvents, the linear polyethylene oxide (PEO). The main difference is that the hetero atoms Si and P, respectively, carry two sidegroups which can easily be modified by direct substitution or addition reactions with precursor polymers (so-called polymer analogous reactions), by ring opening polymerization of substituted cyclic phosphazenes or cyclic siloxanes, or by polymerization of differently substituted monomers [21,22]. The two substituents at a P or Si atom may be the same or may differ. Polyphosphazenes and polysiloxanes, functionalized with varying organic sidechains at the phosphorous and silicon atoms, respectively, are the chemical background of this overview. They are frequently denoted as graft-copolymers, in particular if extended sidechains are used, or as comb polymers. The broad choice of sidegroups offers an enormous variability of properties and a straightforward approach to fine tune the physical and chemical properties. Hence, a particular feature of our work on polysiloxanes and polyphosphazenes was

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Scheme 2. Popular lithium salts as often used in polymer electrolytes.

to apply synthetic strategies for a flexible and optimized functionalization of polymers intended for ion conducting salt-in-polymer membranes. The bonding angles of the inorganic backbones −P=N− and −Si−O− are less restricted as compared to carbon based chains. This is the reason for a high segmental mobility and rather low glass transition temperatures. Mostly, short polyethylene oxide side chains (or: oligoethylene oxide) have been used in ion conducting polyphosphazenes and polysiloxanes. Examples are shown in Scheme 1. Therefore, in all such polymers, the main interaction occurs between the oxygen atoms of the polymer sidechains and the lithium ions whereas the anions remain loosely associated in the vicinity. Thus, the lithium ion mobility is lower than that of the anions, giving rise to a lower cation transference number [23–25]. A disadvantage of polyethylene oxide based polymer electrolytes is the low value of the relative permittivity as compared to the polar solvents of conventional liquid electrolytes used in lithium batteries. In order to achieve a high level of dissociation into cations and anions, the choice of the anions for the lithium salts is an important issue as will be demonstrated on various polymer electrolytes in the following. Single charged highly symmetric anions with low basicity are to be preferred. Larger anions that can delocalize the charge over a larger volume are even better. Scheme 2 shows some of the most most frequently used lithium salts in polymer based electrolytes. Note that LiPF 6 cannot be used in the context of polysiloxanes as formation of fluoride ions leads to degradation accompanied by formation of volatile SiF 4 .

2.2 Polymer electrolytes based on polyphosphazenes A great deal of the chemistry of organo-functionalized polyphosphazenes and of corresponding synthetic procedures has been investigated by H. A. Allcock and coworkers starting in the 70s [21]. A common, traditional way of polyphosphazene synthesis was based on the thermally initiated ring opening

Salt-in-Polymer Electrolytes for Lithium Ion Batteries

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Table 1. Synthetic routes to polyphosphazenes. Synthetic route

Reactands/product data

Thermal ring opening Allcock et al. [29]

N3 P3 Cl6 (250 ◦ C), high M n , broad PDI

Condensation polymerization De Jaeger et al. [30]

Cl3 P=NP(O)Cl2 , M n = 105 g/mol, PDI = 1.5–3.0

Condensation polymerization Numerous investigators [31]

PCl5 + NH4 Cl, medium M n , very broad PDI

Living anionic polymerization Flindt et al. [32]

(RO)3 P=NSiMe3 , medium M n , PDI = 1.3–2.4

Living cationic polymerization Manners, Allcock et al. [33,34]

Cl3 P=NSiMe3 , controlled M n = (103 –106 g/mol), narrow PDI

M n – number average molar mass in g/mol (= molar mass); PDI – polydispersity index = quotient M w /M n of number-average and weight-average molar masses

polymerization of hexachloro-cyclotriphosphazene, i.e. [NPCl2 ]3 . The latter is available in very good quality via reaction of PCl5 and NH 4 Cl. Several other routes to polyphosphazenes are known, a compilation is given in Table 1 [21,26]. A particularly interesting method (last one in Table 1) was reported in 1997 consisting of a cationic living polymerization of phosphazene monomers with PCl 5 as chain growth initiator [27]. However, at the beginning, it was quite difficult to synthesize the monomeric phosphazene with the necessary purity according to the available procedures [28]. In 2002, a remarkable progress was achieved by Wang et al., with a one pot synthesis of the phosphazene monomer Cl3 P=NSi(CH3 )3 which is illustrated in Scheme 3 [35,36]. This route is characterized by a remarkably high yield and purity of the monomer which can be used directly and without isolation for a subsequent polymerization after addition of PCl5 as initiator. For the work on polyphosphazenes as presented in the following, the monomer synthesis of Scheme 3 was applied as the starting step towards the preparation of our polymeric precursor (NPCl2 )n [37,38]. The yield of the polymerization normally reached 80–90% of the monomer and the average molar mass was well controlled by the molar ratio PCl5 /phosphazene monomer [37,38]. Molar masses of 105 –106 g/mol imposed no problem. The freshly prepared precursor (NPCl2 )n acted as a source for numerous differently functionalized polyphosphazenes. All were obtained using the nucleophilic substitution reaction with corresponding alcohols and amines as illustrated in Scheme 4. Scheme 5 shows a compilation of some substituents which have been applied in our experiments with ion conducting polyphosphazenes. With some bulky substituents, it is difficult to achieve a complete substitution of the chlor-

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Scheme 3. Synthesis of the chlorinated polyphosphazene by the living polymerization route which is used as a versatile precursor for preparation of organo-functionalized polyphosphazenes [37,38]. The molar mass is controlled by the ratio of the monomer to the initiator PCl 5 (av. M n = 105 –106 g/mol [37]).

Scheme 4. Preparation of functionalized polyphosphazenes from the precursor polymer (NPCl 2 )n by polymer substitution reactions from alcohols and amines (primary and secondary).

ine atoms in the precursor (NPCl2 )n . This was observed for some secondary amines, long chain alcohols and branched side chains. In such cases, one has to complete the chlorine substitution in a second step by using small size primary amines or alcohols. This is the reason why the polyphosphazene, denoted as BMEAP in Scheme 5, was prepared with two kinds of amine side chains, a branched one and the linear −NHCH2 CH2 CH3 (cf . Scheme 5). Scheme 6 illustrates the variability and the great influence of the particular substitution pattern in polyphosphazenes on their physical properties. The polymers in Scheme 6 were synthesized with the aim to prepare and investigate proton conducting polyphosphazene membranes after introducing acidic sulfonate groups by a sulfonation of the aromatic rings [18]. The

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Scheme 5. Examples for substituents which were used to prepare functionalized polyphosphazenes for polymer electrolyte investigations with dissolved lithium salts; the abbreviations for the corresponding polyphosphazenes below the chemical formulas are used in the following text.

Scheme 6. Illustration of the variable physico-chemical properties of polyphosphazenes by changing or mixing the substituents [18,19]; the three polymer samples have average molecular weights around 105 g/mol.

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Fig. 1. Comparison of the ionic conductivities of MEEP with two different salts; the results prove a much higher dissociation of LiTFSI as compared to LiTf which explains why LiTFSI is so often used for salt-in-polymer electrolytes.

photo in Scheme 6 shows a crystalline appearance of the poly[bis-phenoxyphosphazene] (BPP), the mixed statistical substitution (1 : 1) in PMBP results in a liquid product (very low Tg ), whereas the third polymer, PTFEP, is a flexible plastic polymer. In the following, several examples will be discussed, which show the detailed influence of the substitution pattern of some polyphosphazenes on the ionic conductivity with dissolved lithium salts. The first example in Fig. 1 shows the temperature dependence of the ionic conductivity of MEEP samples which have been prepared by the living cationic polymerization [37,38]. The average molar mass was 105 g/mol. The polymer was a transparent highly viscous liquid. Two observations can be made: first of all, there is a clear influence of the particular choice of the anion in the lithium salt. LiTf shows almost an order of magnitude lower conductivities as compared to LiTFSI, which is explainable by the higher dissociation of LiTFSI. Second, the maximum conductivities range between 10−3 and 10−4 S/cm at room temperature for LiTFSI, which is rather high for a salt-in-polymer system. Blonsky et al. reported a value of 2.2 × 10−5 S/cm for a similar LiTf concentration at 30 ◦ C which nicely agrees with Fig. 1 [6]. Figures 2 and 3 show two cases of differently substituted polyphosphazenes with a very short and a rather long sidechain. The question was: Is there a high sensitivity of the measured conductivities with respect to the length of the sidechains? Figure 2 contains conductivity data of JAPP, a polyphosphazene with long polypropylene oxide sidechains bound to the phosphorous of the polyphosphazene via a terminal amino group (cf . Scheme 5) [39]. The average molar mass of the side group alone is 500 g/mol. The conductivity is only

Salt-in-Polymer Electrolytes for Lithium Ion Batteries

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Fig. 2. Polyphosphazene functionalized with jeffamine, a molecule consisting of a long polypropylene oxide sidechain with a terminal amino group [39].

Fig. 3. Conductivity of a liquid solution of lithium triflate in poly-[bis-methoxy-phosphazene]. The glass transition temperature is rather low (−80 ◦ C) [41].

a factor of three lower than that of MEEP (with the same lithium salt) whereas polymer electrolytes based on pure polypropylene oxide show conductivities clearly below 10−5 S/cm) [40]. Obviously, the high segmental mobility of the polyphosphazene backbone is favorable, as well, for an increased mobility of the attached polypropylene oxide sidechains and, hence, for the ion mobility. BMP in Fig. 3 with the methoxy groups, on the other hand, shows one of the shortest possible functional groups at the phosphorous. The solvating capability for lithium ions may be reduced due to the lower number of donor atoms

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Fig. 4. Temperature dependence of the conductivities of the polymer electrolyte membranes LiSO3 CF3 /BMEAP, NaI/BMEAP, and LiSO3 CF3 /BMEAP + dispersed Al2 O3 [38].

which may also reduce the dissociation of the salt into cations and anions. However, the mobility of the polymer segments is enhanced. SANS measurements support this: the BMP molecules in the liquid polymer behave like statistical coil molecules [41]. Therefore, it is not surprising that the conductivities of salt solutions in BMP are as high as those of MEEP with dissolved lithium triflate. Comparing the results for the three different types of side chains of the samples in Figs. 1–3 supports the conclusion that the influence of the sidechains on the conductivities is less important than a high segmental mobility of the backbone. Accordingly, there was hope that a branched instead of a linear sidechain would not depress the conductivity, but could help to gain mechanical stability of the salt-in-polymer solutions by chain entanglement. In order to test that, a short secondary amine with additional ether oxygen atoms was chosen for the substitution (denoted as BMEAP in Scheme 5) [38]. To guarantee a complete removal of the chlorine atoms of the precursor polymer, a primary amine was added in the final stage of the synthesis. The resulting polymer structure with a mixed substitution is illustrated in Fig. 4, too. As expected, a considerable mechanical stability was obtained. The obtained salt-in-polymer membranes were transparent and elastic. The conductivities, however, were rather low. As can be seen in Fig. 4, the room temperature conductivity for a solution with 10 wt. % lithium triflate amounted to less than 10−6 S/cm. It is evident that a drastic reduction of the entire segmental motion lead to a considerable decrease of the ionic mobilities, too. Obviously, the lithium cations remain well associated within the small bags formed by the sidechains. This was confirmed by a detailed NMR analysis of the interaction between lithium ions and the BMEAP polymer [42].

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It is well established from many investigations on polymer electrolytes and many other ion conducting systems that, under favorable conditions, the average ionic conductivity can considerably increase after dispersing nanoparticles within the polymer electrolyte. Since the first observations of this effect with PEO based polymer electrolytes [44], it has been intensely investigated by many authors [45–48]. Figure 4 shows the result of the dispersion of 4 wt. % Al2 O 3 nanoparticles in the LiTf/BMEAP system. The ionic conductivity at constant concentration of LiTf was increased by more than one order of magnitude. Its value at room temperature was almost comparable to the conductivity of a corresponding LiTf/MEEP solution. Note that in conductivity studies of polymer electrolytes, the purity is an important issue. The presence of water has to be excluded by a careful choice of materials and a thorough drying process of as-prepared polymer electrolytes. The determination of remaining water is usually done by Karl-Fischer titration. Another way to determine the water content is by using NMR [49,50]. The examples presented in this overview are all characterized by careful preparation and drying processes prior to the electrochemical measurements. Details can be found in the cited publications.

2.3 Polysiloxane based electrolytes Compared to the polyphosphazenes, the polysiloxanes possess very similar properties making them attractive for designing stable polymer electrolytes with high ionic conductivities. In particular, they exhibit high segmental mobility, good chemical stability, and like the polyphosphazenes, there is an excellent possibility to modify them with different sidechains by a universal reaction starting from poly[hydromethyl-siloxane] (PDMS) as a precursor polymer. PDMS is available as a cheap product from basic silicon chemistry. Here, PDMS was used with an average length of 35 monomer units. The Si−H groups of PDMS (cf . Scheme 7) were substituted via a platinum catalyzed hydrosilylation. It usually occurs under mild conditions and proceeds with high yield, if the introduced sidechains are not too bulky. Polysiloxanes have been prepared with different sidechains. Scheme 7 shows an example with an oligoethylene oxide. The length of the sidechains was varied over a certain range as denoted in Scheme 7, too. Figure 5 presents temperature dependent conductivity data for two different salts dissolved in OPS(4) where (4) stands for the number of oxygen atoms in the secondary oligoether sidechains. Again, as already observed for polyphosphazenes in Fig. 1, the symmetric anion of LiBOB facilitates dissociation yielding a rather high conductivity as compared to LiTf. A drawback of LiBOB, however, is its limited solubility (around 12 wt. %). The charge/discharge processes in a corresponding battery usually leads to a concentration gradient of the salts. Therefore, the concentration polarization during charging/discharging can cause precipitation of solid LiBOB and dam-

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Scheme 7. Functionalization of PDMS by Pt catalyzed hydrosilylation; Pt(dvs) denotes the applied Karstedt’s catalyst [43].

Fig. 5. Ionic conductivity of an oligoether functionalized polysiloxane with two different lithium salts (due to the higher molar mass of LiBOB, the molar concentration of LiBOB is lower by a factor of 0.68).

ages the electrode structures. Scheme 8 compiles three further substitution concepts which have been investigated within the polysiloxane class of polymer electrolytes. In most cases, maximum conductivities around 2 × 10−4 S/cm were reported [51–54]. To conclude, the ionic conductivity of salts in polyphosphazenes and polysiloxanes with ether sidechains shows maximum room temperature values around 2 × 10−4 S/cm. At present, LiTFSI seems to be the best choice in

Salt-in-Polymer Electrolytes for Lithium Ion Batteries

19

Scheme 8. Further concepts with differently functionalized polysiloxanes as published by other authors [51–54].

many cases. Nevertheless, the polysiloxanes and polyphosphazenes as described above are viscous liquids and cannot be used as dimensionally stable membranes. Obviously, in that case, one cannot speak of solid polymer electrolytes (SPE), an expression sometimes used for polymeric electrolytes. As the example BMEAP (cf . Fig. 3) showed, if sidechains with stronger intermolecular interactions support the formation of self-standing polymer membranes, the segmental mobility will be suppressed more or less leading to a low ion conductivity. Hence, an important task is to find a compromise between contradicting postulates, a good segmental mobility (supporting high conductivity) and a good mechanical stability. This is the topic of the following sections.

2.4 Mechanically stable polymer electrolyte membranes with high conductivity A straightforward way to stabilize a liquid polymer and to obtain a selfstanding membrane is chemical cross-linking of the polymer molecules. There are numerous chemical reactions appropriate for that purpose. For polysiloxanes, a good approach is the sol–gel process, i.e. a catalyzed solvolysis of alkoxysilyl groups in the presence of water or alcohol. It is followed by a condensation reaction forming oxygen bridges between the silicon atoms of different alkoxysilyl group. This type of reaction is also frequently used for the preparation of inorganic-organic hybrid materials, often termed as ORMOCERS (“organically modified ceramics”), where sol–gel processes are applied to couple organic molecules bearing terminal alkoxysilyl groups with hydroxo groups of in-situ formed silica particles [55]. ORMOCERS with dissolved salts have already been investigated as inorganic–organic hybrid electrolytes for lithium ion cells [55]. Sol gel reactions are usually carried out in the presence of acids or bases as catalysts, or

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Scheme 9. a) Preparation of oligoether functionalized polysiloxanes T x OPS with additional trimethoxysilyl groups (denoted by T) for sol–gel cross-linking [43], The chain length of the backbone was typically n = 30–35 (average M n = 2.5 × 104 g/mol, Tg = 85 ◦ C). The length of the oligoether sidechain was m = 3–5. PE3 to PE7 denote the different cross-linker contents, b) cross-linking via acid catalyzed sol–gel process, c) cross-linking via a DBTL catalyzed reaction (DBTL = dibutyl tin dilaurate).

with active metal complexes as catalysts. Scheme 9 shows the reactions used to prepare cross-linked, polyether grafted polysiloxanes. It is a straightforward application of the hydrosilylation to introduce two types of sidechains for solvation of cations and for sol–gel cross-linking in a one-pot reaction. An increasing fraction x/n of Si−O−Si bridges after cross-linking leads to a decrease of the segmental mobility and the ionic conductivity. The appearance of the polymer changes from liquid to that of a solid membrane. Hence, the percentage of cross-linking groups for a good compromise between mechanical stability and a sufficient segmental mobility had to be optimized. Figure 6 shows results for the temperature dependent ionic conductivities of

Salt-in-Polymer Electrolytes for Lithium Ion Batteries

21

Fig. 6. Ionic conductivities of salt-in-polymer electrolyte membranes based on sol–gel crosslinked T x OPS (HCl catalyzed) with various concentrations x of the trimethoxysilyl groups (PE37 denote samples with different percentages of cross-linking groups, cf. Scheme 7) [43].

Fig. 7. Comparison of the conductivities two T 0.1 OPS membranes of the same polysiloxane (HCl catalyzed) with different lithium salts [56]. As the molar mass of LiTFSI is higher compared to LiTf by a factor of 1.8, the molar concentrations of the LiTFSI solutions at the same weight percentage are a factor of about 0.5 lower.

polysiloxanes prepared with different fractions of cross-linking groups (HCl catalyzed). The general observation was that the conductivities at constant salt concentration start to decrease markedly for x/n > 0.1 which is obviously due to the drastic decrease of the segmental motion. Figure 7 compares the results of membranes with two different salts showing again the superior conductivity

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Fig. 8. DC conductivity of a T 0.1 OPS membrane as a function of the LiTf concentration x (DBTL as cross-linking catalyst) [57].

with LiTFSI. In spite of the cross-linking, the conductivities are still reasonably high. Dibutyl tin laurate (DBTL) as catalyst gave the best mechanical stability and elasticity of the membranes. This is due to the higher rate and efficiency of cross-linking (nearly 100% of the Si–OCH 3 react). It is not surprising, however, that sol–gel treated polysiloxane membranes when cross-linked with DBTL normally gave markedly lower conductivities than those cross-linked with HCl as comparison between Figs. 7 and 8 shows for the same LiTf concentration (cf . also data for PE 4 in Fig. 6 with the 10 wt. % sample in Fig. 8). The conductivity is lower by almost one order of magnitude. As already mentioned in the context of the results for BMEAP (cf . Fig. 4), there were numerous investigations where authors tried to increase the conductivities by dispersing nanoparticles in the polymer electrolytes. In all these cases, the factors responsible for the observed increase of conductivity due to nanoparticles are well-known. A specific adsorption of anions or cations can occur at the particle surfaces depending on the Lewis acid or base character of the surface chemistry. The specific interaction leads to surface charges on the particles, increases the concentration of counterions and supports the dissociation of ion pairs in the neighbouring polymer matrix. This may then change the ionic transference numbers, too. Another well-known fact is that dispersed particles can increase the free volume in the polymer matrix around them, thus giving rise to more segmental mobility and possibly higher ionic mobility, too. However, for polysiloxane based polymer electrolytes which already show relatively high conductivities, any further enhancement of the conductivity after adding nanoparticles is rather limited. Sometimes, one even observes a decrease as in the case of T 0.1 OPS with LiTf after addition of Al2 O 3 nanopar-

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23

Fig. 9. Overview over the maximum room temperature conductivities of various polysiloxane based polymers [56]. The first value on the left for LiTf/MEEP is taken from the work of Blonsky et al. for comparison [6].

ticles [57]. To conclude this section, Fig. 9 gives an overview of the room temperature conductivities of numerous investigated salt-in-polymer electrolytes. It nicely summarizes our conclusions that the conductivities of salt-in-polymer electrolytes (without further additives except the salt!) are restricted to maximum values of around 10−4 S/cm as long as we speak of high molecular weight polymers (> 103 g/mol).

2.5 Extended salt-in-polymer concepts using additives and hybrid materials The results discussed in the preceding sections make evident that an increase of the performance of non-liquid polymer electrolytes demands additional concepts beyond the simple salt-in-polymer approach which relies on amorphous low Tg polymers and salts with excellent dissociation. Therefore, it is inevitable to consider the introduction of additional components and materials that can remove the restrictions of the mere salt-in-polymer materials. The traditional way to soften a rigid polymer network and to increase its local mobility is the addition of small molecules as plasticizers. They fill and enlarge the space between the polymer segments and lower the intermolecular interaction of the polymeric network. In the same way, such dissolved polar molecules will interact with dissolved ions and, in this way, compete with the donor atoms of the polymer chain regarding the solvation of ions. The mobility of ions in corresponding associates between ions and low molecular weight additives is considerably higher than that of ions interacting with polymer sites only. Further, the interaction of highly polar molecular additives with the salt ions will also facilitate the dissociation of more ion pairs. One should also remember that proton conducting polymer electrolytes used for low temperature fuel cells already apply this principle for a long

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Scheme 10. Additives based on Lewis acidic esters of boric acid which have been applied in gel polymer membranes with cross-linked polysiloxanes (T 0.1 OPS) [57].

time. The proton conducting polymers usually contain a great deal of water molecules within channels of the polymer network. The water molecules act as shuttles for the transport of protons between immobile acidic sulfonate groups of the polymer network. Accordingly, dissolving a larger concentration of small polar molecules in a salt-in-polymer electrolyte is an approach to decouple the ion transport from the slow polymer mobility and should lead to a significant increase of the conductivities This general strategy is closely connected to concepts of polymer based gel electrolytes (30 to 50 wt. % of low molecular weight components) or salt-in-polymer membranes with dissolved ionic liquids both leading to a swollen network with much higher ionic conductivity [58–62]. In order to stabilize the network it is necessary to apply cross-linking or to use polymers that favor chain entanglement. The most simple approach is dissolve liquid aprotic polar solvents such as propylene carbonate, ethylene carbonate and others which are already well known as components of current liquid lithium battery electrolytes. But in order to develop electrolytes with higher safety, incombustible molecular additives with low volatility are preferred. Scheme 10 shows two kind of boric ester additives with short oligoethylene oxide sidechains that were investigated as additives in cross-linked polysiloxanes [57]. Similar experiments combining related esters with other polymer electrolytes can be found in the literature [64–66]. The electron deficient boron atom acts as a Lewis acid and will therefore interact with Lewis bases and, in particular, with anions. One expects that its presence should increase the dissociation of ion pairs due to the specific interaction with the anions. Further advantages of these boron containing additives are low vapor pressure, high chemical stability, good solubility in the polymer as well as a good plasticizing function. They are available in large amounts and therefore good candidates for modified polymer electrolytes in lithium ion cells. Figure 10 shows results for the temperature dependent conductivities of a sol–gel cross-linked polysiloxane membrane with two different boric acid ester additives. Note that the conductivity with these additives rises to values

Salt-in-Polymer Electrolytes for Lithium Ion Batteries

25

Fig. 10. Arrhenius plot of cross-linked salt-in-polysiloxane electrolytes based on T 0.1 OPS with 15 wt. % lithium triflate: (a) without additives, (b) with 30 wt. % boric acid ester B2, and (c) with 30 wt. % boric acid ester B3 [57].

higher than 0.1 S/cm. Recent experiments showed that this effect may be further increased by higher amounts of the boric acid esters. Indeed, the boric acid esters themselves can be used as solvents for lithium electrolytes [67]. They are characterized by high electrochemical stability. A further quite different concept beyond the simple salt-in-polymer approach refers back to the phosphazene derived polymers. Hexachlorocyclotriphosphazene, (NPCl2 )3 , is a well available industrial chemical. In close analogy to the organo substituted polyphosphazenes, one can also start (NPCl2 )3 and modify this molecule by the same nucleophilic substitution reactions as used with the chlorine substituted polyphosphazene precursor (cf . Schemes 3 and 4). Hence, a huge variety of different organo substituted cyclic phosphazenes are obtainable. In particular, it is possible through the use of suitable sidechains to form three-dimensional networks as shown in the following. These networks can then act as hosts for incorporation of salts, plasticizers, solvents and further additives as shown in the following [18,21,63]. Scheme 11 shows three examples of synthesized cyclic triphosphazenes. The phosphorous atoms of CVEEP, CVEP and CVMEEP carry substituents with terminal vinyl groups which can be activated for a radical initiated crosslinking (e.g. with organic peroxides as thermally activated radical initiators). In this way, it is easy to build up a stable polymeric network from these substituted cyclic phosphazenes. One can control the rigidity and the degree of cross-linking by choosing mixed substitutions with optimized percentage of cross-linking groups (similar to the strategy applied for optimization of the T x OPS based salt-in-polymer electrolytes, described in the preceding section). The mixed substituted CVMEEP, for instance, contains a statistical distribution

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Scheme 11. Cross-linkable cyclic triphosphazenes as a different route to mechanically stable polymer electrolytes [63].

of two kinds of substituents as denoted in Scheme 11. Half of the substituents contain cross-linkable vinyl groups. The advantage of a reduced number of cross-linkable groups is that the resulting membranes are less rigid which is beneficial for the mobility of the polymeric network and, thus, of the salt ions. This was indeed verified, although the conductivities of such membranes with dissolved salt were not superior to the sol–gel cross-linked polysiloxanes (see results for CVEEP and CVEP in Figs. 11 and 12) [63]. The network resulting from cross-linking of the cyclic triphosphazenes is illustrated in Scheme 12 together with a photo of a membrane produced in this way. The procedure leads to very stable polymeric membranes. Incorporation of salts or other additives is easily possible in-situ during the cross-linking reaction. In parallel, Chen-Yang et al. developed a similar electrolyte membrane concept based on functionalized cyclic triphosphazenes [68]. An earlier work of Inoue et al. used a related chemistry with a graft-copolymer of polystyrene and functionlized triphosphazenes [69].

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Scheme 12. Illustration of the network formed by cross-linking of CVEEP, the inserted photo gives an impression of the as-prepared membranes [63].

Fig. 11. Comparison of the conductivities of solutions of LiTf in a closely cross-linked CVEEP network and in a hybrid membrane containing 50% MEEP besides the network forming CVEEP [63].

Figure 11 shows the conductivity of a CVEEP membrane with LiTf. Figure 12 gives data for a CVEP membrane with LiTFSI. Again, the material with LiTFSI yields much better conductivities. However, our idea in developing these cyclomatrix networks was to apply them for a hybrid concept with liquid MEEP. As the conductivity of liquid MEEP and OPS cannot be retained after

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Fig. 12. Comparison of the conductivities of LiTFSI solution in the strongly cross-linked CVEP network, in liquid MEEP alone, and in hybrid materials with MEEP molecules entrapped within the CVEP network.

cross-linking, we prepared hybrid systems by confining MEEP within the cyclomatrix network of CVEEP or CVEP. Figures 11 and 12 contain results on a hybrid system consisting of the above described substituted cyclic triphosphazenes as network former and a considerable percentage of free MEEP as low Tg polymer (see formula in Scheme 5). One observes an increase of the conductivity by up to two orders of magnitude [63]. The polymer hybrid concept turns out to be an attractive idea to achieve the desired properties “high mechanical stability and high conductivity” by combining the properties of different components. MEEP induces a high local segmental mobility and thus guarantees an optimized conductivity. As Fig. 12 demonstrates, the conductivities of salt solutions in CVEP/MEEP and in pure liquid MEEP are not much different. The great advantage lies in the achieved high mechanical stability of the CVEP/MEEP hybrid membrane. This proves that dividing mechanical stability and high conductivity between two different components in a hybrid material is indeed a very good strategy.

3. Transport and electrochemistry 3.1 Mechanistic considerations and limitations of the ionic mobilities The results in Figs. 6–8 are typical for the temperature dependence of the total conductivity of polymer electrolytes at low and medium temperatures which clearly deviates from an Arrhenius plot. Often, the empirical Vogel–Tammann– Fulcher equation (VTF equation) is used to describe and to fit the temperature

Salt-in-Polymer Electrolytes for Lithium Ion Batteries

dependence of the ionic conductivity as given by   −B A0 exp σ(T ) = T T − T0

29

(1)

A 0 , B and T0 are empirical constants. A consistent theoretical model to calculate these three parameters and to interpret them in terms of a transport mechanism is not available. However, a quantitative interpretation of the temperature dependent ion conductivity of polymers has been developed within the MIGRATION concept of Funke and coworkers [57,70–72]. It offers an excellent access to a quantitative description of the microscopic dynamics of ion movement in glasses and polymers. A fundamental result of the MIGRATION concept is that the conductivity σ DC at low frequencies can be written as the product of the high frequency conductivity σ HF (T ) and a temperature dependent function W∞ (T ) according to σ DC (T ) = σ HF (T ) W∞ (T )

(2)

Here, W∞ (T ) denotes the fraction of successful hops for a given temperature. The low frequency range used for measurements of the DC conductivity with impedance spectroscopy typically comprises 10 kHz to 0.1 Hz. Analysis of the high frequency range yields the frequency dependent complex conductivity which has been analyzed in detail for several polysiloxane based polymer electrolytes recently [57]. In molten salts and salt-in-polymer electrolytes, where the ion-sites are not predefined and where the network is not immobile, the HF conductivity is Arrhenius activated, while the DC conductivity is not. σ HF (T ), accordingly it can be written as   α E∗ (3) σ HF (T ) = exp − T RT Besides α, the pre-exponential factor, E ∗ is the activation energy required for an elementary displacive step. The model of Funke et al. finally yields the following temperature dependence for the DC conductivity:   ∗  E∗ E /K − γ exp (4) Tσ DC (T ) = α exp − RT RT where in addition to the short range energy barrier E * , the parameters γ and K determine the temperature dependence. Figure 13 shows results of analyzing data from extended frequency dependent conductivities for cross-linked T 0.1 OPS with 15 wt. % LiTf in terms of the MIGRATION model Eq. (4) [57] (same sample as in Fig. 8). The three parameters of Eq. (4) are obtained as: E * = 0.258 eV, ln(α) = 5.487, γ = 14.9 × 10−3 , and K = 2.1.

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Fig. 13. Temperature dependent DC conductivity and high frequency conductivity derived from frequency dependent measurements followed by a detailed analysis using the MIGRATION concept [57].

As already mentioned, the investigations on polysiloxane and polyphosphazene based electrolytes support the conclusion that it is impossible for cross-linked salt-in-polymer membranes to achieve higher conductivities than those obtainable for the liquid polymer systems. Only an extension towards composite or hybrid systems leads to higher values by decoupling the ion mobility from the polymer. It is useful to discuss the background. The limit is caused by the restricted local thermal motion of polymer segments in polymers with high molecular weights and the absence of fast diffusion of the entire polymer molecule. As long as the mobile ions interact with the polymer network, one cannot circumvent this problem. According to Walden’s rule for electrolyte solutions, the ionic mobility of liquid electrolytes is expected to change nearly proportional with the inverse of the solvent viscosity according to 1 σ ion ∝ csalt η solvent

(5a)

This also holds for polymeric solvents, as long as they are amorphous, not cross-linked and their temperature is above Tg . However, an additional correction factor has to be introduced, if the salt dissociation is not complete. Nevertheless, it is useful for a rough estimate in the case of almost liquid polymers such as low Tg polysiloxanes. Liquid hexamethyl-disiloxane, for instance, has a viscosity which is similar to that of water or propylene carbonate at ambient temperature. It is therefore not surprising that salts dissolved in tri- or tetrameric siloxanes with oligoether sidechains reach good conductivities of around 5 × 10−4 S/cm [73]. The viscosity of polymer melts scales with a power of the molar mass. Statistical

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31

models suggest η polymer melt ∝ M p

with

p ≈ 3.4

(5b)

Hence, an increase in molar mass bei one order of magnitude from 102 g/mol to 103 g/mol should increase the viscosity by about three to four orders of magnitude, lowering the ionic mobility by a corresponding factor. A lower degree of dissociation will lead to a further decrease. This is supported by the experiences with polyphosphazenes and polysiloxanes. Polymer electrolytes based on polysiloxanes (Mn = (6–11)× 103 g/mol) tended to show lower viscosities than the polyphosphazenes (Mn = 2 × 104 to 5 × 105 g/mol). Therefore, the slightly higher ionic conductivities of typical polysiloxanes (containing the same salt) can be explained by the lower molar mass. Accordingly, the conductivity of salt-in-polymer solutions can be increased using polymers with relatively low molar mass (if not cross-linked!). This is indeed the case as examples from recent publications for electrolytes with functionalized low molecular weight siloxanes show [51,54,55,73–77]. The problem of missing mechanical stability cannot be circumvented in this way. But a combination of low molecular weight solvents with cross-linked host polymer networks in hybrid membranes seems to be a promising strategy.

3.2 Ionic transference numbers Apart from the total ionic conductivity, two further electrochemical parameters of polymer electrolytes are as important as the total conductivity for the use in lithium ion cells, i.e. the transference numbers of lithium ions and the electrochemical stability window. The total conductivity as measured by standard impedance spectroscopy (1–10 kHz) does not reveal the relative contribution of cations and anions to the charge transport. This information has to be determined by an independent measurement. Several techniques have been developed and published for polymer electrolytes [78–83]. The transference number of lithium ions expresses the amount of charge transport due to lithium ions under steady state conditions in lithium ion cells. If the mobilities of cations and anions of a lithium salt LiX were identical for all concentrations, the transference number should be 0.5, i.e. the efficiency of lithium ion transport during a steady state discharging of a lithium cell is half of that calculated from the total conductivity alone. The internal resistance is a factor of two larger than calculated from the total conductivity. The following definition of a lithium transference number is usually found in textbooks and publications: tLi+ =

σ Li+ u Li+ = σ Li+ + σ X− u Li+ + u X−

(6)

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Local electroneutrality, i.e. equal local concentrations of cations and anions cLi+ = cX− , holds in good electrolytes (except in space charge regions which are neglected here). The definition in Eq. (6) characterizes a homogeneous electrolyte as a whole. If complete dissociation of the dissolved salt LiX into Li+ and X− can be assumed, the ion concentrations are given by the net salt concentration, cLi+ = cX− = csalt , for incomplete dissociation cLi+ = cX− = αcsalt , with the degree of dissociation expressed by α. The concentration cLiX of undissociated ion pairs then corresponds to cLiX = (1 − α)csalt . Under charge flow, in general, a concentration gradient exists in the electrolyte. Then, the value of tLi+ becomes a function of the position within the electrolyte, if the mobilities depend on the concentrations of the ions. Under these conditions, Eq. (6) is a local definition. However, for lithium ion cells, it is more interesting to describe the transference of lithium ions under the non-equilibrium conditions of steady-state charging/discharging. Corresponding experiments mostly compare the total current just after switching-on (time t = 0) a polarizing voltage in an initially homogeneous symmetrical cell (with the electrolyte between two Li metal electrodes) with the steady-state current reached for long times, i.e. t → ∞. The steady-state is characterized by a vanishing anion current (if no decomposition of the electrolyte occurs) so that the total current corresponds to a net lithium ion transport. Hence, an experimental definition of a local transference number under these conditions is: tLi+ =

i tot,t→∞ (i Li+ )t→∞ = i tot,t=0 (i Li+ + i X− )t=0

with

(i X− ) t=∞ = 0

(7)

where i tot denotes the total electrical current density and i Li+ and i X− are the partial ionic current densities of the cations and anions. tLi+ becomes identical with Eq. (6) in the limit i tot → 0. In a charging or discharging process of a lithium ion cell, only the lithium cations take part in the electrode reactions. Under this condition, the boundary condition requires that the partial anion current density vanishes at the electrode/electrolyte interface. In the particular case of a steady state charging/discharging, this will hold throughout the entire electrolyte and implies the presence of a salt concentration gradient as shown in the following. The general expressions for the local partial ionic current densities are i Li+ = −

σ Li+ gradμ ˜ Li+ , F

i X− =

σ X− gradμ ˜ X− F

(8)

F denotes Faraday’s constant, μ ˜ X− denote the electrochemical poten˜ Li+ and μ tials of cations and anions. Additional conditions are the expressions for the total electrical current density i tot and the chemical potential of the salt μ LiX given by i tot = i Li+ + i X− ,

μ LiX = μ ˜ Li+ + μ ˜ X−

(9)

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From the four equations in Eqs. (8) and (9), one derives i Li+ = −F D˜ salt grad csalt + tLi+ i tot i X− = +F D˜ salt grad csalt + (1 − tLi+ )i tot ↑ diffusion

(10)

↑ migration

The chemical diffusion coefficient D˜ salt of the salt (also: interdiffusion coefficient or ambipolar diffusion coefficient of the salt) in Eqs. (10) explicitly corresponds to the following expression as the derivation from Eqs. (8) and (9) shows:   u Li+ u X− RT d ln a LiX ˜ (11) D salt = u Li+ + u X− F d ln csalt As usual, the thermodynamic activity of the salt, a LiX , is a function of the chemical potential μ LiX of the salt according to μ LiX = μ◦LiX + RT ln a LiX . where μ◦LiX denotes the standard value of the chemical potential. The steady-state condition of a vanishing anion current yields the following expression for the salt concentration gradient in the electrolyte: grad csalt =

(1 − tLi+ ) i tot 1 = ˜ u Li+ D salt F

 RT

i Li+  d ln a LiX d ln csalt

(12)

if i X− = 0, i tot = i Li+ (steady state) As mentioned above, one has to distinguish between the net salt concentration csalt and the concentration cLiX = (1 − α)csalt of undissociated LiX ion pairs. In general, at high concentration gradients, ionic mobilities and transference numbers will depend on the position in the electrolyte between the electrodes. Hence, the concentration gradient, in general, will not be a linear function of the position. The calculation of an averaged transference number for higher current densities in Eq. (7) has to be done by integrating over the electrolyte thickness. Equation (8) states, as expected, that the undesired concentration polarization at high current densities is minimized by choosing an electrolyte with the highest possible lithium cation mobility, or for tLi+ → 1. However, the common experience with salt-in-polymer electrolytes is, if no additives or low molecular weight solvents are used, that the anion mobility is considerably higher than that of the lithium cations. Apart from that, one observes a considerable concentration of undissociated ion pairs [23,24,66,84]. The typical transference numbers of lithium ions in such salt-in-polymer electrolytes are markedly below 0.5, typically 0.2–0.3 [84]. Results obtained with pulsed field gradient NMR even showed that the diffusion of ion pairs is much higher than that of free anions and cations (cf . results and extended discussion

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with respect to polysiloxane based electrolytes in Sect. 4 of Kunze et al. [24]). Under these conditions, the net lithium ion transport in the steady state may be dominated by a coupled countertransport of ion pairs and anions (which show considerably faster diffusion than the lithium cations as proved by NMR measurements for polysiloxanes [23,24] and PEO [66]). Under these conditions, mass conservation leads to the following net partial current densities of cations and anions which include the a contribution of neutral ion pairs LiX (an extended treatment of the background within the framework of irreversible thermodynamics can be found in [85,86]): (i Li+ ) net = i Li+ + i LiX , (i X− )net = i X− − i LiX D LiX grad μ LiX with i LiX = FcLiX RT

(13)

The consequence of the possibility of neutral ion pair transport is to open up an additional path for a net Li+ ion transport by a transport of LiX in one direction and a coupled transport of anions X− in the opposite direction. Using the Eqs. (13) as a starting point, the following expressions for the transference number and for the chemical diffusion coefficient result for the transference number in electrolytes with incomplete dissociation and non-negligible fast ion pair diffusion: i tot,t→∞ (i Li+ ) net,t→∞ u Li+ u X− + u X− u LiX + u Li+ u LiX = = i tot,t=0 (i Li+ + i X− ) net,t=0 (u Li+ + u X− ) (u LiX + u X− ) F D LiX with the abbreviation: u LiX ≡ RT     RT d ln a LiX u Li+ u X− D˜ salt = u LiX + u Li+ + u X− F d ln csalt tLi+ =

(14)

(15)

Equations (14) and (15) yield in the limit of a very small, negligible mobility of free lithium ions   RT d ln a LiX u LiX ˜ tLi+ ≈ (16) , D salt ≈ u LiX u LiX + u X− F d ln csalt if u Li+ u X− , u LiX Using Eq. (16) instead of Eqs. (7) and (11) in the condition for vanishing anion current density, shows that, in principle, it seems to be an alternative solution towards fast lithium transport and high lithium transference, if a polymer with very fast diffusion of both, cation–anion pairs and free anions, could be found. This, however, has not yet been investigated. To conclude, for estimating the properties of polymer electrolytes in a battery, the transference numbers of lithium ions have to be known. The total conductivities of amorphous salt-in-polymer electrolytes without additives

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35

range between 10−4 and 10−5 S/cm. With transference numbers below 0.3, non-negligible concentration gradients will appear which considerably depress the efficiency of polymer electrolytes under steady-state conditions during charging/discharging. Increased salt concentration gradients in a high voltage battery may cause the risk of a serious loss of charge capacity due to precipitation of solid salt within the porous electrode structure on the high salt concentration side. One of the most important tasks for the near future is therefore to develop composite and hybrid electrolytes with high lithium transference. Interestingly, an increase of the transference number of lithium ions has been achieved by dispersing nanoparticles, or by dissolving crown ethers or other molecules with specific ionic interactions [48,87,88]. A completely different approach was suggested and investigated by Bruce et al. which relies on ordered domains of PEO based electrolytes with fast transport of lithium ions along helix channels formed by the PEO backbone [89,90]. Several investigations have proved a favorable influence of ordered domains in PEO [91,92].

3.3 Stability of polymer electrolytes based on polysiloxanes and polyphosphazenes in lithium ion cells Electrochemical stability in the potential range of interest is a required key property for a lithium battery electrolyte. The electrochemical stability is defined as the voltage range (vs. Li/Li+ reference) where the electrolyte is stable vs. oxidation or reduction at the electrodes. Values of 4.0–4.5 V are absolutely necessary. As practically no solvent molecule is thermodynamically stable in such a large activity gradient (corresponding to 70–78 decades of the lithium activity), the materials rely on kinetic stability due to the formation of electrode/electrolyte interlayers (SEI = solid/electrolyte interphase) or simply on metastability and slow reaction kinetics. One of the future strategies to achieve higher energy densities in lithium batteries aims at even higher cell voltages. Hence, development of novel solvents, lithium salts and additives with enhanced high voltage stability is an important task. The electrochemical stability of the organo-functionalized polysiloxanes and polyphosphazenes was investigated under the conditions of lithium cells. A conventional closed three electrode cell was used consisting of an inert working electrode (Ni, Pt) according to Li|electrolyte membrane|Ni (or Pt)

(10)

Metallic lithium was applied as material for the counter and reference electrodes. The polymer electrolyte was measured in the form of a cross-linked elastic membrane with a thickness around or below 100 μm. The interface between the electrolyte membrane and the electrodes was kept under light pressure by spring loading. Several different materials were tested as working

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Fig. 14. Cyclic voltammetry of a polysiloxane membrane, T 0.1 OPS, containing 0.6 mmol/gpolymer LiBOB (cross-linked with Si(OMe) 3 -sidegroups in 10% of the monomeric siloxane units).

electrodes. The final materials were chosen in order to minimize any corrosion or dissolution of the working electrode during the experiments. Cyclic voltammetry curves were taken at 70 ◦ C in order to minimize the electrolyte resistance. Figure 14 shows a typical curve of cyclic voltammetry at 70 ◦ C for a polysiloxane based membrane with dissolved LiBOB (cf . Scheme 2). It is contains separate voltage scans in the high and the low potential range, each one starting at a medium potential near 3 V (vs. Li/Li+ ). As the chemical diffusion coefficients of the salts within the polymer are low, the sweep rates were kept small. The peaks around 0 V in Fig. 14 are caused by lithium plating on nickel. The reverse peaks near 0 V show the re-dissolution of lithium ions into the electrolyte. Anodic oxidation sets in at about 4.6 V accompanied by exponential current increase for potentials larger than 5 V. This reaction is due to the oxidation of the oligoether sidechains and is in agreement with results of other authors for PEO and graft-copolymers of PEO with other backbones. As can be inferred from Fig. 14, polysiloxane based electrolytes are stable with the currently used lithium anode and cathode materials (e.g. LiFePO4 : 3.3–3.6 V). Figure 15 shows corresponding results for a polymer electrolyte made from a cross-linked MEEP membrane. In this example, LiDFOB was the dissolved salt (cf . structure of LiDFOB in Scheme 2). It becomes evident that the oxidation limit of the MEEP membrane extends to even higher potential values as compared to the polysiloxane. This observation was valid for all comparable experiments with polysiloxanes and polyphosphazenes. Thus, one can conclude that the oxidation of the sidechains at the polyphosphazenes is more hindered and slower. It is not easy to explain that as the oligomeric ether sidechains were the same. Possibly, the oxidation products at the interface form a better stabilizing film with polyphosphazene than with polysiloxane.

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Fig. 15. Cyclic voltammetry of a polyphosphazene membrane (MEEP) containing 0.8 mmol/ gpolymer LiDFOB (cross-linked with the help of 5 wt. % benzophenone activated by UV irradiation).

The peaks at the low potential end of the voltage scale around 0 V in Fig. 15 are sharper, because the membrane thickness and therefore also the polymer electrolyte resistance were rather small as compared to the polysiloxane in Fig. 14. Both examples in Figs. 14 and 15 give a clear statement that the electrochemical stability of polysiloxanes and polyphosphazenes is suitable for an application in lithium batteries. At present, an increasing number of experimental results are available which prove that hybrid or gel type electrolytes prepared with polyphosphazenes and polysiloxanes can attain sufficient ionic conductivities with the help of low molecular weight additives and show excellent cycling stability. They are easily produced as thin films with excellent adhesion on the current anode and cathode materials of lithium cells.

4. Final remarks One of the conclusions of the research on polyphosphazenes and polysiloxanes is that the intrinsic limitations of ionic transport due to coupling to the polymer mobility limits the conductivities at room temperature to values around 0.1 mS/cm or only slightly higher. For liquid polysiloxanes and polyphosphazenes with low Tg , a possibility exists to reduce the viscosity by choosing rather low molar masses (significantly below 103 g/mol). Of course, the consequence is that one leaves the field of polymers. In order to use the advantage of polymer chemistry with respect to mechanical stability and safety, the necessarily enhanced intermolecular interaction or cross-linking inevitably slows down the molecular mobility and with it the ion mobility. Nevertheless, as the results described in this article showed, electrolytes made of cross-linked polymer membranes can be optimized by making use of composite and hy-

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brid concepts. The dispersion of nanoparticles alone is not enough to achieve conductivities and ion mobilities as those found in liquid electrolytes. But, incorporation of low molecular weight additives has a great influence and can decouple the ion transport from the polymer network in the background. In this context, concepts of nano- or micro-structured polymer networks such as foams or ordered pore channels become quite interesting if combined with a low viscosity additive. To conclude, polymer based electrolyte concepts remain attractive for application in batteries. There is a good perspective that suitable electrolytes will be designed for the next generation of lithium batteries.

Acknowledgement This work was part of the research program A2 within the collaborative research center “SFB 458”, funded by the Deutsche Forschungsgemeinschaft. We thank K. Funke, R. Banhatti, H. Eckert, C. Cramer-Kellers, M. Schönhoff, A. Heuer, R. Pöttgen, B. Krebs, T. Nilges, N. Stolwijk, L. van Wüllen and D. Wilmer for helpful discussions, thanks also to all colleagues in the SFB 458 for the excellent collaboration. Finally, we would like to acknowledge the collaboration with D. Richter, R. Zorn, and W. Pyckhout-Hintzen on SANS experiments in Jülich, the collaboration with S. Passerini and M. Winter regarding the electrochemical analysis of polymer electrolytes in lithium ion cells, and we thank H. Gores (University Regensburg) and G. Röschenthaler (Jacobs University Bremen) for preparing and making available a number of novel lithium salts.

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Lithium-Transition Metal-Tetrelides – Structure and Lithium Mobility By Rainer Pöttgen1 , ∗, Tim Dinges1 , Hellmut Eckert2 , Puravankara Sreeraj1 , and Hans-Dieter Wiemhöfer1 1 2

Institut für Anorganische und Analytische Chemie, Universität Münster, Corrensstr. 30, 48149 Münster, Germany Institut für Physikalische Chemie, Universität Münster, Corrensstr. 30, 48149 Münster, Germany (Received June 10, 2010; accepted July 22, 2010)

Lithium / Tetrelides / Ionic Mobility / Crystal Chemistry Lithium-transition metal (T)-tetrelides (tetr. = C, Si, Ge, Sn, Pb) are an interesting class of materials with greatly differing crystal structures. The transition metal and tetrel atoms build up covalently bonded networks which leave cavities or channels for the lithium atoms. Depending on the bonding of the lithium atoms to the polyanionic network one observes mobility of the lithium atoms. The crystal chemistry, chemical bonding, 7 Li solid state NMR, and the electrochemical behavior of the tetrelides are reviewed herein.

1. Introduction Elemental lithium is the most efficient anode material for a lithium ion battery from the point of view of cell voltage. However, metallic lithium has some distinct disadvantages with respect to safety and lithium reduction. The melting (180.5 ◦ C) and boiling points (1347 ◦ C) of lithium are comparatively low and electric shortening in such a lithium ion battery may cause irreversible damage. Furthermore, using elemental lithium, dendrite and whisker growth occurs upon lithium reduction, and penetration of the separation polymer foils can cause electric shortening. In order to overcome these disadvantages, a variety of binary intermetallic lithium compounds has been tested as anode material. Prominent examples are the lithium aluminides, indides, silicides, stannides, and antimonides. Many of these compounds can be classified as Zintl phases. An extensive overview of the crystal chemistry of the binary phases was given by Nesper [1]. In such * Corresponding author. E-mail: [email protected] Z. Phys. Chem. 224 (2010) 1475–1504 © by Oldenbourg Wissenschaftsverlag, München

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phases the lithium atoms transfer their valence electron to the main group element atoms enabling the formation of a polyanionic partial structure (Zintl anions). Due to the electrostatic interactions between the Zintl anions and the lithium cations, the melting point of these phases drastically increases with respect to elemental lithium, a safety advantage. However, the use of binary or even multinary intermetallic lithium compounds comprises two severe disadvantages: (i) the density of the material drastically increases and (ii) the lithium activity (cell voltage) decreases. An overview of the electrochemical data of the binary compounds with respect to battery application is given by Huggins [2]. The key subjects for lithium alloy electrode materials are (i) a preferably low density, (ii) cheep components, (iii) an enhanced melting point in order to overcome the melting of lithium, (iv) a high lithium-to-metal ratio at the end of charge (high capacity), (v) a low volume expansion during lithiation, and (vi) a good cycling performance. Another approach concerns the addition of a third component, a 3d, 4d, or 5d transition metal. This way one obtains a two- or three-dimensional polyanionic network formed by the transition metal and X (X = element of the 3rd , 4th , or 5th main group) component where the lithium atoms fill cages or channels. The pioneering synthetic work in this field was done around fourty years ago in the groups of H.-U. Schuster and A. Weiss. In that time, only structural aspects of this interesting class of compounds had been considered. Later on, these compounds have attracted renewed interest with respect to their use as anode materials in lithium ion batteries. Since then different approaches have been applied to these materials. Groups working in the field of synthetic solid state chemistry used classical phase analyses in order to search for new ternary compounds, while in the field of electrochemistry several binary T y X z intermetallics have been used as so-called ‘alloy-electrodes’ and their lithiation behavior has been tested electrochemically. Besides the many Li3 Bi related cubic phases with group III and group IV main group elements reported by the Schuster and Weiss groups (Table 1), the Li–Cu–Al [3,4] and Li–Ag–In [5–7] system gained renaissance when searching for new light weight and efficient anode materials. Recent research with the group V main group elements mainly focused on the 3d transition metal phosphides with respect to electrochemical lithium insertion ([8–11] and references therein), leading to comparatively lithiumrich materials like Li 6 FeP 2 [12] or Li8−y Mn y P 4 [13]. Also binary 3d transition metal antimonides have been intensively investigated as negative electrodes for lithium ion batteries ([14–16], and references therein). In the last ten years we have investigated the lithium-transition metaltetrelides in more detail within the synthetic subgroup of the collaborative research center SFB 458 Ionic Conductivity in Materials with Disordered Structures. The crystal chemical peculiarities of these tetrelides are reviewed herein together with 7 Li solid state NMR results and electrochemical data. The basic crystallographic data of the tetrelides synthesized in our group and those

Space Group

Fm3¯ m Im3¯ P6/mmm C2/m P6/mmm P63 /mmc Fm3¯ m Im3¯ P6/mmm P31 Cc R3¯ m P 3¯ m1 hex. P 6¯ 2m P4/mbm P21 /m P4/mbm I 4¯ 3m I23 Pnnm I4mm P43 21 2

Compound

Li x T y Si z LiNi2 Si Li13 Ni9 Si18 LiNi6 Si6 Li0.6 Ni 5.37 Si6 Li13 Ni40 Si31 Li18.75 Ni5 Si32 LiCu 2 Si Li13 Cu 14.22 Si12.78 LiCu 3 Si2 Li7 Cu7 Si5 Li14.1 Cu6.8 Si7.1 Li119 Cu 145 Si177 Li2 ZnSi Li2 ZnSi LiYSi LiY2 Si2 Li0.29 Zr1.83 Si0.88 LiRh 2 Si2 Li3 Rh4 Si4 Li13.7 Rh8 Si18.3 Li13 Pd12 Si12 LiPdSi3 Li4.82 Pd2.90 Si2.28 555.3 1274.1 846.1 812.2 1709.2 1287.0 577.6 1293.3 1708.3 1417.4 1412.0 1313.7 424.7 424.7 702.3 710.5 370.1 698.1 727.3 1306.9 1833.9 412.0 679.1

a

a a a 374.28 a a a a a a 2462 a a a a a 366.9 a a a 1278.6 a a

b

a a 756.6 1106.9 784.8 2144.6 a a 785.3 1352.7 915.1 4164 822.4 822.4 421.2 414.4 758.1 274.6 a a 440.6 1047.5 3774.1

c

0.041 0.072 0.038 0.041 0.049 0.096 0.054 0.0268 0.056 0.055 0.0291 0.155 0.061 0.045 0.0408 0.0306 0.0104 0.0447 0.0430 0.064 0.0392

90 90 103.70 90 90 90 90 90 90

R-value

90 90 90 111.41 90 90 90 90 90 90 100.22 90 90

β

[18] [21] [31] [37] [19] [20] [24,27] [21,22] [23] [32] [26] [33] [25] [59] [29] [30] [34] [36] [38] [100] [39] [40] [38]

Reference

Table 1. Basic crystallographic data of the known lithium-transition metal-tetrelides. For structures refined from single crystal diffractometer data also the residuals are listed.

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45

LiAg2 Si LiAg2 Si Li8 Ag 3 Si5 Li3 Ag 2 Si3 Li2 AgSi 2 Li8 Ag 3 Si3 Li2 AgSi 2 Li13 Ag5 Si6 Li2 La 2 Si3 Li3 Ir 4 Si4 LiIr2 Si Li4 Pt 3 Si Li2 AuSi Li8 Au 3 Si5 Li x T y Ge z Li2 MnGe Li2 MnGe LiFe6 Ge6 LiFe6 Ge5 LiFe6 Ge4 LiCo2 Ge LiCo6 Ge 6 LiNi2 Ge LiNi6 Ge6 LiCu2 Ge

Compound

Table 1. Continued.

404.5 404.5 606.7 605.5 605.0 606.7 1082.2 439.06 450.4 732.4 576.4 693.5 607.8 603.7 610.8 610.8 874.4 504.8 504.5 567.3 504.8 567.4 873.4 589.2

P4/mbm tetr. P6/mmm R3¯ m R3¯ m Fm 3¯ m P6/mmm Fm3¯ m P6/mmm Fm3¯ m

a

Pm3¯ m Fm3¯ m tetr. P42 /nnm F 4¯ 3m P42 /nnm orthorh. R3¯ m Cmcm I 4¯ 3m Fm3¯ m R32 Fd3¯ m Fd3¯ m

Space Group

a a a a a a a a a a

a a a a a a 604.9 a 1882.9 a a a a a

b

633.1 635.7 803.3 4364 1966 a 772.9 a 779.7 a

a a 618.8 616.4 a 618.6 430.6 4229.3 689.7 a a 1626.1 a a

c

90 90 90 90 90 90 90 90 90 90

90 90 90 90 90 90 90 90 90 90 90 90 90 90

β

0.087 0.136

0.108

0.078

0.077

0.0118 0.0262 0.075

0.0416

R-value

[46] [48] [50] [51,52] [51,52] [49] [31] [18] [31] [24,27]

[44] [43] [22] [44] [44] [43] [43] [41] [42] [38] [38] [45] [28] [22]

Reference

46 R. Pöttgen et al.

Space Group

¯ F 43m ¯ F 43m Fd3¯ m hex. hex. Fm3¯ m ¯ F 43m hex. hex. R3c P 6¯ m2 ¯ F 43m P 6¯ 2m Pnma R3¯ m P21 /m Fm3¯ m Fm3¯ m Fm3¯ m B2/m Fm3¯ m Pm3¯ m P42 /nnm Fm3¯ m P42 /nnm

Compound

Li2 CuGe Li2 CuGe Li2 Cu3 Ge Li2.5 CuGe Li1.5 CuGe Li2 ZnGe Li2 ZnGe Li1.25 ZnGe Li1.14 ZnGe1.37 Li8 Zn2 Ge 3 LiZnGe Li16.995 Zn0.0052 Ge 4 LiYGe LiY4 Ge 4 Li4 ZrGe2 Li0.5 Zr 2 Ge0.5 LiRh 2 Ge Li2 PdGe LiPd2 Ge Li2 Pd2.7 Ge 2.3 LiAg2 Ge Li3 Ag3 Ge2 Li3 Ag2 Ge3 LiAg2 Ge Li8 Ag3 Ge5

Table 1. Continued.

596.8 595.2 706 424.05 428.1 614.2 611.4 427.7 417.3 755.5 427.75 1884.2 706.5 714.4 477.8 388.4 584.7 603.2 601.1 734.19 632.6 404.5 616.9 632.6 617.8

a a a a a a a a a a a a a a 1479.9 a 361.3 a a a 670.57 a a a a a

b a a a 818.19 968.1 a a 937.3 676.8 2444.9 936.53 a 423.3 775.7 1840.1 771.6 a a a 423.90 a a 621.3 a 629.1

c 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 103.50 90 90 90 111.421 90 90 90 90 90

β

0.062 0.097 0.066

0.024 0.022 0.0197

0.030 0.0186 0.0318 0.073

0.017

R-value

[24,60] [28] [53] [54] [54] [56,59,62,68] [55] [59] [59] [55] [55] [58] [29] [63] [61] [61] [35] [65] [65] [64] [57] [44] [44] [44] [47]

Reference

Lithium-Transition Metal-Tetrelides – Structure and Lithium Mobility

47

Li2 CdGe LiLaGe2 Li2 LaGe Li2 La 2 Ge 3 Li2 PtGe Li2 AuGe Li2 HgGe Li x T y Sn z LiV2 Sn Li11 V3 Sn4 Li2 MnSn LiCoSn6 LiNi2 Sn Li2 CuSn Li2 CuSn LiCu2 Sn LiCu3 Sn2 Li2 ZnSn Li7.33 Zn4.56 Sn4.11 LiYSn LiRuSn4 LiRhSn4 LiRh3 Sn5 Li4 Rh 3 Sn5 Li0.45 Rh3 Sn6.5

Compound

Table 1. Continued.

641.5 792.2 455.6 448.7 599.5 617.0 640.9 621.292 470.82 650.8 620.2 596.3 628.2 626.2 430.3 509.5 665 644.8 929.6 662.61 658.73 538.9 812.9 930.91

Fm3¯ m P 3¯ m1 tetr. Ibca Fm3¯ m F 4¯ 3m Fm3¯ m P63 /mmc P6/mmm Fm3¯ m F 4¯ 3m P63 mc I4/mcm I4/mcm Pbcm Pnnm Im3¯ m

a

Fm3¯ m Pnma P63 /mmc Cmcm Fm3¯ m F 4¯ 3m Fm3¯ m

Space Group

a a a 621.0 a a a a a a a a a a 976.6 2258.4 a

a 403.2 a 1893.6 a a a

b

a 1710.04 670.2 3699.6 a a a 763.7 951.6 a a 734.6 1116.98 1136.4 1278.5 450.7 a

a 1094.6 775.9 703.6 a a a

c

90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90

90 90 90 90 90 90 90

β

0.043 0.096 0.0432 0.0241 0.0383 0.0219 0.0135

0.0351

0.098

0.024

0.1234 0.1198

0.0271

R-value

[81] [81] [48] [17] [49] [28] [24] [69] [75] [56] [71] [77] [72] [72] [73] [76] [74]

[56,68] [66] [67] [67] [38] [28] [56,68]

Reference

48 R. Pöttgen et al.

Space Group

Im3¯ m Im3¯ m Fm3¯ m P 3¯ m1 Fm3¯ m ¯ F 43m ¯ F 43m P4/mbm ¯ F 43m Fm3¯ m Fm3¯ m Fm3¯ m P31m Fm3¯ m ¯ F 43m Cmcm Fm3¯ m I4/mcm Im3¯ m Im3¯ m Fm3¯ m ¯ F 43m Ia3¯ Ia3¯

Compound

Li0.64 Rh 3 Sn6.3 Li0.8 Rh 3 Sn6.2 LiRh 2 Sn Li8 Rh7 Sn8 LiPd2 Sn Li2 PdSn LiMgPdSn Li1.42 Pd2 Sn5.58 Li2 AgSn LiAg2 Sn LiAg2 Sn Li2 AgSn Li17 Ag 3 Sn6 Li2 CdSn Li2 CdSn LiLaSn2 Li2 IrSn LiIrSn4 Li0.62 Ir 3 Sn6.38 Li0.66 Ir 3 Sn6.34 Li2 PtSn Li2 PtSn Li2.27 Pt2 Sn3.73 Li2.43 Pt2 Sn3.57

Table 1. Continued.

929.09 927.31 615.5 894.4 626.4 631.4 642.0 662.61 655.2 659.9 659.2 656.5 806.3 669 672.7 444.6 624.3 657.34 932.21 931.16 627.8 626.1 1269.7 1266.6

a a a a a a a a a a a a a a a a 1809 a a a a a a a a

b a a a 1107.3 a a a 843.39 a a a a 850.9 a a 452.2 a 1130.4 a a a a a a

c 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90

β

0.0461 0.1131 0.0483

0.0155 0.0177 0.0295

0.045

0.0450

0.064 0.044 0.0388

0.0288 0.0335 0.0143 0.0352

R-value [74] [74] [38] [38] [65] [65] [70] [78] [28] [24] [103] [24,60] [79] [56] [71] [80] [65] [72] [74] [74] [65] [84] [84] [84]

Reference

Lithium-Transition Metal-Tetrelides – Structure and Lithium Mobility

49

Li3 Pt2 Sn 3 LiMgPtSn Li2 AuSn Li2 AuSn LiAuSn LiAu3 Sn4 Li2 AuSn2 Li x T y Pb z Li8.96 Zn3.20 Pb 3.84 Li0.6 ZrPb0.4 Li0.5 ZrPb0.5 Li2 PdPb LiPd2 Pb Li2 AgPb Li8.49 Cd 3.59 Pb3.92 Li2 AuPb

Compound

Table 1. Continued.

1264.3 639.7 641.7 643.8 467.08 448.31 455.60 663.4 296.79 343.13 644.9 638.4 667.8 682.5 660.1

F 4¯ 3m P 6¯ m2 Pm3¯ m Fm3¯ m Fm3¯ m F 4¯ 3m F 4¯ 3m F 4¯ 3m

a

Ia3¯ F 4¯ 3m Fm3¯ m F 4¯ 3m P63 /mmc P63 mc I41 /amd

Space Group

a a a a a a a a

a a a a a a a

b

a 476.26 a a a a a a

a a a a 603.7 2055.7 1957.4

c

90 90 90 90 90 90 90 90

90 90 90 90 90 90 90

β

0.0632 0.0561

0.0150 0.0302 0.0681

0.0192

R-value

[71] [61] [61] [65] [65] [28,85] [71] [28]

[84] [70] [24,60,70] [28] [82] [82] [83]

Reference

50 R. Pöttgen et al.

Lithium-Transition Metal-Tetrelides – Structure and Lithium Mobility

51

reported in literature are listed in Table 1. A first overview on some of the stannides presented herein has been published in 2002 [17]. In the following chapters we briefly discuss the crystal chemistry of the Li x T y X z intermetallics and then turn to the properties, mainly with respect to lithium mobility.

2. Syntheses conditions Lithium-transition metal-tetrelides can be synthesized in X-ray pure form from the elements in sealed inert metal tubes (niobium or tantalum). Lithium is used in the form of rods in order to reduce the reactive surface. Reaction with moist air causes contaminations with LiOH and Li3 N in the form of a dark cusp at the surface. The transition metals were used as wires (∅1–2 mm) or powders (typically 200 mesh). The group IV elements were taken in the form of lumps or granules. Lithium pieces were mixed with the transition metals and the group IV element in the stoichiometric ratios and arc-welded in niobium or tantalum tubes [86] under an argon pressure (purified over titanium sponge at 900 K, silica gel and molecular sieves) of about 800 mbar. The sealed metal tubes were subsequently enclosed in evacuated silica tubes to prevent oxidation and then annealed in tube furnaces. Alternatively the metal tubes can be heated in a water-cooled sample chamber of a high-frequency furnace [87]. The samples can be separated quantitatively from the tubes. No reaction of the samples with the container material was observed. Most compounds were stable in air over several weeks. For further details we refer to the original papers. Standard characterization of the samples was performed via X-ray powder diffraction (Guinier technique), ICP-OES, and metallography in combination with EDX. Where single crystals were available, structure refinements based on single crystal diffractometer data were carried out. Selected compounds have been studied by powder neutron diffraction.

2.1 Lithium-transition metal-acetylides With carbon as main group element, so far only LiAgC2 and LiAuC2 have been structurally characterized [88–90]. These compounds are acetylides with carbon–carbon triple bonds. The silver, respectively gold atoms build up infinite linear {AgC2 } and {AuC2 } chains which are arranged in the form of a hexagonal rod packing. Between the rods the lithium atoms are coordinated to three C 2 pairs in a side-on fashion. Similar to binary Li2 C2 [91], also these highly ionic acetylides are extremely moisture sensitive.

2.2 Lithium-transition metal-silicides The basic crystallographic data of the diverse Li x T y Si z silicides are listed in Table 1. Especially the ternary systems with nickel, copper, and silver have intensively been investigated. Several silicides crystallize with cubic structures

52

R. Pöttgen et al.

Fig. 1. The crystal structure of LiNi2 Si. Lithium, nickel, and silicon atoms are drawn as medium grey, filled, and open circles, respectively. The [Ni 2 Si] network is emphasized.

which derive from the Li3 Bi type by an ordered arrangement on the three crystallographically independent sites. As an example we present the structure of LiNi2 Si in Fig. 1. The lithium and silicon atoms build up a rocksalt-like substructure in which the tetrahedral voids are filled by the nickel atoms. This corresponds to the structure of the Heusler phase. In addition to the silicides, also the other tetrelides and compounds with group III and group V occur with a similar structural arrangement. Most compounds had already been reported in the early work by Weiss and Schuster [24,28]. Besides the compositions LiT2 Si, also several silicides Li 2 TSi have been reported (Table 1). All of these structures can be considered as superstructures of the well known CsCl type through site occupancy variants in a 2 × 2 × 2 supercell. An overview of the different variants is given in the textbook by Müller [92]. Several of the silicide structures are quite complex with large unit cells (Table 1). These silicides show pronounced mixed occupancies, i.e. T–Si as well as Li–Si mixing besides partially occupied silicon and T sites (for details we refer to the original literature), leading to extended homogeneity ranges. These structural features might be good prerequisites for lithium mobility, however, only few phases had been tested in this respect (vide infra). Various ternary silicides with silver as transition metal component have also been reported. They show homogeneity ranges as well and for some compounds the exact composition is still not known. A structurally very interesting compound of this family is Li13 Ag 5 Si6 [41] which shows a complicated stacking sequence within a wurtzite-related network of the silver and silicon atoms besides some Ag/Li mixing within the tetrahedra. This is favorable for use of Li 13 Ag 5 Si6 as an anode material (vide infra). Besides the many disordered materials we exemplarily describe some crystallographically ordered structures hereafter in more detail. With rhodium as transition metal we observe the silicides LiRh 2 Si 2 [36] and Li3 Rh 4 Si4 [38]. Both compounds have an identical Rh–Si ratio but show different threedimensional [RhSi] networks (Fig. 2) with strong covalent Rh–Si bonding (as is evident from electronic structure calculations). This is underlined by shorter Rh–Si distances at 244–248 (LiRh 2 Si2 ) and 227 pm (Li3 Rh 4 Si 4 ), which com-

Lithium-Transition Metal-Tetrelides – Structure and Lithium Mobility

53

Fig. 2. The crystal structures of LiRh 2 Si2 and Li3 Rh4 Si4 . Lithium, rhodium, and silicon atoms are drawn as medium grey, filled, and open circles, respectively. The [RhSi] networks are emphasized.

pare well with the sum of the covalent radii [93] of 242 pm. In addition one also observes shorter Rh–Rh contacts within the two networks. In tetragonal LiRh2 Si2 [36] lithium-filled channels extend in the c direction. Temperature dependent 7 Li solid state NMR data were indicative of restricted one-dimensional lithium mobility producing only partial line narrowing of the static NMR lineshape and a well-defined quadrupolar splitting, in qualitative agreement with this structural feature. A different situation occurs for Li3 Rh 4 Si 4 [38]. This cubic structure leaves larger channels in all three directions. At room temperature Li3 Rh 4 Si4 crystallizes with the Na3 Pt 4 Ge4 type [94], however, with a split position for the lithium atoms. Around 220 K the stoichiometric sample shows a structural phase transition to a rhombohedral lock-in phase with lithium ordering. If the lithium content deviates from Li3 per formula unit we observe different modulated structures at low temperature. In total, one observes a quite complicated microstructure. This picture is confirmed by detailed temperature-dependent 7 Li solid state NMR spectra, to be discussed below. LiPdSi3 [40] shows a site occupancy variant of the tetragonal BaNiSn 3 type [95] structure. Most likely due to size restrictions, the small lithium atoms occupy the square-pyramidal 2a site (245–246 pm Li–Si), while the palladium atoms take the 2a barium site with a square-prismatic PdLi 4 Si2 4 coordination (Fig. 3). For an overview of the many BaAl 4 /ThCr 2 Si2 related superstructures and ordering variants we refer to a review article [96]. As an example for a rare earth metal containing silicide we present the LiYSi structure [29] in Fig. 4. LiYSi crystallizes with a site occupancy variant of the ZrNiAl structure [97,98]. The yttrium and silicon atoms build up a three-dimensional [YSi] network which leaves strongly distorted hexagonal channels for the lithium atoms. The latter connect to four silicon atoms of the

54

R. Pöttgen et al.

Fig. 3. The crystal structure of LiPdSi3 . Lithium, palladium, and silicon atoms are drawn as medium grey, filled, and open circles, respectively. The [LiSi3 ] network is emphasized.

Fig. 4. The crystal structure of LiYSi. Lithium, yttrium, and silicon atoms are drawn as medium grey, filled, and open circles, respectively. The [YSi] network is emphasized.

[YSi] network with Li–Si distances of 265 and 278 pm. Between the channels one observes also shorter Li–Li distances of 279 pm, somewhat smaller than in bcc lithium (304 pm) [99]. These shorter Li–Li distances express the ionic character of lithium (smaller size of Li+ ). The lithium-rich silicide Li 4 Pt3 Si [45] exhibits a peculiar crystal structure. As emphasized in Fig. 5, the silicon atoms have trigonal-prismatic platinum co-

Lithium-Transition Metal-Tetrelides – Structure and Lithium Mobility

55

Fig. 5. Projection of the Li4 Pt3 Si structure along the c-axis. Lithium and platinum atoms are drawn as medium grey and black circles, respectively.

ordination and always two prisms are condensed via a common triangular face to a double unit. The latter further condense with neighbouring units via common corners, leading to a three-dimensional network which leaves channels for the two crystallographically independent lithium atoms. As discussed below, the hopping motion of lithium between these sites can be detected by 7 Li magic-angle spinning NMR.

2.3 Lithium-transition metal-germanides Many of the Li–T–Ge phase diagrams show cubic ternary compounds of compositions Li2 TGe and LiT2 Ge. The structural chemistry of these germanides is similar to those of the isotypic silicides (vide ultra). A structurally peculiar material among the germanides is the lithium-rich compound Li8 Zn2 Ge3 [55]. The zinc and germanium atoms built up planar [Zn2 Ge3 ] heterographite layers (Fig. 6) which are separated by the lithium atoms. The stacking sequence of the layers is ABCDEF. Such defect hexagonal layers also occur in Yb 3 Si 5 [101] and the stannide Li17 Ag 3 Sn 6 [79] (vide infra). A side product of the Li8 Zn 2 Ge3 synthesis concerns the solid solution Li17−x Zn x Ge4 [58]. These germanides can be considered as zinc-doped variants of binary cubic Li17 Ge4 [102]. The zinc-doping leads to a small increase in the lattice parameter from 1875.6 [102] to 1884.2 pm [58]. In the lithium-poor parts of the Li–T–Ge phase diagrams the germanides LiFe6 Ge4 and LiFe 6 Ge5 [51] have been reported. Originally these germanides were described with monoclinic unit cells in space groups C2/m. Analyses of the positional parameters pointed to the higher symmetric space groups

56

R. Pöttgen et al.

Fig. 6. Cutout of one [Zn2 Ge 3 ] heterographite layer in the crystal structure of Li8 Zn2 Ge3 [55].

R3¯ m [52]. Both structures are relatively complex stacking variants of Zr 4 Al3 and CeCo 3 B2 -related slabs which also occur in the germanide MgFe6 Ge6 . With an even higher germanium content LiFe 6 Ge6 [50] has been obtained. This germanide is structurally related to the CoSn type.

2.4 Lithium-transition metal-stannides A huge number of the stannide structures (Table 1) has been studied on the basis of X-ray single crystal data. A first overview of the stannide structures was given in an earlier review article [17]. Again, besides the compounds with comparatively complex crystal structures, also diverse cubic stannides Li2 TSn and LiT2 Sn have been reported. In the present chapter we concentrate only on some representative examples out of the Li–T–Sn family of compounds. The most detailed study for the cubic phases was performed for the lithium insertion and Sn/Li substitution in binary PtSn 2 . In a first step the octahedral sites can be filled with lithium. Due to an attractive Pt–Li interaction the a lattice parameter drastically decreases (Fig. 7). In a next step, part of the tetrahedral tin sites can ce substituted by lithium. At the composition Li3 Pt 2 Sn 3 [84] a fully ordered superstructure occurs. The end member of this series is the lithium-rich stannide Li2 PtSn. Most likely many of the cubic phases in the silicides, germanides, and stannides series show similar structural behavior which deserves meticulous reinvestigation. Also the various lattice parameters given for the lithium-silver-silicides might be a hint for such ordered variants. Tin-lithium substitution is not limited to Li3 Pt2 Sn 3 [84] and the respective solid solution Li3−x Pt2 Sn 3+x . A binary stannide Rh3 Sn 7 does not exist, however, upon tin-lithium substitition a broad homogeneity range Li x Rh 3 Sn 7−x occurs [74]. Similar behavior has been observed for the series Li x Ir 3 Sn 7−x (Table 1), but here, the binary stannide Ir 3 Sn 7 is known, while a small portion of lithium is needed to stabilize the rhodium compound. The transition metal atoms in the two solid solutions Li x Rh 3 Sn 7−x and Li x Ir 3 Sn 7−x have square antiprismatic tin coordination. This is also the case

Lithium-Transition Metal-Tetrelides – Structure and Lithium Mobility

57

Fig. 7. Course of the cubic lattice parameter vs. lithium portion on fluoride sites of the solid solution Li3−x Pt2 Sn3+x , PtSn2 and Li2 PtSn. The open circle marks the composition with the lowest lithium content possible of the solid solution where no Sn/Li mixing on the tetrahedral sites occurs. Filling of tetrahedral sites corresponds to a replacement of tin atoms.

Fig. 8. The crystal structures of LiRhSn4 and Li1.42 Pd2 Sn5.58 . The layers of condensed squareantiprisms are emphasized. For details see text.

for several of the binary transition metal stannides. In the latter compounds, the square antiprisms are condensed via common edges, forming mono or double blocks. Interestingly such blocks occur in a variety of ternary lithiumcontaining stannides like LiTSn 4 (T = Ru, Rh, Ir) [72] and Li1.42 Pd 2 Sn 5.58 [78] (Fig. 7). In the structures of the LiTSn4 stannides, no Sn/Li mixing has been observed, while the palladium containing phases deserve a certain degree of Sn/Li substitution. Refinement of several crystal structures always resulted in the same, optimized composition.

58

R. Pöttgen et al.

Fig. 9. The crystal structure of tetragonal Li2 AuSn2 . Lithium, gold, and tin atoms are drawn as medium grey, black filled, and open circles, respectively. The three-dimensional network of condensed AuSn 4/2 tetrahedra is emphasized.

Completely different crystal chemistry has been observed for the stannide Li2 AuSn 2 [83] which crystallizes with its own structure type. Each gold atom has tetrahedral tin coordination and these AuSn 4/2 tetrahedra are condensed via common corners within the ab plane (Fig. 9). Adjacent layers are further condensed via Sn–Sn bonds, leading to a three-dimensional network. Due to its peculiar crystal chemistry (Li 2 AuSn 2 crystallizes with space group I4 1 /amd), the tetrahedral network leaves many channels that exist in the a and b direction as well. This offers excellent structural prerequisites for lithium mobility (vide infra).

2.5 Lithium-transition metal-plumbides So far only few lithium-transition metal-plumbides have been reported (Table 1). Except Li0.6 ZrPb0.4 [61] with hexagonal LiRh type structure, they all crystallize with cubic structures which derive from the CsCl type. A severe problem for the plumbides is the refinement of the lithium positions. Besides the strongly scattering lead atoms, lithium is hardly reliably detectable by X-ray diffraction. All plumbides had only been studied on the basis of powder X-ray diffraction. Yet no properties have been studied.

Lithium-Transition Metal-Tetrelides – Structure and Lithium Mobility

59

2.6 Lithium structure, bonding, and mobility: results from solid state NMR spectroscopy Many of the intermetallic Lix T y X z compounds listed in Table 1 show the necessary structural conditions for facilitating lithium mobility, i.e. partially occupied lithium sites, vacancies in the lithium substructure, or mixed occupied sites. In order to get information on the degree of lithium mobility and in order to check the distinguishable lithium sites, many of the tetrelides have been studied by temperature dependent 7 Li solid state NMR. Owing to the favorable properties of the 7 Li isotope (I = 3/2), NMR presents an element-selective, inherently quantitative structural tool for characterizing the local environments and the dynamics of the lithium atoms. Figure 10 summarizes the basic principles of this technique. The four Zeeman levels are shifted by the interaction of the nuclear electric quadrupole moment with the electric field gradient present at the lithium site. The shift depends on both |m| and the orientation of the electric field gradient relative to the magnetic field direction. As a result the two outer Zeeman transitions (3/2|− 1/2| and −3/2| − −1/2|) depend on the orientation of the electric field gradient tensor relative to the magnetic field and are thus subject to strong inhomogeneous broadening in polycrystalline or amorphous samples. Under the influence of magic angle sample spinning (MAS) this powder pattern is converted to a spinning sideband manifold. From the frequency range and intensity distribution of this manifold one can extract the nuclear electric quadrupolar coupling constant C Q (which characterizes the size of the field gradient) and the asymmetry parameter η Q (which characterizes its deviation from cylindrical symmetry). Thus NMR can provide detailed numerical information regarding the local distortions of lithium environments in crystalline compounds. In general, however, the interest is focused on the central |1/2 − | − 1/2 transition, which is unaffected by the quadrupolar interaction in first order, and thus yields highly resolved distinct resonance signals for crystallographically inequivalent lithium sites under MAS conditions. Thus 7 Li MAS-NMR carries the potential of differentiating between various local environments present in the sample and providing detailed infomation about populations and site occupancies. Examples for this application include work on Li2 CuSn [75], Li2 AuSn [75], and Li4 Pt 3 Si [45]. Since the precise 7 Li resonance frequency is strongly affected by the interaction of the lithium nuclei with delocalized spin density of electrons near the Fermi edge (Knight shift), the signals also provide important information about the local electronic properties of the lithium atoms in these compounds. The data are usually given in terms of the isotropic chemical shift δ iso = (ν − ν ref )/ν where ν ref denotes the resonance frequency of a suitable reference standard, usually 1 M LiCl solution. Chemical shifts near zero ppm indicate that the lithium atoms in the intermetallic compounds are highly ionized, wheras high values indicate substantial localized electron density at the lithium atoms. Table 2 summarizes the chemical shift

60

R. Pöttgen et al.

Fig. 10. Zeeman energy level diagram for 7 Li NMR (spin −3/2), green, blue and red colors represent the three Zeeman transitions and spinning sideband patterns are shown in black.

Table 2. 7 Li MAS-NMR chemical shifts δiso of intermetallic lithium compounds. Compound

δiso (ppm)

Li mobility information from NMR

Reference

LiRh2 Si2 Li3 Rh4 Si4 Li3 Ir4 Si4 Li4 Pt3 Si LiCoSn6 Li2 CuSn LiCu2 Sn LiCu3 Sn2 LiRuSn4 LiRhSn4 LiIrSn4 Li4 Rh3 Sn5 LiAg2 Sn LiAuSn Li2 AuSn Li2 AuSn2

26.1 42 63 126.2; 113.8 13.2 38.4; 1.6 28.8 6.8 76.8 14.5 9.5 19.4; 4.9 5.3 9.8 35.4; 27.8 9.7

1D (T > 170 K) 3D (T > 120 K) – 2-site exchange (T > 300 K) – 2-site exchange (T > 300 K) 3D (T > 350 K) – no Li mobility up to 450 K no Li mobility up to 450 K no Li mobility up to 450 K no Li mobility up to 440 K 3D (T > 450 K) no Li mobility up to 470 K 2-site exchange (T > 350 K) 3D (T > 300 K)

[36] [38] [38,106] [45] [75] [75] [38] [75] [105] [105] [105] [38] [103] [104] [75] [83]

data on all of the lithium transition metal tetrelides measured thus far, indicating a wide range. Most compounds exhibit shifts in the range from 0 to 40 ppm, indicating a high ionicity of the lithium atoms. Besides this larger group, three compounds, i.e. Li3 Ir 4 Si 4 , LiRuSn 4 , and Li4 Pt3 Si, exhibit significantly higher shifts with extreme values of 114 and 126 ppm for the two signals of Li4 Pt 3 Si. For the latter compounds the data suggest a higher degree of metallic character for the lithium atoms. Still, in all cases the shifts are significantly

Lithium-Transition Metal-Tetrelides – Structure and Lithium Mobility

61

Fig. 11. Temperature dependent MAS-NMR spectra of Li4 Pt3 Si, revealing site exchange effects on the NMR timescale.

below the value measured for metallic lithium 260 ppm) indicating that significant electron transfer from the lithium to the other constituent atoms takes place. Besides providing useful structure and bonding information, lithium NMR can also give quantitative insights into the local dynamics of the lithium ions. Table 2 summarizes the results obtained so far. If the lithium atoms transfer between crystallographically distinct sites at a hopping rate comparable to the resonance frequency difference between these sites (typically 100–1000 Hz), the spectra are influenced by exchange broadening and signal coalescence effects, and in the fast hopping limit, a sharp line at the average isotropic shift appears. An example is shown for the compound Li4 Pt3 Si in Fig. 11. While the two distinct lithium environments, present in a 3 : 1 ratio in this compound are clearly differentiated at room temperature, signal averaging caused by lithium motion becomes evident above 350 K and activation energies can be extracted by lineshape simulations of temperature dependent MAS-NMR spectra. Similar effects have been observed in Li2 CuSn [75] and Li2 AuSn [75], The above approach is possible only if the samples contain distinct lithium species with resolved MAS resonances. A more general method of characterizing lithium ion dynamics is to study the static NMR spectra as a function of temperature. In the rigid lattice limit (typically realized at 77 K), the width of the central |1/2 − | − 1/2 transition is dominated by strong homonuclear 7 Li–7 Li magnetic dipole–dipole coupling, resulting in typical linewidths near 5000–10 000 Hz. If, however, the lithium atoms move among multiple sites with transfer rates on the order of the 7 Li NMR linewidth (5–10 kHz),

62

R. Pöttgen et al.

Fig. 12. Temperature dependent static 7 Li solid state NMR spectra of Li 3 Rh 4 Si4 [38] and Li2 AuSn 2 [83].

this interaction is averaged out, producing a sharp signal in the motional narrowing limit. Thus, a simple static spectrum recorded at room temperature allows a rapid assessment of the lithium ion mobility in a given compound. In addition, the activation energy can be estimated from detailed tempera-

Lithium-Transition Metal-Tetrelides – Structure and Lithium Mobility

63

Fig. 13. Temperature dependent full width at half height of the 7 Li static NMR spectrum of LiRh2 Si2 . Note the plateau value of the NMR linewidth at high temperatures indicating residual anisotropy. The inset shows the static NMR spectrum at 300 K, which shows prominent residual quadrupolar splitting. The sharp component arises from the Li3 Rh4 Si4 by-product.

ture dependent linewidth data, using the approximate Waugh–Fedin formula E a (kJ/mol) ≈ 0.156 × To , where To is the onset temperature of motional narrowing. Finally, isotropic mobility on the frequency scale of the quadrupolar splitting (∼ 50–100 kHz) leads to complete averaging of the static quadrupolar broadening and the disappearance of the corresponding spinning sideband pattern in the MAS NMR spectra. As an example we present the static 7 Li solid state NMR data of Li3 Rh 4 Si4 [38] and Li2 AuSn 2 [83] in Fig. 12. The drastic line-narrowing with increasing temperature readily indicates lithium motion on the NMR timescale. From the onset temperatures of motional narrowing we obtain activation energies of of 27 (Li2 AuSn 2 ), 33 (LiAg 2 Sn), and 19 (Li3 Rh 4 Si4 ) kJ/mol. These low values classify the above three ternary intermetallic lithium compounds as potentially suitable anode materials for lithium batteries. If the lithium mobility is dimensionally restricted, such as is the case for hopping between distinct sites, or lithium motion within 1D channels or 2D planes, typical NMR signatures are observed: in this case the motional narrowing is incomplete, resulting a constant plateau value of the static linewidth (typically several kHz) in the high-temperature limit. Furthermore, a welldefined quadrupolar splitting is observed, reflecting the residual electric field gradient experienced by the lithium nuclei. Both of these features have been observed in LiRh 2 Si2 (see Fig. 13) [36].

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Fig. 14. 11.7 T 7 Li MAS-NMR spectrum (5 kHz) of a Li3 Rh 4 Si4 sample measured at 178 K. In spite of the obvious motional narrowing effect seen in Fig. 10, multiple resolved MAS-NMR peaks are observed.

While showing clear motional narrowing effects on the NMR timescale above 120 K, the compound Li3 Rh 4 Si 4 represents a special case from a structural point of view. This compound undergoes a phase transition at 230 K, leading to an incommensurate phase, in which still substantial lithium mobility exists according to the small static 7 Li NMR linewidths. The high-resolution MAS-NMR lineshapes of this phase are rather complex indicating a multitude of local environments. In apparent contradiction to the high lithium mobility observed in the static spectra, the multiple lithium resonances observed in the MAS spectra reveal the absence of site exchange on the timescale of 100 Hz (see Fig. 14). Based on these results, the distinct 7 Li resonances must come from highly mobile lithium atoms, which are, however, localized in spatially separated regions. The NMR spectra are thus inconsistent with a uniform superstructure, and rather indicate that the Li3 Rh 4 Si4 samples are better formulated as Li3−δ Rh 4 Si4 , leading to a distribution of different domains with slightly different lithium contents and environments.

3. Electrochemical characterization In parallel to the temperature dependent NMR data, some selected compounds have been analyzed by electrochemical techniques (GITT (Galvanostatic Intermittent Titration Technique) and PITT (Potentiostatic Intermittent Titration

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Fig. 15. A typical PITT plot of ln[I(t)] versus t of Li2+x AuSn 2 at 25 ◦ C. The slope of the linear part of the curve for long times gives the chemical diffusion coefficient DLi according to the shown equation [109]. The inset shows a PITT plot for Li 1−x Ag2 Sn along with the verification of linearity of the Cottrell equation [108].

Fig. 16. Cyclic voltammogram (the first 2 cycles) for Li2 AuSn 2 and Li1−x Ag 2 Sn (inset).

Technique) measurements) in order to study the chemical diffusion, thermodynamic activity and electrochemical capacity as an electrode material in lithium cells. LiIrSn4 [107] and LiAg 2 Sn [108] were investigated by a combined neutron scattering and electrochemistry approach. In view of the low scattering power of lithium for X-rays, neutron diffraction was used for a refinement of the lithium occupancy parameters. The electrochemical measurements on Li1−x Ag 2 Sn showed a chemical diffusion coefficient, D Li of 10−6 –10−7 cm2 s−1

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Table 3. Chemical diffusion coefficient, DLi in anode materials at 25 ◦ C. Compound

DLi (cm2 s−1 )

Reference

Sn Li0.7 Sn Li2.33 Sn Li4.4 Sn Li1+x IrSn4 Li1+x Ag 2 Sn Li2+x AuSn2 α-Si α-Si thin film nano-Si Si (Bulk) Cu 6 Sn 5 LiZn LiCd LiBi LiHg3 Li3 Pb Al Al−Sn α-LiAl β-LiAl MCMB Natural graphite Artificial graphite Graphite powder

6 × 10−7 (6–8) × 10−8 (3–5) × 10−7 (1.8–5.9) × 10−7 10−7 –10−9 10−6 –10−7 10−6 –10−9 (2–3) × 10−13 10−11 –10−14 10−12 –10−13 3 × 10−14 10−9 –10−12 10−9 –10−10 10−9 –10−10 2.9 × 10−11 1.3 × 10−8 9.8 × 10−11 6 × 10−12 3.2 × 10−8 10−10 –10−14 10−9 –10−12 10−6 –10−12 10−7 –10−11 10−7 –10−9 10−8 –10−14

[110,114] [114] [114] [114] [107] [108] [109] [115] [116] [117] [118] [119] [110] [110] [120] [120] [120] [114] [114] [120] [120] [121,122] [121,122] [121,122] [121,122]

and a nearly reversible insertion/deinsertion in the voltage range between 1.6 and 2.1 V. Coulometric titration, chronopotentiometry, chronoamperometry and cyclic voltammetry were used to study the kinetics of lithium ion diffusion in LiIrSn 4 . The range of homogeneity (Li1+Δδ IrSn 4 , −0.091 ≤ δ ≤ +0.012) without any structural change in the host structure, and the high values of chemical diffusion coefficient (10−7 –10−9 cm2 s−1 ) point to good conduction behaviour. Compound electrodes made from Li2+x AuSn 2 [109] were characterized with respect to their kinetics and charging/discharging behaviour at ambient temperature. Figure 15 represents the typical behaviour of these ternary intermetallics during PITT measurements. PITT and GITT measurements result in values of the chemical diffusion coefficient D Li at room temperature in a range between 10−7 and 10−8 cm2 s−1 . For small lithium non-stoichiometry x, a maximum value of D Li max. = 1.5 × 10−6 cm2 s−1 was measured. Cyclic voltammetry (CV) measurements were also carried out inorder to check the reversibility of lithium insertion in these materials as shown in Fig. 16. The cyclic voltammogram for Li2+x AuSn 2 cycled

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in the range 0 to 3 V at a scan rate of 10 mV s−1 shows reversibility of the anode reaction, but a fast irreversible fading of the lithium insertion with repeated cycling [109]. CV for Li1−x Ag 2 Sn was carried out at 5 mV s−1 . The voltage range for a nearly reversible insertion/deinsertion reaction lies between 1.6 and 2.1 V [108]. Li2+x AuSn 2 had an initial discharge capacity of 251 mA h g−1 falling to 60 mA h g−1 [109] after 10 cycles. Although compounds like Li2+x AuSn 2 are not anode materials with high lithium capacity, the unique structure of these intermetallics and the faster lithium diffusion than found in many other anode materials are interesting results which make it worthwhile to continue the investigations in this class of intermetallic compounds. Electrochemical data have also been collected for Li 13 Ag 5 Si6 . This silicide behaves fairly well as an anode material with a reversible specific capacity around 800 mA h g−1 [41]. The chemical diffusion coefficient of Li 2+x AuSn 2 is much higher than values reported for typical binary intermetallics, such as LiZn and LiCd (10−9 – 10−10 cm2 s−1 ) [110]. High values comparable to those of this work were found for the binary lithium stannides LiSn (10−8 cm2 s−1 ) [111] and Li4.4 Sn (10−7 cm2 s−1 ) [112]. The standard material Lix C 6 currently used in commercial lithium batteries is known to have chemical diffusion coefficients of lithium between 10−8 and 10−14 cm2 s−1 [113]. This broad range is typically explainable by the large variability of the particle size, orientation and defect density of graphite. Overall, the data about chemical diffusion coefficients in newer lithium anode materials is rather limited. Table 3 compiles values for some interesting systems published so far with emphasis on binary compounds. Finally we need to mention the optical properties of such Li x T y X z intermetallics. Besides the many silvery compounds, depending on the valence electron concentration, several compounds show intrinsic color. This has systematically been investigated in the Schuster group [123,124] many years ago. A very interesting example is the solid solution Li2−x Mg x PdSn, where the color changes from brass yellow via copper red to red violett with increasing x.

4. Conclusions Lithium-transition metal-tetrelides are an interesting class of compounds with respect to crystal chemistry and lithium mobility. Although many compounds with quite expensive noble metals have been studied, they are important model compounds in oder to understand the structure–property relationships. The rigid character of the two- or three-dimensional [T y X z ] polyanionic networks seems to be favorable for lithium incorporation in cavities and/or channels and for lithium mobility as well. The determined diffusion coefficients determined thus far on some of the new compounds studied are quite promising. For characterizing the ionic mobility in more detail, the portfolio of tem-

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perature dependent solid state NMR lineshape analysis has been incorporated into the standard characterization protocol of new intermetallic lithium compounds. Our investigations clearly showed that some of the key requirements for lithium alloy electrodes, i.e. enhanced temperature stability and good lithium mobility hold for several of the lithium-transition metal-tetrelides. Further phase analytical studies with the much cheeper and more abundant 3d transition metals are in progress in order to develop electrode materials with lower density.

Acknowledgement This work was supported by Deutsche Forschungsgemeinschaft through SFB 458 Ionenbewegung in Materialien mit ungeordneten Strukturen – Vom Elementarschritt zum makroskopischen Transport. The work reported here was the basis of the doctoral theses of Drs. Zh. Wu., P. Sreeraj, and T. Dinges.

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Silver(I)-(poly)chalcogenide Halides – Ion and Electron High Potentials By Tom Nilges1 , 2 , ∗, M. Bawohl2 , O. Osters2 , S. Lange2 , and J. Messel2 1 2

Institut für Anorganische und Analytische Chemie, Universität Münster, 48149 Münster, Germany Department Chemie, Synthese und Charakterisierung innovativer Materialien, Technische Universität München, Lichtenbergtrasse 4, 85747 Garching, Germany (Received July 20, 2010; accepted in revised form August 30, 2010)

Ion and Electron Conductors / Thermoelectrics / Ion Dynamics in Solids Materials with high dynamics in the solid state are of potential interest in semiconductor science and for a great variety of energy applications. In most of the cases the majority of physical properties of such compounds are directly related to their electronic structure. Optimization of properties is therefore correlated with direct or indirect control of this parameter. Compounds with highly mobile ions like the coinage metal cations and the heavy chalcogenide anions can easily be modified chemically or physically to fine tune their electronic structures. The range of adjustable properties lasts from ion conductivity, magneto resistance, thermoelectricity to a reversible redox-driven switch of semiconductivity. Today, the strong demand on clean energy production, the efficient energy transport and energy storage is a major goal for present and oncoming generations. Stable, mixed-conducting materials will play a major role in this process. The ongoing development of coinage metal chalcogenide halides during the last 50 years reflects the fundamental interest in this field. Many interesting and sometimes unexpected properties have been determined recently which substantiates the great potential of this class of materials. Especially the new class of coinage metal polychalcogenide halides is of potential interest due to their drastic modulations of physical properties driven by a fundamental tuning of its electronic structures. Thermopower and thermal diffusivity, two major properties to tune thermoelectricity can be varied in such a way that drastic changes can be addressed in very small temperature ranges close to room temperature. Herein we report on the recent progress of polychalcogenide and chalcogenide halides with the mobile d 10 ions Ag and Cu putting the main focus on silver compounds.

1. Introduction Today, the exponentially growing demand of energy forces us to develop new ideas and to optimize current processes in science and technology [1]. The * Corresponding author. E-mail: [email protected] Z. Phys. Chem. 224 (2010) 1505–1531 © by Oldenbourg Wissenschaftsverlag, München

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broad spectrum lasts from alternative routes for energy conversion, favorably without the production of green house gases, via the storage of energy in highly efficient batteries to a powerful way of energy transport on a small and large distance scale [2]. This energy challenge requires suitable and available materials in the future [3]. Mixed conducting materials can contribute to most of these fields in various ways. Mixed electron and ion conducting compounds are of general interest in applications where the conduction of matter and electronic charge carriers is needed [4]. During the past 25 years the understanding of electronic properties, defect chemistry and physics of mixed conductors were studied [5,6] leading to a fundamental understanding of the transport phenomena. The variety of compounds is not only restricted to crystalline materials, glasses or polymers are also of general interest. Crystalline inorganic semiconductors are widely used in photovoltaic applications as chalcopyrite type copper indium gallium diselenide cells (CIGS) [7] with high efficiencies [8]. In solar cell technology and thermoelectrics the optimization of mixed conductors is badly needed to improve the efficiency of the respective process [9]. Nevertheless mixed conductors play a key role in data storage applications as phase change materials [10] or as resistivity switching memories [11,12]. Quaternary telluride alloys containing coinage metals like Ag5 In5 Sb60 Te30 [13] are frequently used in DVD-RW applications and coinage metal chalcogenides like Cu2 S or Ag2 S [14,15] are successfully tested as solid state resistivity switching materials [16,17]. The binary silver chalcogenides are also promising materials with high magneto resistances [18–20] capable to design magnetoresistive sensors [21]. According to these aspects the question arises if ternary compounds, like for instance the coinage metal (poly)chalcogenide halides, do also have valuable properties in this sense. In this review we will focus on ternary coinage metal (poly)chalcogenide halide compounds combining two general properties: the high d 10 ion dynamic in the solid state and the tendency of heavy chalcogenide ions to perform primary and secondary interactions. Both types of ions are widely used in state of the art thermoelectric materials and quantized conductance atomic switches (QCAS) (e.g. Ag2 S/Ag [16]). Especially tellurium tends to form secondary interactions (e.g. in transition metal ditellurides [22]) which were examined in the element [23] and in a huge number of binary and multinary compounds. In such compounds linear or nonlinear chains (e.g. RbUSb0.33 Te6 nonlinear chain [24]), planar nets (e.g. LnTe3 [25] and LnSeTe2 [26]) and more complex structure units (e.g. ALnTe8 A = alkli metal, Ln = lanthanoid [27]) are realized. A nice overview is given by Papoian and Hoffmann reflecting the most prominent structure–property relations [28]. Thermoelectric materials are usually narrow band gap semiconductors featuring the right compromise between a high Seebeck coefficient, high electrical and low thermal conductivity. The most prominent criteria for enhanced thermoelectrics include covalent bonding, heavy constituent elements and complex

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crystal structures of low symmetry. Recent examples are the antimony containing derivatives of pavonite AgSb x Bi3−x S 5 showing a high figure of merit ZT = S 2 σT/κ of 2 at 800 K [29]. A second class of quaternary thermoelectrics is AgPb m SbTe2+m , the so called LAST-m compounds [30] with figure of merits ranging between 1 to 2 at 800 K. Successful efforts were made in the groups of Kanatzidis, DiSalvo and Kleinke to find new thermoelectric compounds containing alkaline earth metals. Examples are Ba(CM)2 Te2 (CM = coinage metal) [31,32], A 2 BaCu 8 Te10 (A = K, Rb, Cs) [33], Ba3 Cu 14−δ Te12 [34], Ba7 Au 2 Te14 [35] or Ba3 Cu 17−x (Se,Te) 11 [36]. The high mobility of the coinage metals often helps to reduce the thermal conductivity below the values of commercially thermoelectrics as shown for Ba3 Cu 14−δ Te12 [34]. In Ba(CM) 2 Te2 , a continuous increase of the silver content can significantly increase their Seebeck coefficient from 88 μV/K [31] (BaCu 2 Te2 ) to 995 μV/K (BaAg 2 Te2 ) [32]. On the other hand an opposite trend was found for the electrical conductivity in that case. New materials with new building principles and new electronic features, especially in bulk form for high Δ − T applications are one possible way out of this dilemma. The ongoing developments in coinage metal (poly)chalcogenide halides may point in the right direction and the polychalcogenide halides are promising candidates to contribute to this topic in the future.

2. Coinage metal chalcogen halides, chalcogenide halides and polychalcogenide halides Early developments in the field of coinage metal chalcogenide halides were dealing with the super ion conductors in the late 1960’s which cumulated in a plethora of different phases featuring highly disordered substructures and complex electronic behaviors. Short after the 1970’s the first coinage metal chalcogen halides emerged but their ion conductivities could not reach the values of the fast super ion conductors like RbAg 4 I 5 [37–39]. Unfortunately, the low concentration of potentially mobile charge carrying cations per formula unit lead to a relatively low ion conductivity compared to the chalcogenide halides. Nevertheless, the mixed conduction with reasonable high ion conductivity opened up new perspectives in electrochemistry and -physics. The latest substantial step in this development was the discovery of the first coinage metal polychalcogenide halide in 2007, bringing together two main features of the first two groups, the high silver mobility on the one hand and the covalent chalcogen interactions on the other. This new facet combines the two sets of substance classes according their chalcogenide substructures which contain isolated chalcogenide ions and covalently bonded chalcogen units. As a result, some intriguing properties emerged combining favorable properties of both classes of materials discussed before. A brief overview is given in Fig. 1.

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Fig. 1. Phase diagrams of Cu, Ag and Au chalcogenide halides excluding oxides and focussing only on the phase diagram triangle chalcogen Q−(CM)X−(CM)2 Q. All ternary coinage metal chalcogen(ide) halides outside this triangle are shown. The copper halide thiometallates are only quasi-ternary featuring complex thiometallate anions like [TeS2 ]− or [TeS3 ]2− .

Beside isolated Q2− ions (Q = chalcogenide) a selection of stable (helical chains and rings) and non-stable (linear chains) subunits of the covalentlybonded heavy chalcogens (e.g. tellurium [40]) have been synthesized with coinage metal and halide ions, dividing this substance class in three main subclasses: the coinage metal chalcogen halides, the chalcogenide halides and the polychalcogenide halides. Each class will be discussed in Sects. 2.1–2.3 putting a focus on new findings and recent developments of chalcogenide and polychalcogenide halides. In the following we will call the whole class of materials (CM)QX with CM = coinage metal, Q = chalcogen and X = halogen by neglecting the accurate composition. Ternary (CM)QX compounds exist only for a few number of compositions which are mainly localized on the borderlines of the phase diagram triangle chalcogen Q−(CM)X−(CM)2 Q. A strictly covalent-bonded chalcogen substructure is realized for Cu, Ag and Au compounds on the quasi-binary section (CM)X−Q. In this case the chalcogen substructure is characterized by chains and rings (Cu and Ag), closely related to the structures of the elements, and of [Q2 ]2− -dumbbells (Au). These compounds are discussed in Sect. 2.1 On the quasi-binary section (CM)X−(CM)2 Q only a few silver compounds exist. Surprisingly, no Cu or Au compounds have been reported so far on this section. In contrast to the chalcogen halides the chalcogenide halides accommodate isolated Q2− ions only. A brief overview is given in Sect. 2.2. During the past few years the triangle chalcogen Q−(CM)X−(CM)2 Q was explored and first the quaternary copper halide thiometallates (CuI)3 Cu 2 TeS 3 [41], (CuCl)Cu 2 TeS 3 [42] and (CuBr)Cu 1.2 TeS 2 [43] were reported. This class of compounds was reviewed recently [44] and is not in the focus of this review.

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Recently, the first ternary compounds were found within the phase diagram triangle (Q)−(CM)X−(CM)2 Q for CM = Ag. These materials will be discussed in Sect. 2.3. Ternary coinage metal compounds outside this triangle are rare and only reported for Au with positively charged chalcogen incorporated in complex ions like in AuCl3 XCl4 (X = S–Te) [45] or AuCl 3 SCl2 [46]. Those compounds are also not in the focus of this review. A brief summary of all existing (CM)QX compounds within the phase triangle (Q)−(CM)X−(CM)2 Q is given in Table 1.

2.1 Covalent chalcogen substructure – coinage metal chalcogen halides 2.1.1 Chalcogen chains, rings and dianions Helical chains, comparable with the ones realized in the heavy chalcogen elements, and Q2 dumbbells can be found in (CM)XQ and (CM)XQ2 (see Fig. 2). The helical chains can be found in the case of Cu and Ag compounds while a separation into Te2 dumbbells is present in the case of gold compounds. By far the highest number of compounds are realized with (CM) = Cu and Q = Se and Te (see Table 1) followed by four gold compounds and only one silver representative. Neutral chalcogen rings have been stabilized in (CM)XQ3 for the coinage metals CM = Cu and Ag, X = Br, I and Q = Se or Te. The formation and stabilization of Te6 rings in (AgI) 2 Te6 , an former unseen structure unit in the element chemistry of Te, represents a spectacular result after a kinetically controlled hydrothermal synthesis. A brief summary of compounds containing helical chalcogen chains and cyclic chalcogen rings can be found in the literature [47] and is therefore not in the focus of this report.

2.2 Ionic chalcogen substructure – coinage metal chalcogenide halides Compounds with purely ionic chalcogenide substructures featuring isolated Q2− ions only are exclusively known for the coinage metal silver. Less than a handful compounds are located on the quasi-binary section AgX to Ag2 Q. The ternary compounds on this section are mixed conductors combining the physical features of the ion conducting silver halides and the mixed conducting silver chalcogenides. Examples of fully characterized compounds are Ag3 QX (Q = S–Te, X = Br, I) [48–50], Ag5 Q2 X (Q = Te; X = Cl, Br) [51–53] and Ag 19 Te6 Br 7 [54]. The crystal structures of all compounds can be easily rationalised using a topological approach to describe the anion substructures (see Fig. 3, solid lines represent a distance close to or slightly above the sum of the respective van der Waals radii of the ions). According this formalism the anisotropy of the silver distribution and the anisotropic diffusion can be explained and under-

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Table 1. Ternary and quasi-ternary coinage metal chalcogen(ide) halides. The chalcogen substructure is classified in 1 = covalently bonded Q (chalcogen halide), 2 = isolated Q2− ions (chalcogenide halide), 3 = isolated Q2− and covalently bonded Q ((poly)chalcogenide halide), 4 = complex chalcogen ions; (CM) = coinage metal. Compound

Representatives

(CM)XQ

CuClTe CuCl 1−x Br x Te (x = 0–1) CuBrTe CuBr1−x I x Te (x = 0–1) CuITe Cu 1.035 ITe Cu 0.973 ITe AgITe AuITe AuSeBr

1 1 1 1 1 1

Helical Helical Helical Helical Helical Helical

1 1 1

Helical ∞1 [Te] chains Te dumbbell Se dumbbell (3.072(6) Å)

[106] [107] [108]

CuClSe2 CuBrSe2 CuClSeTe

1 1 1

[109] [110] [111]

CuBrSeTe

1

CuISeTe

1

CuClTe2 CuClS0.94 Te1.06 CuBrTe2 CuBrS 0.92 Te1.08 CuITe2 AuClTe2 AuITe2

1 1 1 1 1 1 1

Helical ∞1 [Se] chains Helical ∞1 [Se] chains Ordered helical ∞1 [SeTe] chains Ordered helical ∞1 [SeTe] chains Ordered helical ∞1 [SeTe] chains Helical ∞1 [Te] chains Helical ∞1 [STe] chains 1 ∞ [Te] chains Helical ∞1 [STe] chains 1 ∞ [Te] chains Te dumbbell Te dumbbell

[112] [113] [114] [113] [114] [115] [115]

(CM)XQ3

CuBrSe3 CuBrSe2.36 S0.64 CuISe3 CuISe2.6 S0.4 CuISe1.93 Te1.07 AgISe3 AgITe3

1 1 1 1 1 1 1

Se6 rings (Se/S)6 rings Se6 rings (Se/S)6 rings (Se/Te)6 rings Se6 rings Te6 rings

[116,117] [118] [119] [118] [118] [120,121] [120,121]

(CM)3 QX

Ag 3 SBr Ag 3 SBr 1−x I x Ag 3 SI Ag 3−x Cu x SBr (x > 0.25) Ag 3−2x Cd x SBr (x > 0.10) Ag 3 SBr 1−x Cl x

2 2 2 2

Isolated Isolated Isolated Isolated Isolated Isolated

S2− S2− S2− S2− S2− S2−

[122,123] [124] [122,123,125,126] [76] [77] [75]

Ag 5 Te2 Cl Ag 5 Te2 Cl 1−x Br x Ag 5 Te2 Br Ag 5 Te2−y Se y Cl (y = 0–0.7) Ag 5 Te2−z S z Cl (z = 0–0.3)

2

Isolated Isolated Isolated Isolated Isolated

Te2− Te2− Te2− Se2− and Te2− S2− and Te2−

[51,52,63] [70,71,127] [53] [64,127] [70,128]

(CM)XQ2

(CM)5 Q2 X

Chalcogen substructure

2

2 2

Chalcogen unit

1 ∞ 1 ∞ 1 ∞ 1 ∞ 1 ∞ 1 ∞

[Te] [Te] [Te] [Te] [Te] [Te]

chains chains chains chains chains chains

Lit.

[96,97] [98] [99–103] [98] [103] [104,105]

[111] [111]

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Silver(I)-(poly)chalcogenide Halides – Ion and Electron High Potentials Table 1. Continued. Compound

Representatives

(CM)19 Q6 X7

Ag 19 Te6 Br7 Ag 19 Te6 Br5.4 I1.6 Ag 19 Te5 SeBr7 Ag 19 Te6 Br7−x Cl x (x ≤ 1.25) Ag 19 Te6 Br7−y I y (y < 1.6) Ag 19 Te6−a Sa Br7 (a ≤ 0.4) Ag 19 Te6−b Seb Br7 (b > 1) Ag 19−z Cu z Te6 Br7 (z = 2)

2 2 2 2 2 2 2 2

Isolated Isolated Isolated Isolated Isolated Isolated Isolated Isolated

(CM)6 QX4

Ag 6 TeBr4

2

Unknown

(CM)10 Q4 X3

Chalcogen substructure

Ag 10 Te4 Br3 Ag 10 Te4 Br3−x Cl x (x = 1.6) Ag 10 Te4 Br3−y I y (y = 0.2) Ag 10 Te4−x S x Br3 (x = 0.2)

3 3 3 3

Ag 10 Te4−y Se y Br3 (y = 0.4)

3

(CM)23 Q12 X

Ag 23 Te12 Cl Ag 23 Te12 Br

(CM)QX5

Chalcogen unit

Te2− Te2− Se2− and Te2− Te2− Te2− S2− and Te2− Se2− and Te2− Te2−

Lit.

[54] [54] [54] [129] [129] [129] [129] [129] [78]

2−

Isolated Te and linear Te chains Isolated Te2− and linear Te chains Isolated Te2− and linear Te chains Isolated Te2− /S2− and linear Te/S chains Isolated Te2− /Se2− and linear Te/Se chains

[80–82] [88] [88] [89]

3 3

Isolated Te2− and linear Te chains Isolated Te2− and linear Te chains

[90] [90]

[SCl 2 ][AuCl 3 ]

4

Dichlorosulfonium(II)

[130]

(CM)QX7

[SCl 3 ][AuCl 4 ] [SeCl 3 ][AuCl 4 ] [TeCl 3 ][AuCl 4 ] [TeBr3 ][AuBr4 ](Br2 )0.5

4 4 4

Trichlorosulfonium(IV) Trichloroselenium(IV) Trichlorotellurium(IV) Tribromotellurium(IV)

[131] [132] [133] [134]

(CM)5 Q4 X3

(CuI)3 Cu 2 TeS3

4

Trithiotellurate(IV)

[41]

(CM)2.2 Q3 X

(CuBr)Cu 1.2 TeS2

4

Dithiotellurate(VI) Dithiotellurate(V)

[43]

(CM)3 Q4 X

(CuCl)Cu 2 TeS3

4

Trithiotellurate(IV)

[42,135]

[89]

stood. Silver is preferably located around the chalcogenide substructures or at intersections of those. The halide substructure acts as a silver ion separator between the chalcogenide subunits [53,54]. This feature is shown in the bottom part of Fig. 3. Polymorphism, caused by the substantial silver ion mobility, is a dominant feature of the silver(I) chalcogenide halides directly affecting the electrical properties of all compounds. The activation energies and conductivities change significantly in the different systems. Ag 3 SI is a trimorphic anti-perowskite type compound with phase transitions at 520 and 157 K [50,55]. The superionic α-phase, stable above 520 K is characterized by randomly distributed anions over the bcc anion substructure and can be quenched to room temperature (α* -phase). A remarkably high ionic conductivity of 1.3 × 10−1 S cm−1 results at 300 K after the quenching process [50]. In contrast the anion ordered β-phase has a two orders of magnitude

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Fig. 2. Chalcogen substructures of coinage metal chalcogen halides. Three different chalcogenide substructures are realised. Chalcogen (grey spheres), halogen (white spheres with cross), coinage metal (black spheres).

lower ionic conductivity at the same temperature. MD simulations and neutron diffraction experiments in the β-phase showed that silver tends to preferably occupy empty sites towards the face centred rather than the body centred sites which can be translated as an occupation of sites in the neighbourhood of chalcogenide rather than halide ions [56,57]. Band structure calculations substantiated this finding due to the much weaker coupling of the Ag-d bands with the halide- p than with the chalcogen- p bands [58]. This observation corresponds to the behaviour of the silver(I) chalcogenide halides shown in Fig. 4. Ag 3 SI was subject to intensive studies of the non-periodic local motion of the

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Fig. 3. A topological approach of the anion substructures for coinage metal chalcogenide halides. Chalcogenide positions are connected by black lines, halide ions are drawn as white spheres or connected with grey lines and silver ions are represented by transparent dark grey spheres (50% displacements). Lines represent approximately the sum of the van der Waals radius of neighboured ions.

cations and anions [59] and intensive experiments were performed to tune their physical properties by partial and full ion exchange. These results are discussed later on in this section. Ag3 SBr shows a phase transition at 123 K which separates the high temperature anti-perowskite type phase from the orthorhombic low temperature modification [60,61]. Ag3 SBr is non-stoichiometric with a stoichiometry parameter x of 0.04 in Ag 3−x SBr [62]. Ag 5 Te2 Cl is a trimorphic compound with phase transition temperatures of 240.9(2) and 334.2(5) K [63] and can be prepared in a conventional melting, quenching and annealing route. The high temperature α-phase is characterized by a quasi molten silver substructure, while ordering is present in both low temperature phases. It decomposes peritectically at 790(3) K [64]. Silver is preferably located at the cross sections of two interpenetrating honeycomb nets of telluride ions (nets are defined according the previously mentioned topology principle) and the halide ions form linear strands pointing in the same direction than the silver strands. During the α–β phase transition the tellurium nets become corrugated. Surprisingly, after the order/disorder phase transition from the β- to the α-phase silver tends to form a pronounced attractive interaction within their

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Fig. 4. Total (σtot ) and partial ionic (σion ) conductivity of silver(I) (poly)chalcogenide halides. Thick solid lines represent the ternary compounds, open spheres σion data.

substructure. This interaction positively influences the electronic structure in such a way that the thermoelectric performance (or the figure of merit) can be varied over almost two orders of magnitude within a very small temperature window of approximately 50 K (350 to 400 K) [65]. The occurrence of attractive interaction substantiates the general concept of low-dimensionality as a positive aspect for the improvement of thermoelectric performance. This concept will be briefly discussed in the next chapter dealing with polychalcogenide halides. Based on solvothermal and hydrothermal reactions as already applied to prepare coinage metal mercury chalcogenide halides like (CM)HgQX [66–68], CuHg2 S 2 I [68] or Ag 2 HgSI 2 [69] different kinetically controlled synthesis strategies were tested to stabilize new phases in the title system. Ag5 Te2 Br [53] was prepared using an ammono-thermal reaction route from a pre-reacted and quenched ideal mixture of the AgBr, Ag and Te. A synthesis strategy like in the case of Ag 5 Te2 Cl was not successful. At room temperature, Ag5 Te2 Br is isostructural to the chloride compound and polymorphism seems to be highly probable. The chalcogenide [64,70] and halide [70,71] substructure can be substituted over a wide range by other homologues of the same group. The partial exchange has a direct influence on the phase transition temperatures of the different polymorphs and the total electrical conductivity. A detailed analysis of the joint probability density functions (jpdf) between the different silver positions led to a detailed insight in the preferred diffusion pathways in these compounds. Two- and three-time correlation spectroscopies were performed for Ag 5 Te2 Cl [72] in order to understand the silver dynamics of this substance class in more detail.

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Ag 19 Te6 Br 7 is dimorphic showing an order–disorder phase transition at 235 K [54]. The crystal structure of the low temperature phase β-Ag 19 Te6 Br 7 is not known but impedance spectroscopic investigations substantiated an orderdisorder transition of the silver substructure featuring a small conductivity jump of one quarter of magnitude at this temperature. The compounds vary their conductivity with an increase in structural complexity and a decrease in dimensionality. There is no obvious trend according the structural topology and the related ion mobility of these compounds at least at temperatures around room temperature. A cubic symmetry, like in the case of the Ag3 SX compounds in their highly ion conducting state, seemed to be the most favorable structure for high conductivities according the number of vacant silver sites to be occupied during their movement. Unfortunately, the high symmetric and in most cases also high conducting polymorphs of superionic conductors are often stable only above room temperature (e.g. AgI, Ag 2 Q, Ag3 SBr, Ag 3 SI). Therefore it is not a surprise, that Ag 19 Te6 Br 7 , being in the disordered state at room temperature, exceeds the conductivity of β-Ag3 SBr (3.2 × 10−3 Ω−1 cm−1 , 300 K) and β-Ag 3 SI (10 × 10−3 Ω−1 cm−1 , 300 K) by two orders of magnitude. The exchange of anions or cations in the silver(I) polychalcogenide halides leads to a substantial change in the physical properties of the compounds. In some cases a full exchange of either the anions or cations was possible with retention of the crystal structure (e.g. Ag 5 Te2 Cl to Ag5 Te2 Br) but most of the compounds only show the tendency for a partial anion or cation substitution. In general the ions can be exchanged up to a certain extend before the structure collapses and the binary phases become more stable. Table 1 summarizes the substituted phases also stating their range of stability. The halide ion in polymorphic Ag 3 SI and Ag3 SBr can be partially substituted by other halides [73–75]. This substitution slightly affects the βAg 3 SX conductivity and causes a certain decrease of the γ –β phase transition. Ag3 SBr can be substituted in the halide substructure by iodide and chloride. A partial chloride substitution is possible up to a substitution grade of 50% [75]. The order–disorder phase transition of the ternary phase is raised slightly above 123 K but non-reacted binary phases were present in all studies. A partial exchange of silver by copper and cadmium was also performed [76,77] but no quantitative values were given for the conductivity or activation energy change. The transition temperature did not change during this exchange and most of the examinations were performed using impure samples. A significantly different structural behaviour was found for the Ag19 Q 6 X 7 phases. The Ag 19 Te6 Br 7 structure type is stable up to Ag 19 Te6 Br 7−x Cl x (x ≤ 1.25), Ag 19 Te6 Br 7−y I y (y ≤ 1.25), Ag 19 Te6−a S a Br 7 (a ≤ 0.4), Ag 19 Te6−b Seb Br 7 (b > 1) and Ag 19−c Cu c Te6 Br 7 (c = 2). In the case of the iodide and selenide substitution completely different compounds with new crystal structures were found at higher grades of substitution. The new compounds are stable in the

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Fig. 5. Structure variations dependent on the grade of substitution in Ag19 Q6 X7 phases. The anion substructure can be illustrated by nets where lines represent the sum of the van der Waals radii of the respective ions. Dependent on the grade of substitution the anion substructure is arranged by different sets of structure units. The halide substructure acts as a separator for the silver ions with a pronounced gap of the silver distribution in each compound (the Ag substructure of Ag19 Te6 Br 7 itself is shown in Fig. 3). Black spheres represent chalcogenide, white octands halide and grey spheres silver ions. Displacement parameters are shown at 90% probability.

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compositional ranges of y = 1.25–1.75 in Ag 19 Te6 Br 7−y I y and b = 1–1.25 in Ag 19 Te6−b Seb Br 7 . Two crystal structure determinations were performed for Ag 19 Te6 Br 5.4 I 1.6 and Ag 19 Te5 SeBr 7 [54]. Obviously the variability of the Ag 19 Te6 Br 7 structure type is restricted to the grade of substitution and it can be tuned by a partial cation and anion exchange. The topological formalism for the description of the anion substructure is a powerful tool to illustrate the structural features of the different compounds (Fig. 5). Different subunits in the anion substructures are coordinated in a systematic way to form linear and corrugated subnets. While a linear arrangement of a honeycomb chalcogenide(Q)-net interpenetrated by Q 2 dumbbells, a Kagom´e halide(X)-net and a corrugated Q-net centred by X ions are alternately stacked in Ag 19 Te6 Br 7 , a reorientation and corrugation of this units takes place in the quaternary compounds. According this formalism it also becomes obvious that the halide-nets act as a separator for the silver ions between the remaining anion substructures. A full substitution to the ternary end members of the solid solutions were not observed neither for cation nor anion exchange applying the high temperature melting and annealing route. There are no substantial differences in the conductivities of the ternary and quaternary compounds (Fig. 4). Two postulated compounds are mentioned in literature where no detailed structure information is available. Ag 6 TeBr 4 and Ag 3 TeBr were first reported as peritectic compounds during an evaluation of the quasi-binary section AgBrAg 2 Te [78]. Later on Ag 6 TeBr 4 could not be reproduced after a deeper investigation of this section of the phase diagram [51] nor by different groups working in this field since then. Blachnik postulated the occurrence of Ag 3 TeBr, crystallizing in a hexagonal metric without fundamental structure analysis. Interestingly, an electrochemical characterisation was performed later on a compound having the same postulated composition [79]. We have recently shown that Ag 3 TeBr cannot be prepared by the thermodynamically controlled reactions stated in the afore mentioned articles and also more kinetically controlled mineralisator reactions or solvothermal synthesis did not lead to success [54]. Ag 3 TeBr seemed to be Ag 19 Te6 Br 7 in all cases.

2.3 Covalent and ionic chalcogen substructures – coinage metal polychalcogenide halides Ag 10 Te4 Br 3 is the first representative of a silver(I) polychalcogenide halide with covalently bonded [Te 2 ]2− units and isolated Te2− ions. It represents a new class of materials located between coinage metal chalcogen halides, characterized by covalently bonded chalcogen structure motives, on the one hand and chalcogenide halides, consisting of isolated chalcogenide ions, on the other. A brief summary of selected physical properties is given in Table 3. In Fig. 6a one can see the fully ordered room temperature γ -structure of Ag 10 Te4 Br 3 illustrated by a topological approach.

Transition temperatures/K

157 (γ –β) 520 (β–α) α phase stable at room temperature by quenching of α

126–133 (γ –β) 508 (β–α)

241(1) (γ –β) 334(1) (β–α)

n. m.

235(1) (β–α)

Compound

Ag3 SI

Ag3 SBr

Ag5 Te2 Cl

Ag5 Te2 Br

Ag19 Te6 Br 7

n. m.

n. m.

n. m.

703

–

Decomposion/ melting point/K

n. m. 0.32 (β) 0.19 (α)

2.1 × 10 (β, 201 K) 9.3 × 10−4 (α, 263 K)

0.44 (β) 0.14 (α)

1.5 × 10−3 (β, 323 K) 4.3 × 10−1 (α, 469 K) −5

–

> 10−5 (γ , > 126 K) 3.2 × 10−3 (β, 300 K)

n. m.

–

Activation energy/eV

10−3 (β, 300 K) 1.5 (α, > 520 K) 1.3 × 10−1 (α , 300 K)

Conductivity (phase, T )/S cm−1

Table 2. Selected thermoanalytic and conductivity data of silver(I) chalcogen(ide) halides. n. m. = not measured.

–

[53]

[63]

[122,136–138]

[50,57,61]

Lit.

86 T. Nilges et al.

Transition temperatures/K

none (100 to decomp. T )

none (100 to decomp. T )

290 (δ–γ ) 317 (γ –β) 355–410 (β–α)

n. m.

Compound

Ag23 Te12 Cl

Ag23 Te12 Br

Ag10 Te4 Br 3

Ag20 Te10 X2 n. m.

n. m.

690

734

Decomposion/ melting point/K

n.m. (δ) 0.60 (γ ) 0.30 (β) 0.15 (α)

1.5 × 10−4 (δ, 223 K) 3.1 × 10−2 (γ , 314 K) 1.0 × 10−1 (β, 350 K) n. m. (α)

n. m.

n. m.

1.2 × 10−5 (312 K) 1.8 × 10−2 (447 K)

n. m.

n. m.

Activation energy/eV

n. m.

Conductivity (phase, T )/S cm−1

Table 3. Selected thermoanalytic and conductivity data of silver(I) polychalcogenide halides. n. m. = not measured.

[139]

[81]

[90]

[90]

Lit.

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Fig. 6. (a) Crystal structure section of γ -Ag10 Te4 Br 3 . All positions are fully occupied and all Te4 units are linearly coordinated by a silver ion within the 63 Te net. Chalcogen (black spheres), halogen (white spheres), silver (grey, non-connected spheres). (b) Peierls distortion mechanism after the removal of the coordinating Ag ions. Crystal structure sections of the γ -, β- and α-Ag10 Te4 Br 3 structure type (chalcogen (black spheres), halogen (white spheres), silver (transparent, non-connected spheres)).

Ag 10 Te4 Br 3 is a mixed conductor [80] showing three phase transitions in a very narrow temperature window of about 100 K [81]. The electronic conductivity extends the ionic conductivity by one order of magnitude. Beside an order–disorder transition of the silver at 317 K the most prominent feature is a reversible and temperature driven redox-switch starting at 355 K. Respon-

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Fig. 7. Seebeck coefficient and DSC data for Ag10 Te4 Br 3 [82]. A pronounced peak in the Seebeck coefficient with two times a change in the sign of the value was observed for Ag10 Te4 Br 3 . This feature is consistent with the broad phase transition between 355 and 410 K for the β–α transition found during DSC and phase analyses.

sible is a mobile polyanionic tellurium strand which performs a Peierls-type distortion towards a linear arrangement and therefore offers band gap tuning during this process. The band gap width is temporary reduced during this process enabling the p-type material to perform n-type conduction. This switch causes an unseen thermopower drop of 1400 μV K−1 at 390 K which is accompanied by an also reversible change of its semiconducting properties from p- via n- back to p-type conduction [82]. The different stages of disorder in the tellurium substructure are shown in Fig. 6. At the phase transition from β-Ag 10 Te4 Br 3 to α-Ag 10 Te4 Br 3 the Seebeck coefficient changes sign from +310 μV K−1 in the β-phase to −940 μV K−1 during the transition. On a further increase of temperature the Seebeck coefficient again changes sign to values around +540 μV K−1 in the α-phase. Solid state Te-NMR proved the occurrence of a Peierls-type like Te-mobility during this transition which is responsible for a pronounced modulation of the electronic structure throughout the phase transition. At the point of the β–α transition two different substructures are mobile (Ag and the partially covalent-bonded Te) which represents an absolutely rare case for a solid material. Such a high mobility has a certain impact on the thermoelectric properties. At the point of maximum thermopower at 390 K a figure of merit ZT −1 with of 1.7 × 10−2 results (the figure of merit Z is defined as Z = σS 2 κ tot κ tot = total thermal conductivity). The Seebeck coefficient in relation to the differential scanning calorimetry (DSC) curve is shown in Fig. 7. The thermal diffusivity is favorably low and ranges from 1.8 × 10−7 m2 s−1 at 323 K to 2.3 × 10−7 m2 s−1 at 473 K. These values are comparable with non-metallic compounds like polymers and glasses. Thermal conductivities of Ag 10 Te4 Br 3

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are found to be κ tot = 0.27 to 0.43 W m−1 K−1 in the temperature range of 250 to 390 K [82]. While isolators [83] and semiconductors can perform isolator–metal transitions or p–n (n–p) switches, such behaviour is unique. There is almost no significant volume jump during these transitions which makes this system of potential interest to room temperature applications using the high ion mobility and/or its switching properties [84]. Due to the reasonable thermopower drop at relatively low temperatures there is a chance to optimize this compound for thermoelectric applications. At present stage the thermal conductivity reaches the values of the best micro-structured thermoelectric materials like quantumdot and superlattice systems [30,85–87]. The unique combination of two independent and mobile substructures seemed to be a new and additional mechanism to significantly reduce the thermal conductivity in a crystalline solid. Some major advantages are the insensitivity to light or moisture and the fact that Ag 10 Te4 Br 3 can be produced in reasonable amounts using conventional solid state reactions. β-Ag 10 Te4 Br 3 exceeds the conductivity of Ag3 SBr by one order of magnitude. The high temperature α-phase almost reaches the well known super ion conductor RbAg 4 I 5 and stays only one order of magnitude below this prominent compound. Chemical tuning of properties can be achieved by partial substitution of the anions. The important phase transition at 355 to 410 K with a peak maximum at about 385 K can be shifted in both directions. A chloride substitution leads to an increase of this maximum to 396 K (Ag10 Te4 Br 1.8 Cl1.2 ), while in the case of iodide (Ag 10 Te4 Br 2.8 I 0.2 ) the opposite trend was found featuring a reduction to 376 K [88]. Even more pronounced is the shift of this effect on the chalcogenide side [89]. The introduction of the lighter homologues of tellurium reduces the broad β–α transition to 350 K (Ag10 Te3.8 S 0.2 Br 3 ) and to 357 K (Ag 10 Te3.6 S 0.4 Br 3 ) in maximum. The influence of anion substitution on the Seebeck coefficient is shown in Figs. 8 and 9. A substitution within the chalcogenide substructure can significantly reduce the ability for an effective Peierls distortion or even can completely destroy such a process. A different situation can be observed if the chalcogenide substructure kept untouched and the halide substructure is modified. The biggest change in the thermopower can be found in the substitution with chlorine. First experiments point towards significant increase of the thermopower drop from 1400 μV K−1 for the ternary compound to values around 1800 μV K−1 for Ag10 Te4 Br 2.6 Cl0.4 . Concrete details will be reported soon. If bromide is partially exchanged by the heavier homologue iodide the thermopower drop is reduced but the pnp-switch (change in sign of the thermopower; see Fig. 9 bottom part) remains intact. This is in contrast to the chalcogenide substituted phases where the pnp-switch is completely suppressed. It has to be stated at this point that each minimum of the Seebeck coefficient drop is consistent with the maximum of the β–α phase transition ef-

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Fig. 8. Seebeck measurements of selected Ag10 Q4 X3 compounds after chalcogen substitution. Data of Ag10 Te4 Br 3 are given for comparison.

fect observed by DSC measurements. The drop is therefore directly related to the mobility in the silver and tellurium substructure driving the phase transition at this point. Ag 23 Te12 X with X = Cl, Br are two polychalcogenide halides also containing a linear telluride strand [90,91]. The crystal structure of this phase is shown in Fig. 10. Surprisingly, the lower content of halide in this systems compared

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Fig. 9. Seebeck measurements of selected Ag10 Q4 X3 compounds after halogen substitution. Data of Ag10 Te4 Br 3 are given for comparison.

with the Ag 10 Q 4 X 3 samples changed the ion dynamics drastically. In the measured temperature range of 100 K to their peritectic decomposition temperatures at 690 K (Br) and 734 K (Cl) the silver did not show any ordering phenomena and kept disordered. The partial ion conductivity of Ag23 Te12 Br is two orders of magnitude lower than the comparable ion conductivity of Ag 10 Te4 Br 3 . If these compounds can chase the superior properties of the Ag 10 Q 4 X 3 ones remains an open question at the moment. Experiments in this direction are currently underway.

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Fig. 10. Crystal structure of Ag23 Te12 X (X = Cl, Br). Tellurium forms puckered 63 -nets which interpenetrate each other. Strands of structurally frustrated Te4 units and halide ions run parallel to the c axis. The silver atoms are distributed over a large number of partially occupied sites.

More recently, the newest example of a silver(I) polychalcogenide halide, Ag 20 Te10 BrI [92], was reported also featuring Peierls-distorted Te strands and a highly mobile silver substructure. While the covalently bonded anion substructure also contains Peierls-like distorted Te chains the remaining noncovalently bonded anion substructure is characterized by another variation of corrugated and interpenetrated honeycomb nets. In contrast to the latter example chalcogenide and halide ions are both forming the anion net. Compared with the previous compounds in this system the anion substructure is slightly different but it can also be described using a topological approach [93] by connecting anions at distances close to the van der Waals contacts. Corrugated and interpenetrating honeycomb nets formed by telluride and halide ions are forming the anion substructure and silver is distributed statistically within these units. The phonon scattering is that optimized in the compound that the thermal diffusivity reaches values around 1.2 × 10−7 m2 s−1 . The mentioned topologic approach can be generalized to other classes of compounds (e.g. sodium polytellurides) [92], to minerals like stuztite (Ag 4.53 Te3 ) [94] and stutzite analogues [95].

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Fig. 11. Crystal structure section of Ag20 Te10 BrI. Telluride and halide ions (occupancy Br : I = 0.5 : 0.5) form two different types of puckered 63 -nets which interpenetrate each other. The halide ions occupy one defined position within these nets. Strands of structurally frustrated Te4 units and halide ions run parallel to the c-axis. The silver atoms are distributed over a large number of partially occupied sites (chalcogen (black spheres), halide (white spheres with cross), silver (grey spheres)).

3. Concluding remarks Recent progress in synthesis strategies and a systematic exploration of the ternary phase fields coinage metal (Cu, Ag, Au) chalcogen and halogen led to the discovery of new coinage metal chalcogenide halides and the new class of materials called coinage metal polychalcogenide halides. All compounds are characterized by a pronounced complexity in their structures demanding the development of a topologic description for an illustrative and simple access to the substances. Based on this topology concept a prediction and synthesis planning of new phases is possible which still leads to the discovery of new phases in these phase fields. All compounds show a high coinage metal mobility resulting in high ion conductivities close to the best known ion conductors today. This mobility is the origin for an intriguing polymorphism and a rich structure chemistry including severe crystallographic problems like multi domain twinning or nonharmonic behaviour of the coinage metal distribution. Activation energies for the ion transport through the solid are in the order of 0.1–0.4 eV putting them into the class of super ion conductors. Within the new class of materials called the silver(I) polychalcogenide halides a second unexpected and powerful physical property emerged, in addition to the previously described ones, which influences the structural and electronic features drastically. The occurrence of a mobile Peierls-like distorted partially covalent-bonded chalcogen chain within the anion substructure of these compounds has a direct influence on the electronic structure and the

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electric and thermoelectric properties. A combination of a high coinage metal ion mobility and the occurrence of low dimensional covalent interactions in parts of the structure creates a reversible and tuneable modulation of the electronic properties. Important thermoelectric features like the thermal diffusivity and the thermopower can now be switched and tuned efficiently. This tuning led to extremely low thermal diffusivities and a former unseen drop of the thermopower of about 1800 μV K−1 within very small temperature windows. Exactly this interplay defines a new concept for the design and improvement of next generation thermoelectric materials and its performance beside the well established ones like quantum well formation or nano structuring via phase separation. While the latter two aspects are focused on the optimization of the electrical conductivity or the thermal conductivity the modulation of the thermopower opens up a new aspect for the ongoing improvement of the thermoelectric figure of merit. The modulation of the electronic structure directly affects the semiconducting properties of the polychalcogenide halides. During a defined phase transition a reversible pnp-switch has been observed which represents the first example of such a process in a single compound. A single compound pnpeffect opens up new perspectives for data storage and computer technology after a further improvement of the electronic characteristics and an implementation in technological processes.

Acknowledgement This work was financed by the German science foundation DFG via the collaborative research centre SFB 458 during the years 2006 to 2009 as the subproject A5. We thank our cooperation partners within the SFB 458 Prof. Dr. H.-D. Wiemhöfer (project A2), Prof. Dr. H. Eckert (project B7), Dr. H.-W. Meyer and PD Dr. D. Wilmer (project B2), Dr. J. Vannahme, Dr. J.-M. Deckwart, Dipl.-Chem. M. Janssen, Wilma Pröbsting and our national and international partners Prof. Dr. P. Schmidt (Lausitz University, Germany), PD Dr. R. Weihrich (University of Regensburg, Germany), Dr. B. Chevalier, Dr. R. Decourt and Prof. J.-L. Bobet (Institute de Chimie de la Matière Condens´ee de Bordeaux (ICMCB), France) for their fruitful and continuous help and substantial scientific contributions during the past five years. The generosity of Prof. Dr. H.-L. Keller donating us two impedance analyzers and additional equipment after his retirement is gratefully acknowledged.

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115. H. M. Haendler, D. Mootz, A. Rabenau, and G. Rosenstein, J. Solid State Chem. 10 (1974) 175. 116. H. M. Haendler and P. M. Carkner, J. Solid State Chem. 29 (1979) 35. 117. T. Sakuma, T. Kaneko, T. Kurita, and H. Takahashi, J. Phys. Soc. Jpn. 60 (1991) 1608. 118. A. Pfitzner and S. Zimmerer, Z. Kristallogr. 212 (1997) 203. 119. W. Milius and A. Rabenau, Mater. Res. Bull. 22 (1987) 1493. 120. M. Wagener, H.-J. Deiseroth, B. Engelen, C. Reiner, and S. T. Kong, Z. Anorg. Allg. Chem. 630 (2004) 1765. 121. H.-J. Deiseroth, M. Wagener, and E. Neumann, Eur. J. Inorg. Chem. 24 (2004) 4755. 122. B. Reuter and K. Hardel, Angew. Chem. 72 (1960) 138. 123. B. Reuter and K. Hardel, Z. Anorg. Allg. Chem. 340 (1965) 168. 124. X. Lian, H. Honda, K. Basar, S. Siagian, T. Sakuma, H. Takahashi, T. Baer, H. Kawaji, and T. Atake, J. Phys. Soc. Jpn. 76(1–4) (2007) 114603. 125. E. Perenthaler, H. Schulz, and H. U. Beyeler, Naturwissenschaften 48 (1981) 161. 126. J. J. Didisheim, R. K. McMullan, and B. J. Wuensch, Solid State Ionics 18/19 (1986) 1150. 127. T. Nilges and S. Lange, Z. Anorg. Allg. Chem. 630 (2004) 1749. 128. T. Nilges and S. Lange, Z. Kristallogr. 22(Suppl.) (2005) 173. 129. T. Nilges and J. Messel, Z. Anorg. Allg. Chem. 634 (2008) 2185. 130. V. B. Rybakov, L. A. Aslanov, S. V. Volkov, Z. A. Fokina, and V. F. Lapko, Zh. Neorg. Khim. 36 (1991) 2534. 131. P. G. Jones, D. Jentsch, and E. Schwarzmann, Acta Crystallogr. C44 (1988) 210. 132. P. G. Jones, R. Schelbach, and E. Schwarzmann, Acta Crystallogr. C43 (1987) 607. 133. P. G. Jones, D. Jentsch, and E. Schwarzmann, Z. Naturforsch. 41b (1986) 1483. 134. C. Freire Erdbruegger, D. Jentsch, P. G. Jones, and E. Schwarzmann, Z. Naturforsch. 42b (1987), 1553. 135. A. Pfitzner, S. Reiser, T. Nilges, and W. Kockelmann, J. Solid State Chem. 147 (1999) 170. 136. T. Sakuma and S. Hoshino, J. Phys. Soc. Jpn. 49 (1980) 678. 137. J. Kawamura and M. Shimoji, Solid State Ionics 3/4 (1981) 41. 138. G. Staikov, M. Nold, W. J. Lorenz, A. Froese, R. Speck, W. Wiesbeck, M. W. Breiter, Solid State Ionics 93 (1997) 85. 139. O. Osters, Dilpoma thesis, University of Münster (2009).





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NMR Multi-Time Correlation Functions of Ion Dynamics in Solids By Christian Brinkmann1 , Sandra Faske1 , Barbara Koch1 , and Michael Vogel2 , ∗ 1 2

Institut für Physikalische Chemie, Westfälische Wilhelms-Universität Münster, Corrensstraße 30, 48149 Münster, Germany Institut für Festkörperphysik, Technische Universität Darmstadt, Hochschulstraße 6, 64289 Darmstadt, Germany (Received June 25, 2010; accepted in revised form September 2, 2010)

Solid-State Electrolytes / Ion Dynamics / Nuclear Magnetic Resonance / Multi-Time Correlation Functions / Dynamical Heterogeneities We show that NMR multi-time correlation functions provide interesting new insights into the nature of lithium and silver ion dynamics in solids. For solid ion conductors, they usually probe the elementary jumps of the long-range charge transport. NMR two-time correlation functions yield rates and activation energies of the ionic hopping motion. Moreover, they reveal that the ionic relaxation exhibits a strong nonexponentiality in the studied crystals and glasses. NMR three-time correlation functions enable quantitative determination of the origin of the nonexponentiality. For various solid-state electrolytes, the analysis shows that pronounced dynamical heterogeneities govern the ionic hopping motion. If dynamical heterogeneities exist, NMR four-time correlation functions allow one to measure the time scale of exchange processes within the rate distributions. For silver ion dynamics in a crystalline and a glassy material, the results indicate that initially slow ions exhibit random new rates from broad rate distributions after very few jump events.

1. Introduction Despite of an enormous technological importance of solid-state electrolytes, a generally accepted theory of ion transport in solids is still lacking, limiting knowledge-based design of new materials. A development of a fundamental understanding of the transport mechanisms requires a detailed characterization of the ionic hopping motion in solids on a microscopic level. Measurement of NMR multi-time correlation functions (MT-CF) proved very useful for such characterization. * Corresponding author. E-mail: [email protected] Z. Phys. Chem. 224 (2010) 1535–1553 © by Oldenbourg Wissenschaftsverlag, München

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Solid-state NMR studies of ion dynamics have a long-standing history [1]. Traditionally, spin-lattice relaxation and line-shape analyses were used to determine correlation times τ of the ionic hopping motion in crystals and glasses [2–8]. These NMR approaches often yield activation energies that deviate from the activation energy E dc describing the temperature dependence of the dc conductivity [1]. Specifically, due to the complex nature of the ionic hopping motion, the relaxation of the ions is often nonexponential rather than exponential so that NMR spin-lattice relaxation and line-shape analyses can suffer from a lack of knowledge about the exact shape of the correlation functions. In this contribution, we show that NMR stimulated-echo analysis enables straightforward measurement of correlation functions of ion dynamics in solid electrolytes. This approach allows one to correlate the positions of an ion, more precisely, the resonance frequencies of the nucleus associated with the ion, at two, three, or four points in time, enabling detailed insights into the nature of the ionic jump motion [1,9]. Here, we present results from 6 Li, 7 Li, and 109 Ag NMR stimulated-echo studies on various lithium- or silver-ion conducting crystals and glasses to demonstrate this potential.

2. NMR basics 2.1 NMR interactions In NMR studies on solid-state electrolytes, properties of the local environment determine the resonance frequency ω of a nucleus associated with a mobile ion. In general, the environments differ at different positions in the solid matrix and, hence, ionic translational motion results in a time-dependence of the resonance frequency. Therefore, information about ion transport in solids is available when we correlate the frequencies ω(t) at different points in time. The existence of well-defined ionic sites enables straightforward interpretation of NMR data for solid electrolytes. At sufficiently low temperatures, the jump motion between the sites occurs on a much longer time scale than the vibrational motion within the sites, which merely leads to a preaveraging of the NMR interactions. Then, it is often possible to ascribe an inherent resonance frequency to each ionic site so that any time dependence of ω can be traced back to a jump of an ion to a distinguishable site, see below. Different NMR interactions dominate for diverse probe nuclei. Here, the chemical shift (CS), the dipole–dipole (DD), and the quadrupole (QP) interactions are important. The CS interaction describes the shielding of the external magnetic field at the nuclear site due to neighboring electrons, while the DD interaction is a consequence of the dipolar couplings between various nuclear magnetic moments. Nuclei with spin I ≥ 1 exhibit an electric quadrupole moment that couples to a local electric field gradient, leading to the QP interaction. In solids, these interactions are described by second rank tensors. Then, decom-

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position into isotropic and anisotropic contributions leads to [12]: ω(θ, φ) = ωiso + ωaniso (θ, φ)

(1)

The isotropic contribution ωiso exists for the CS interaction, while it is absent for the DD and the QP interactions. Transforming into the principal axis system of the relevant interaction tensor, the anisotropic contribution can be written as [12]: ωaniso (θ, φ) =

δ [3 cos2 (θ) − 1 − η sin2 (θ) cos(2φ)] . 2

(2)

Here, θ and φ are the polar coordinates of the applied magnetic field B0 in the principal axis system of the interaction tensor. Furthermore, the anisotropy parameter δ and the asymmetry parameter η are determined by the principal values of the interaction tensor [12]. While the former characterizes the strength of the interaction, the latter quantifies deviations from axial symmetry. For 109 Ag (I = 1/2), CS interactions are much stronger than DD interactions, which are negligible because of the low gyromagnetic ratio of this nucleus. Thus, when the matrix is rigid, inherent resonance frequencies ω i can be ascribed to the silver ionic sites. Then, the frequency of a nucleus changes when the corresponding ion jumps to a different site and the frequency is restored when the ion jumps back to the initial site at a later time. These effects enable straightforward interpretation of 109 Ag NMR MT-CF, see below. For 6 Li (I = 1) and 7 Li (I = 3/2), the QP interaction dominates, but Li−Li DD interactions can affect the experimental results. Specifically, in addition to site-specific contributions ωi , there are contributions ωd to the resonance frequency, which reflect the DD interactions with the magnetic moments of mobile lithium nuclei in the vicinity and, hence, depend on the occupancies of neighboring lithium sites. Then, the resonance frequency of a nucleus associated with a static lithium ion can change as a consequence of a redistribution of lithium ions at neighboring sites and the resonance frequency is not necessarily identical during consecutive visits of a site. In general, these effects have to be considered when interpreting Li NMR MT-CF, see below.

2.2 NMR multi-time correlation functions 2.2.1 Measurement Measurement of NMR MT-CF was developed to investigate molecular dynamics in viscous liquids and melts [12,13]. In these experiments, appropriate pulses divide the experimental time in short evolution times tp  τ, during which the resonance frequencies of the nuclei are probed, and long mixing times tm ≈ τ, during which the resonance frequencies can change as a consequence of molecular dynamics. Specifically, the basic building block of these

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experiments is the stimulated-echo pulse sequence (pulse − tp − pulse − tm − pulse − tp ). Recently, NMR MT-CF proved very useful to study ion dynamics in solid and polymer electrolytes [1,9–11]. When the height of the echo after the stimulated-echo pulse sequence is measured for various tm and constant tp , twotime correlation functions (2T-CF) can be obtained. In the following, we first focus on nuclei with spin I ≤ 1, including our cases of 6 Li and 109 Ag. For such nuclei, observation of the stimulated-echo decay provides access to the correlation functions [12,13]:      F2cc (tm ; tp ) ∝ cos ω1 tp cos ω2 tp (3)      ss (4) F2 (tm ; tp ) ∝ sin ω1 tp sin ω2 tp Here, ω1 and ω2 are the resonance frequencies of a nucleus during the two evolution times, i.e., at two points in time separated by the mixing time tm . The brackets . . . denote ensemble averages and, throughout this contribution, the nomenclature F(x; y) is used to indicate that the function F(x) parametrically depends on the value of y. When a motion is characterized by nonexponential correlation functions, further insights are available from three-time (3T-CF) and four-time correlation functions (4T-CF). Such higher-order correlation functions can be obtained when appropriate pulse sequences are employed to correlate the resonance frequencies ω1 , ω2 , ω3 , and ω4 during four evolution times separated by three mixing times tm1 , tm2 , and tm3 . These pulse sequences can be regarded as two consecutive stimulated-echo experiments with mixing times tm1 and tm3 , respectively, which are separated by a mixing time tm2 . In principle, variation of all these mixing times for a constant value of tp allows one to map out a full data set F 4 (tm1 , tm2 , tm3 ). However, the most important questions concerning nonexponential relaxation can be addressed without measuring the full data set. A 3T-CF, which allows one to determine the origin of nonexponential relaxation, is obtained when tm3 is varied for constant tm1 ≡ tf ≈ τ and tm2 → 0 (ω2 = ω3 ≡ ω23 ) [1,13]: F 4 (tf , 0, tm3 ) ≡ F3 (tm3 ; tf ) ∝  cos[(ω23 − ω1 )tp ] cos(ω23 tp ) cos(ω4 tp )

(5)

When a distribution of correlation times G(log τ) exists, the time scale of exchange processes within this distribution can be measured by varying tm2 for tm1 = tm3 ≡ tf so as to record the 4T-CF F 4 (tf , tm2 , tf ) ≡ F4 (tm2 ; tf ) ∝  cos[(ω2 − ω1 )tp ] cos(ω3 tp ) cos(ω4 tp )

(6)

In Eqs. (5) and (6), mixing times that act as dynamical filters are denoted as filter times tf , see Sect. 2.2.2. In 7 Li NMR, I = 3/2 leads to more complex spin dynamics so that more elaborate pulse sequences can be required to measure MT-CF [1,14]. For the

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present contribution, it is sufficient to note that the stimulated-echo pulse sequence provides straightforward access to F2ss for I = 3/2 nuclei, too, see e.g. [1,15–17]. All 7 Li and 109 Ag NMR experiments were carried out on a Bruker DSX 500 spectrometer at Larmor frequencies of 2π × 194.3 MHz and 2π × 23.3 MHz, respectively. The 6 Li NMR measurements on LiPO 3 glass were performed using a Bruker CXP 300 spectrometer and a Larmor frequency of 2π × 44.2 MHz. The 6 Li NMR data of Li0.5 Ag 0.5 PO 3 glass were acquired in the laboratory of R. Böhmer at the Technical University Dortmund utilizing a homebuilt spectrometer at a Larmor frequency of 2π × 46.1 MHz. Further experimental details can be found in Refs. [16,18–20]. Throughout this paper, we show normalized 2T- and 3T-CF, e.g., F2 (0) = 1 and F3 (0; tf ) = 1. 2.2.2 Analysis and interpretation NMR MT-CF depend on the length of the evolution time tp . Noting that phases ωtp are correlated in the experiments, it is clear that small fluctuations of the resonance frequency lead to larger effects for long evolution times. The interpretation of NMR MT-CF is determined by the strength of single-particle interactions (CS or QP) with respect to multi-particle interactions (DD). In our Li/Ag NMR studies, the spectral widths associated with the QP/CS interactions, Δωi , and the Li−Li/Ag−Ag DD interactions, Δω d , are particularly relevant. Most straightforward interpretation of NMR MT-CF is possible in the case Δωi tp 1 Δωd tp . Then, the MT-CF are independent of the exact value of tp . In particular, F2ss (tm ) ≈ F2cc (tm ) ≈ F2 (tm ) ∝  cos[(ω2 − ω1 )tp ] ∝ δ(ω2 − ω1 )

(7)

measures the probability that an ion occupies the same site or, for crystals, any periodic site after the mixing time tm . In detail, (i) ions that occupy the same or periodic sites at the beginning and at the end of the mixing time do contribute since, within experimental resolution, the frequencies are identical at equivalent sites, (ii) ions that reside at different sites, on average, do not contribute, and (iii) rearrangements of neighboring ions have no effect, i.e., single-particle correlations functions are measured. In 109 Ag NMR work on glassy and crystalline silver ion conductors, Δω i typically exceeds Δωd by about 3 orders of magnitude so that the above condition can be met [20–27]. Specifically, all 109 Ag NMR MT-CF of the present contribution were measured using an evolution time tp = 100μs. This value is in the range, in which the decays of the studied samples are independent of the evolution time and F2ss (tm ) = F2cc (tm ) [20,21,25], enabling interpretation of the data in terms of Eq. (7). In 6 Li and 7 Li NMR, effects of Li−Li DD interactions and, hence, multi-particle contributions can affect the interpretation of NMR MTCF. For example, in 6 Li NMR studies on glassy lithium ion conductors [1,18], Δωi ≈ 2π × 3 kHz and Δωd ≈ 2π × 0.6 kHz so that the ratio of these quantities

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only amounts to a factor of about 5. Moreover, DD interactions enable spin diffusion, i.e., a transfer of magnetization via flip–flop processes of spins, which leads to an additional damping of the stimulated-echo amplitude. Then, interference of the respective decays due to ionic jumps and spin diffusion can affect the interpretation of experimental data. However, adjustment of the 6 Li/7 Li ratio via isotopic enrichment and temperature-dependent measurements can help to overcome the problem [1,18]. Two fundamentally different scenarios can explain nonexponential relaxation [12,13]. In the limit of purely homogeneous dynamics, all ions obey the same correlation function, which is, however, intrinsically nonexponential due to correlated forward–backward jumps, which are one of the cornerstones of current modeling approaches [28]. In the limit of purely heterogeneous dynamics, all ions are random walkers, i.e., correlated jumps are absent, but a distribution of correlation times G(log τ) exists. Analysis of 3T-CF, see Eq. (5), allows one to quantify the relevance of homogeneous and heterogeneous contributions to the nonexponentiality of 2T-CF. In this approach, the first stimulated-echo experiment of the pulse sequence is used to apply a dynamic filter, which is aimed at selecting a subensemble of slow ions, and the second stimulated-echo experiment is employed to determine whether dynamic filtering was successful by measuring the correlation function of the selected subensemble. For a quantitative analysis, it is useful that expectations for the outcome of F3 can be calculated on the basis of F2 in the limits of purely heterogeneous and purely homogeneous dynamics. For the analysis, it is important to what extent back jumps restore the initial resonance frequency. Recently [29], we showed that an involved analysis is inevitable when the difference of the resonance frequencies during consecutive visits of the same ionic site, δω, is of the order of the inverse evolution time, whereas straightforward interpretation is possible in the limits of exact restoration (δω  tp−1 ) and inexact restoration (δω tp−1 ). In the latter limit, the resonance frequency at an ionic site changes due to dynamical processes in the environment, e.g., via time-dependent DD couplings to other mobile ions, whereas such effects are absent in the former limit. First, we discuss the case of exact restoration, which can be met in 109 Ag NMR work on solid silver ion conductors [23]. Then, NMR MT-CF are sensitive to correlated forward–backward jumps, enabling determination of the relevance of such dynamics. The expectation for purely homogeneous dynamics and exact restoration is given by [23]: 1 (8) [F2 (tf + tm3 ) + F2 (tf )F2 (tm3 )] 2 In the limits of purely heterogeneous dynamics and exact restoration, it is necessary to specify the crossover pathways between different ionic sites in order to calculate an expectation. Moreover, we have to consider that, for crystals, the frequency is restored both when the ion returns to the initial site and when it visits a periodic site, while the latter effect does not exist for glasses, leading F3hom (tm3 ; tf ) =

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to different expectations for these types of materials. For crystals, the dependence on the pathway was shown to be small, resulting in a quasi-model free expectation for purely heterogeneous dynamics and exact restoration [23]: F3het (tm3 ; tf ) = F2 (tf + tm3 ) +

1 [F2 (tf ) + F2 (tm3 ) − F2 (tf + tm3 ) − 1] 2N

(9)

Here N is the number of magnetically non-equivalent sites. For glasses, the expectation for purely heterogeneous dynamics and exact restoration is [23,29,30]    1 (10) F2 (tf + tm3 ) + F2RW (tf , κ)F2RW (tm3 , κ)G(κ)dκ F3het (tm3 ; tf ) = 2 where G(κ) is the distribution of jump rates κ and F2RW (t, κ) is the 2T-CF resulting for a random walker described by a rate κ. The functions F2RW are nonexponential because the probability of a back jump to the initial site is finite even for a random walker in a glass as a consequence of a limited number of neighboring sites. The exact shape of these functions depends on the connectivity of the ionic sites in the glass matrix. Here, we exploit that information is available from molecular dynamics simulations, see below. Next, we deal with the limit of inexact restoration. In this case, NMR MTCF do not probe correlated forward–backward jumps. In particular, these homogeneous contributions do not affect 3T-CF, which are thus expected to agree with the prediction for purely heterogeneous dynamics and inexact restoration [31]: F3het (tm3 ; tf ) = F2 (tf + tm3 )

(11)

This case may apply to 6 Li and 7 Li NMR 3T-CF. For these nuclei, the resonance frequencies at the lithium ionic sites fluctuate when a redistribution of neighboring lithium ions leads to time dependent Li−Li DD couplings. Assuming that Δωd is a measure for δω, we expect that the resonance frequencies of a 6 Li nucleus typically differ by δω ≈ (250 μs)−1 during consecutive visits of the same site. Hence, in 6 Li NMR, the limit of inexact restoration should be fulfilled when using evolution times of the order of several hundreds of microseconds. When a distribution G(log τ) exists, NMR 4T-CF, see Eq. (6), can be used to measure on which time scale the ions exchange their correlation times. For this purpose, the same dynamical filter is applied during the first and the second stimulated-echo experiment of the pulse sequence. Then, ions that are sufficiently slow (τ > tf ) during the first and the second filtering periods contribute to the signal. Thus, F4 (tm2 ) decays when slow (τ > tf ) ions become fast (τ < tf ) during the mixing time tm2 . In studies on solid-state electrolytes, restoration of the resonance frequency can hamper straightforward interpretation, but the

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effect is weak for broad distributions G(log τ) [26]. Then, it is possible to determine the rate memory parameter Q. It is a measure for the average number of jumps, which are necessary until a slow ion exhibits a random new rate from the distribution. Quantitative analysis is possible by comparison of experimental data with theoretical curves F4Q for various rate memories Q ≥ 1, which can be calculated according to [32]: F41 (tm2 ; tf ) = F2 (2tf + tm2 ) +

tm2 Q 1 F4 (tm2 ; tf ) = F4 ; tf . Q

[F2 (tf ) − F2 (tf + tm2 )]2 1 − F2 (tm2 ) (12)

3. Results 3.1 Line-shape analysis Before presenting results of NMR MT-CF, it is instructive to discuss applications of NMR line-shape analysis to ion dynamics in solid electrolytes. Figure 1 shows temperature-dependent 109 Ag NMR spectra of AgI 0.3 -(AgPO 3 ) 0.7 glass, which are typical of results for silver iodide doped silver phosphate glasses [20,22]. We see that the spectra at sufficiently low and high temperatures are well described by a broad Gaussian and a narrow Lorentzian, respectively. While the former line shape revealed that silver ionic hopping motion is slow on the NMR time scale, the latter indicated that fast silver jumps result in motional narrowing. Specifically, at sufficiently low temperatures, the diversity of the local silver environments leads to a wide distribution of CS interaction parameters and, thus, to NMR spectra extending over roughly 15 kHz in high magnetic fields. At sufficiently high temperatures, all silver ions perform fast jumps on the time scale of the inverse spectral width, τ ≈ 10 μs, leading to an averaging over diverse silver ionic environments and, thus, resonance frequencies. Consistent with the present results, 109 Ag NMR Gaussian spectra were reported for silver iodide doped silver borate glasses at low temperatures [27,33]. At intermediate temperatures, the 109 Ag NMR line shapes of glassy silver ion conductors were well described by a weighted superposition of a broad Gaussian and a narrow Lorentzian, indicating that slow (τ 10 μs) and fast (τ  10 μs) silver ions coexist [20,22,27]. In this range, cooling resulted in a continuous decrease of the relative intensity of the Lorentzian, as can be seen for the example of AgI 0.3 -(AgPO 3 ) 0.7 glass in Fig. 1. These observations showed that broad and continuous distributions of correlation times G(log τ) exist for the silver ionic hopping motion in glasses. It was concluded that the silver ionic mobility is higher in iodine rich environments since the position of the Lorentzian component, which results from the fast ions of the distribution, does not coincide with the center of gravity of the Gaussian component,

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Fig. 1. 109 Ag NMR spectra of AgI 0.3 -(AgPO3 ) 0.7 glass at various temperatures. The dashed lines are interpolations with weighted superpositions of a broad Gaussian and a narrow Lorentzian. The vertical doted line marks the position of the low-temperature spectrum. The data were adapted from Ref. [20].

reflecting the slow ions [22]. Furthermore, 6 Li and 7 Li NMR spectra comprising broad Gaussian and narrow Lorentzian components were reported for glassy lithium ion conductors [1,18]. Hence, NMR line-shape analysis revealed that strong dynamical heterogeneities govern ionic hopping motions in glasses. Then, it is not possible to extract reliable correlation times from a simple analysis of the temperature-dependent NMR line width, but it is necessary to take into account the existence of a broad distribution of jump rates [1,18,27]. Specifically, in recent work [18], comparison with results from mechanical and electrical relaxation measurements showed that a simple line-width analysis underestimates the correlation times and their activation energy, see Fig. 2. Nevertheless, detailed analysis of NMR line shapes can yield interesting insights into ion dynamics in solids, in particular, for crystalline materials [1].

3.2 NMR two-time correlation functions Stimulated-echo experiments enable direct measurement of correlation functions of ion jump dynamics in solids, see Sect. 2.2.1. We exploited the potential of 109 Ag NMR 2T-CF to characterize silver ion dynamics in various materials [20–23,25]. It was found that the temperature dependence of 109 Ag NMR

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Fig. 2. 6 Li NMR correlation functions F2ss (tm ; tp ) of LiPO3 glass (50% 6 Li enrichment), obtained from stimulated-echo experiments at 350 K, 310 K, 275 K, and 258 K. All data were corrected for spin diffusion, which leads to a low-temperature decay characterized by τKWW = 0.37 s and β = 0.9. The evolution time was set to tp = 100 μs. The dashed lines are KWW fits with β = 0.27 and C = 0, see Eq. (13). Inset: Arrhenius plot of the mean correlation times τ resulting from the KWW interpolations (2T). The solid line is a fit with an Arrhenius law, yielding an activation energy of E a = 0.66 eV. For comparison, we include results from 7 Li NMR lineshape analysis (LS) and mechanical relaxation studies (MR) [39] on LiPO3 glass. The data were adapted from Ref. [18].

2T-CF follows the Arrhenius law. The resulting values of the activation energy are nicely consistent with E dc , implying that the elementary steps of the longrange charge transport are probed. Moreover, for all studied crystalline and glassy materials, strongly nonexponential correlation functions were observed. The decays were well described by a modified Kohlrausch–Williams–Watts (KWW) function: 

β tm +C (13) F2 (tm ) = (1 − C) exp − τ KWW Here, the stretching parameter β is a measure for the nonexponentiality and a finite residual correlation C can result for crystals because of a finite number of magnetically distinguishable sites. 6 Li and 7 Li NMR stimulated-echo experiments proved a versatile tool to study lithium ionic motion in solids [1,15,17,18,34–38]. For these nuclei, Li−Li DD interactions can facilitate spin diffusion. However, contributions from spin diffusion and ion dynamics to stimulated-echo decays can often be separated based on temperature-dependent experiments, in particular, when the former effect is slowed down by isotopic dilution. Specifically, it is possible

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Fig. 3. 7 Li NMR correlation functions F2ss (tm ; tp ) of the garnet Li 5 La3 Nb2 O12 for an evolution time of tp = 15 μs. The sample was annealed at 900 ◦ C. All data were corrected for spin diffusion, which results in a low-temperature decay described by τKWW = 0.18 s and β = 0.51. The dashed lines are fits with a modified KWW function, see Eq. (13), yielding β = 0.34–0.51 and C = 0.28–0.44. Inset: Arrhenius plot of the mean correlation time τ , as obtained from these interpolations. The solid line is an Arrhenius fit with an activation energy of E a = 0.38 eV. The data were adapted from Ref. [19].

to determine the spin-diffusion contribution at low temperatures when ion dynamics is too sluggish to attenuate the stimulated echo. Exploiting that spin diffusion is largely independent of temperature, its effect upon the stimulatedecho decays at higher temperatures can be removed by division [18]. In Fig. 2, we present 6 Li NMR 2T-CF of LiPO 3 glass, which were corrected for spin diffusion in the described manner [18]. It is evident that the decays are nonexponential and shift to longer times upon cooling. In KWW fits, a stretching parameter β = 0.27 was found to well describe the nonexponentiality of all decays [18]. Employing the Γ function, the mean correlation times τ were calculated from the fit parameters according to τ = (τ KWW /β)Γ(1/β). The temperature dependence of τ is displayed in the inset of Fig. 2. We see that the correlation times from 6 Li NMR 2T-CF are in harmony with that from mechanical relaxation studies [39]. Furthermore, unlike NMR line-shape analysis, NMR stimulated-echo analysis yielded an activation energy that agrees with E dc = 0.66 eV [40]. The fact that 2T-CT give microscopic access to long-range transport parameters was also shown in 7 Li NMR stimulated-echo work on crystalline Li7 BiO 6 [34]. In addition, the good agreement of the present stimulated-echo data with previous mechanical and electrical relaxation data [39,40] justifies the used spin-diffusion correction approach. Figure 3 shows 7 Li NMR 2T-CF of the garnet Li5 La3 Nb 2 O 12 . Again, spindiffusion correction was applied [19]. Previous studies showed that lithium dy-

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Fig. 4. NMR 2T-CF and 3T-CF of the mixed cation glass Li0.5 Ag0.5 PO3 : (a) 6 Li NMR data for the lithium ion dynamics at T = 375 K (tp = 600 μs) and (b) 109 Ag NMR data for the silver ion dynamics at T = 360 K (tp = 100 μs). For both ionic species, F3 (tm3 ) was recorded using a filter time of tf = 5 × 10−3 s. The dotted lines are KWW fits of F2 (tm ). The solid lines are the respective expectations for the 3T-CF in the limit of purely heterogeneous dynamics. For 6 Li, the expectations are calculated according to Eq. (11), i.e., inexact restoration of the resonance frequency is assumed, see Sect. 2.2.2. For 109 Ag, we suppose exact restoration and utilize Eq. (10). The shape of F2RW was obtained from molecular dynamics simulations for LiPO3 glass [30].

namics strongly depends on the annealing temperature for this garnet [19,37]. Here, we focus on a sample annealed at 900 ◦ C [37]. In Fig. 3, we see that the correlation functions exhibit a finite long-time plateau for this crystal. Fitting to Eq. (13) yields a residual correlation C ≈ 0.28 near ambient temperatures. It was suggested that such plateau results from a coexistence of octahedral and tetrahedral lithium sites, leading to a bimodal distribution of correlation times for Li5 La3 Nb 2 O 12 [19,37]. Then, the slow component is static on the experimental time scale at the studied temperatures, resulting in a finite plateau at long times. The hopping motion of the fast component follows an Arrhenius law with an activation energy of E a = 0.38 eV. Moreover, it is governed by nonexponential correlation functions, which are described by stretching parameters β = 0.34–0.51. Hence, strongly nonexponential lithium ion dynamics is not restricted to glasses, but it can also occur in crystals, consistent with previous results [1]. In NMR work on solid electrolytes featuring more than one mobile ion species, the element selectivity of the method can be exploited to separately investigate the dynamical behaviors of the different ionic species. In particu-

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lar, stimulated-echo experiments allow us to directly measure the correlation functions of the respective hopping motions. Figure 4 shows 6 Li and 109 Ag NMR 2T-CF of Li0.5 Ag 0.5 PO 3 glass at comparable temperatures. Comparison of the decays indicates that lithium and silver dynamics occur on similar time scales in this mixed cation glass. However, the correlation functions are more nonexponential for the larger silver ions than for the smaller lithium ions. Specifically, KWW fits yield β = 0.33 and β = 0.46 for the former and the latter ionic species, respectively. More detailed insights into the mixed cation effect of Li x Ag 1−x PO 3 glasses are available when analyzing the dependence of the 6 Li and 109 Ag NMR 2T-CF on temperature and composition. These results will be published elsewhere [41]. Below, we will use 6 Li and 109 Ag NMR 3T-CF to investigate the origin of the nonexponential ionic relaxation in Li0.5 Ag 0.5 PO 3 glass.

3.3 NMR three-time correlation functions The above results of NMR stimulated-echo studies reveal that strongly nonexponential correlation functions often characterize the ionic hopping motion in crystals and glasses. Therefore, it is very important to understand the origin of this key feature of charge transport in solid-state electrolytes. NMR 3TCF allow one to quantify the relevance of homogeneous and heterogeneous contributions to the nonexponentiality, see Sect. 2.2.2 [12,13,23]. In Fig. 5, we present 109 Ag NMR 2T-CF and 3T-CF of γ -Ag 5 Te2 Cl at T = 204 K [25].

Fig. 5. 109 Ag NMR 2T-CF and 3T-CF of polycrystalline γ -Ag5 Te2 Cl at T = 204 K. In all measurements, the evolution time was set to tp = 100 μs. F3 (tm3 ) was recorded using a filter time of tf = 2 × 10−2 s. The experimental 3T-CF is compared with expectations for purely homogeneous dynamics (F3hom ) and for purely heterogeneous dynamics (F3het ), see Eqs. (8) and (9). Exact restoration of the resonance frequency was assumed in the calculations, as indicated by the finite residual correlation of the 2T-CF. Specifically, a KWW fit (dotted line) to Eq. (13) yielded β = 0.36 and C = 0.05. The data were adapted from Ref. [25].

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Again, Eq. (13) enabled a good description of the 2T-CF. While a stretching parameter β = 0.36 indicated the nonexponentiality of the ion dynamics, a residual correlation C = 0.05 was consistent with 20 crystallographically distinct silver sites in γ -Ag 5 Te2 Cl [25]. The observation of a finite long-time plateau showed that the resonance frequency of a silver ion is restored when it returns to the initial site or visits a periodic site during the diffusion process. This is a necessary condition for an observation of homogeneous contributions to the nonexponentiality due to correlated forward–backward jumps, see Sect. 2.2.2. The experimental 3T-CF was compared with the predictions for purely homogeneous dynamics (F3hom ) and purely heterogeneous dynamics (F3het ), which were calculated using Eqs. (8) and (9). While F3hom substantially deviates from the experimental data, F3het agrees with the measured decay, expect for minor deviations at short times. Thus, the nonexponentiality of the NMR correlation function largely results from heterogeneous contributions. Consistently, 109 Ag NMR 3T-CF indicated that the nonexponential silver ionic relaxation in β-Ag 7 P 3 S 11 is due to dynamical heterogeneities rather than to intrinsic nonexponentiality [23]. Likewise, 109 Ag NMR MT-CF revealed that pronounced dynamical heterogeneities govern the silver ionic hopping motion in silver iodide doped silver phosphate glasses [20], in harmony with the results of the above line-shape analysis. Here, we use 109 Ag NMR and 6 Li NMR 3T-CF to separately investigate silver and lithium ion dynamics in the mixed cation glass Li 0.5 Ag 0.5 PO 3 . We point out that this study is the first experimental application of 6 Li or 7 Li NMR 3T-CF to lithium ion dynamics. In Fig. 4, it is evident that the experimental data are well described by F3het for both silver and lithium ions, but the interpretation of this observation differs for the ionic species. In the case of 109 Ag, F3het is calculated according to Eq. (10) assuming exact restoration of the resonance frequency. While it is not possible to directly test this assumption for glassy ion conductors because of the diversity of all sites in a disordered matrix, it is motivated by the observation of a finite residual correlation for crystalline silver ion conductors [23,25] and by the negligible Ag−Ag DD couplings, see Sect. 2.2.2. Hence, the analysis yields strong evidence that the nonexponentiality of the 109 Ag NMR 2T-CF of the mixed cation glass is due to the existence of a broad distribution of ionic jump rates, whereas intrinsic nonexponentiality plays a minor role. In the case of 6 Li, F3het is computed according to Eq. (11), i.e., we assume inexact restoration of the resonace frequency since Li−Li DD couplings lead to tp δω = 2–3 for the used evolution time tp = 600 μs, see Sect. 2.2.2. Then, 6 Li NMR MT-CF should hardly probe correlated forward–backward jumps and, thus, any nonexponentiality of the 2T-CF should result from dynamical heterogeneities, which in turn means that the 3T-CF should coincide with the expectation F3het . In Fig. 4, we see that the experimental 6 Li NMR 3T-CF of Li0.5 Ag 0.5 PO 3 glass indeed agrees with the expectation for purely heterogeneous dynamics. We conclude that, for 6 Li, any nonexponentiality of NMR 2T-CF for sufficiently long evolution times directly

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reflects the distribution of jump rates and analysis of NMR 3T-CF does not allow one to determine the relevance of correlated forward–backward jumps.

3.4 NMR four-time correlation functions Since silver ionic hopping motions exhibit strong dynamical heterogeneities in silver iodide doped silver phosphate glasses [20], it is interesting for how many jumps the silver ions remember their jump rates, i.e., to determine the time scale of rate exchange. NMR 4T-CF enable straightforward measurement of the life time of dynamical heterogeneities [12,13,26]. Figure 6 shows 109 Ag NMR 4T-CF of (AgI) 0.43 -(Ag 4 P 2 O 7 ) 0.57 glass. In general, restoration of the resonance frequency affects the interpretation of NMR 4T-CF. However, it was shown that, in the present case of a very broad rate distribution, observation of a decay still indicates the existence of rate exchange processes [26]. Comparison of the decays of the 2T-CF and the 4T-CF reveals that the ionic jumps and the exchange within the broad rate distribution occur on comparable time scales. Quantitative analysis was possible based on the rate memory parameter Q, see Sect. 2.2.2. It is a measure for the average number of jumps, which are necessary until a slow ion exhibits a random new rate from the distribution [32]. In Fig. 6, we compare the experimental 109 Ag NMT 4T-CF with expectations for different values of Q, which were calculated from the corresponding 2T-CF according to Eq. (12). We see that the theoretical curve for Q = 5 is in reasonable agreement with the experimental data, indicating that, in the studied silver iodide doped silver phosphate glass, initially slow silver ions exhibit random new rates from the distribution after very few jumps. In other words, the

Fig. 6. 109 Ag NMR 2T-CF and 4T-CF of (AgI) 0.43 -(Ag4 P2 O7 ) 0.57 glass at T = 210 K. In all measurements, the evolution times were set to tp = 100 μs. F4 (tm2 ) was recorded using a filter time of tf = 10−2 s. The solid lines are theoretical curves for rate memory parameters Q = 1 and Q = 5, respectively, see Eq. (12). The dashed line is a KWW fit of F2 (β = 0.2). The data were adapted from Ref. [26].

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dynamical heterogeneities are short-lived so that use of an average correlation time is sufficient to describe the charge transport on length scales larger than a few interatomic distances. Consistently, 109 Ag NMR 4T-CF indicated fast rate exchange for silver ion dynamics in β-Ag 7 P 3 S 11 [23].

4. Conclusion We demonstrated that NMR multi-time correlation functions provide detailed insights into the nature of lithium and silver ionic hopping motions in solidstate electrolytes. Stimulated-echo experiments enable straightforward measurement of two-time correlation functions of the ion dynamics. While lineshape and spin-lattice relaxation analyses often underestimate the temperature dependence of ion dynamics, stimulated-echo experiments yield activation energies, which nicely agree with that obtained from dc conductivity and mechanical relaxation studies. Hence, the latter NMR technique probes the elementary steps of the long-range charge transport. NMR two-time correlation functions showed that the ion jump dynamics is nonexponential in all studied crystalline and glassy materials. Although nearly exponential correlation functions were observed for a crystalline lithium ion conductor [17], we conclude that nonexponential ionic relaxation is a key feature of most solid-state electrolytes. NMR three-time correlation functions allow one to quantify the origin of nonexponential ionic relaxation. The analysis indicated that pronounced dynamical heterogeneities govern ion dynamics in the studied crystals and glasses. The detailed interpretation of NMR three-time correlation functions can depend on the studied material and the used nucleus. In this context, it is important whether back jumps restore the initial resonance frequency of a nucleus within the experimental resolution. A straightforward answer to this question exists for crystals, but it does not for glasses. Specifically, due to a finite number of distinguishable ionic sites in crystals, restoration of the resonance frequency is indicated by the existence of a finite plateau of the twotime correlation function at long mixing times, provided the data are recorded using a suitable evolution time such that Eq. (7) applies. Then, NMR multitime correlation functions are sensitive to correlated forward–backward jumps and, hence, their analysis allows one to determine the relevance of such dynamics. For two crystalline silver ion conductors, the analysis showed that intrinsic nonexponentiality, i.e., correlated forward–backward motion, hardly contributes to the nonexponentiality of NMR correlation functions. In studies on glasses, an essentially infinite number of distinguishable ionic sites results in complete decays of the correlation functions irrespective of whether back jumps do or do not restore the initial resonance frequency of a nucleus and, hence, a clear criterion for the sensitivity of NMR multitime correlation functions to correlated forward–backward jumps is lacking. Then, the data do provide information about the existence of dynamical het-

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erogeneities, but they do not necessarily yield insights into the relevance of correlated forward–backward jumps. Specifically, two explanations can rationalize an agreement of experimental three-time correlation functions with the prediction for purely heterogeneous dynamics: correlated forward–backward jumps are either not relevant or they are not probed by the technique. By analogy with results for crystalline materials, it is reasonable to assume that 109 Ag NMR is sensitive to forward–backward jumps and, hence, the data should yield insights into the relevance of such dynamics for glasses, too. By contrast, correlated forward–backward jumps are hardly probed by 6 Li NMR three-time correlation functions, which were measured for the first time in the present contribution. Since DD interactions are not negligible with respect to QP interactions for 6 Li, the resonance frequency at a lithium site can change somewhat due to a redistribution of the lithium ions at neighboring sites, interfering with frequency restoration due to back jumps within the experimental resolution. Then, the nonexponentiality of two-time correlation functions directly reflects the dynamical heterogeneities. Therefore, our 6 Li NMR multi-time correlation functions for single cation and mixed cation glasses indicate the existence of strongly heterogeneous lithium dynamics, but they do not allow us to draw conclusions about the relevance of correlated back-and-forth motion of the lithium ions. NMR four-time correlation functions enable measurement of the life time of dynamical heterogeneities in solid-state electrolytes, if existent. The results for a crystalline and a glassy silver ion conductor indicated that initially slow silver ions show a random new rate from the distribution after very few jumps. Therefore, extended regions where silver jump dynamics is fast and slow, respectively, do not exist in the studied materials. Rather, silver sites featuring different jump rates are intimately mixed. Then, as a consequence of the rate averaging process, a single mean correlation time suffices to describe silver ion migration on length scales larger than a few interatomic distances.

Acknowledgement This work was supported by the Deutsche Forschungsgemeinschaft within the SFB 458 program. We thank R. Böhmer for collaborating with us on the study of mixed cation glasses.

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Mechanisms of Ion Conduction in Polyelectrolyte Multilayers and Complexes ´ ad W. Imre#, Amrtha Bhide, and By Monika Schönhoff∗, Arp´ ∗ Cornelia Cramer Institut für Physikalische Chemie, Westfälische Wilhelms-Universität, Corrensstr. 28/30, 48149 Münster, Germany (Received July 21, 2010; accepted in revised form September 23, 2010)

Complex Coacervate / Layer-By-Layer / Proton Conductor / Polymer Electrolyte / Impedance Spectroscopy This paper reviews the progress made in understanding of the mechanisms of ion conduction in polyelectrolyte multilayers (PEM) and polyelectrolyte complexes (PEC). The basis are experimental conductivity data obtained by impedance spectroscopy as a function of relative humidity and temperature, respectively. Mechanically stable thin films of PEM have interesting perspectives as ion conductors, however, being prepared by self-assembly, their stoichiometry and content of ionic charge carriers is unknown. Therefore PEC act as a model material with a variable stoichiometry and known ion content. Employing poly(sodium 4-styrene sulfonate) (NaPSS) and poly(diallyldimethyl ammoniumchloride) (PDADMAC), we present conductivity spectra of dried polyelectrolyte complexes of type x NaPSS · (1 − x)PDADMAC as a function of temperature and composition, respectively. The dependence of the dc conductivity is discussed along with scaling properties of the spectra. The results show that the conductivity is always determined by the sodium ions, even in PEC with an excess of PDADMAC. The ion dynamics and transport mechanisms are, however, different in PDADMAC-rich than in NaPSS-rich PEC. PEM of different polyionic compounds are investigated in dependence on relative humidity. A general law of an exponential increase of the dc conductivity with relative humidity is found. Absolute values of the conductivity and the strength of the humidity dependence are different for different polyion materials, however, they do not depend on the type of small counterion employed in layer formation. Therefore, it is concluded that in hydrated PEM, protons are the dominant charge carriers. For both PEM and PEC we show that the MIGRATION concept developed by Funke and co-workers can be used for describing the experimental spectra over wide ranges in frequency. This implies that forward-backward hopping motions of small ions play a vital role in solid polyelectrolyte materials. Apart from these potentially successful hops, localized motions of charged particles are found to influence the conductivity spectra as well.

* Corresponding authors. E-mail: [email protected]; [email protected] # Current address: Robert Bosch GmbH, P.O. Box 30 02 40, 70442 Stuttgart, Germany.

Z. Phys. Chem. 224 (2010) 1555–1589 © by Oldenbourg Wissenschaftsverlag, München

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1. Introduction In electrochemical applications the demand for novel materials with tailored properties is immense, in particular in view of the current search for high capacity energy storage devices. Fuel cells and Li ion batteries are competing devices that can fulfill the requirements, provided that suitable materials can be developed. A key element in both devices is the polymer electrolyte membrane which separates the electrodes and is required to exhibit a high ionic (Li+ or protons) and negligible electronic conductivity. At the same time thin film processing and a high mechanical stability are required. In fuel cells since a long time Nafion is the benchmark material, since it shows a microphase separation into hydrophobic, stabilizing domains and hydrated channels, where the interface, carrying free sulfate groups, provides the proton conduction properties [1–3]. In Li ion batteries the classical basis of stateof-the-art polymeric materials is poly(ethyleneoxide) and many modifications thereof attempt to achieve a compromise between high mechanical stability and high ionic conductivity. One example is to apply salt-in-polymer electrolytes made from comb-shaped copolymers, which exhibit short oligoether side chains that can solubilize and transport the ions [4–6], while the backbone can be cross-linked to provide mechanical stability. Typically, such bulk polymers are then prepared as a thin membrane with a thickness in the μm range. In the context of such electrochemical applications polyelectrolyte multilayers (PEM) prepared by layer-by-layer-assembly (LbL) [7] have attracted interest. They are prepared by the alternating adsorption of polycations and polyanions from aqueous solutions. The process is predominantly driven by multiple electrostatic interactions and is therefore very versatile with respect to different charged building blocks that can be employed in multilayer formation. External parameters such as salt concentration [7], pH value [8] or temperature [9] provide control of the layer thickness, which typically lies in the range of one nm per layer. Research in the field of PEM has vastly expanded in the past two decades, as there are numerous potential applications such as containers, sensors, drug delivery, etc. Various review articles provide information about structural aspects and summarize the properties of PEM [10–13]. Aiming at applications in electrochemical energy devices PEM fulfill two of the three main requirements already by their generic material properties: Firstly, PEM layer thickness can be tuned in the nm to μm range. A low film thickness is of advantage to yield low overall resistance and fast loading. Secondly, in spite of the low thickness, PEM are a tremendously stable material, which is an effect of the multiple electrostatic interactions between subsequent layers. Their mechanical properties have been extensively investigated in free-standing geometries, and Young’s moduli of the order of GPa have been found [14,15].

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A challenge remains however to fulfill the third requirement, which is to achieve large ionic conductivities in these films. In several publications dc conductivities of PEM ranging from 10−12 S cm−1 to 10−5 S cm−1 have been reported [16–19]. Studies of the conductivities of PEM started with the seminal work by Durstock and Rubner: They investigated films of PSS/PAH or PAA/PAH, respectively, where PAA denotes poly(acrylic acid), PSS is poly(styrene sulfonate sodium salt), and PAH is poly(allylamine hydrochloride). They found dc conductivities, σ dc , in the range of 10−12 S cm−1 to 10−7 S cm−1 [16], where the maximum of 10−7 S cm−1 was achieved only at strong hydration. Further studies of conductivities in PEM suggested them as potential ion conductive materials for battery applications [17,18]: When employing poly(2-acrylamido-2-methyl-1-propansulfonic acid) (PAMPS) as a polyanion, at high humidity a value of the dc conductivity of σ dc ≈ 10−5 S cm−1 was achieved, which is already in a realistic range for applications [17]. Different polyelectrolyte pairs tested for their conductivity already involved polymers known as ion conductors, such as Nafion as an established proton conductor [17], poly(ethylene oxide) (PEO) as a classical polymer electrolyte [18], or polyphosphazenes which provide flexible main chains as polymer electrolytes [19,20]. However, despite these studies on conductivities of different types of PEM, a fundamental understanding of ion transport properties in PEM is not in sight yet. A major problem in the interpretation of conductivity data is the lack of knowledge about the composition of the films: Since PEM are formed by self-assembly, the compensation of surface charges upon chain adsorption is controlling the stoichiometry of the films. The excess charges of an outermost polyelectrolyte layer might become fully compensated by the oppositely charged segments of the subsequent layer (‘intrinsic charge compensation’), or, if this is sterically not favorable, small counterions might incorporate into the film in order to compensate the polyion charges (‘extrinsic charge compensation’). Thus, PEM are a material of unknown stoichiometry. Though in first approximation the translational entropy of the small counterions would always lead to intrinsic charge compensation being favored, in a number of polyion combinations a deviation from a 1:1 stoichiometry of the polyions has been found and a substantial degree of extrinsic charge compensation by small counterions was concluded [21–23]. In hydrated multilayers even protons can contribute to the conductivity. Several authors employ Nafion in multilayer formation and discuss protons as the dominating charge carriers in the dc conductivity [24–27]. The hydration state of the layer assembly indeed has a strong influence on the conductivity when films are compared in the dry and the completely hydrated state [16,17,24]. A review of the activities up to 2007 is given by Lutkenhaus and Hammond [27]. There, it is argued that protons carry the current, however, an analysis of the contributions of other small counterions is suffering from the lack of systematic knowledge about the composition of PEM.

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As a reference system with a controlled and variable stoichiometry, polyelectrolyte complexes (PEC) are an interesting material: The complexation of polyelectrolytes of opposite charge in aqueous solutions is a longknown phenomenon, often leading to the precipitation of ionically cross-linked polymer complexes. Recently, such complexes have regained interest due to their similarity to PEM. One similarity lies in the same short-range interactions of the charged groups. It was shown that the distance of positive and negative charges in either type of complex is identical [28]. Furthermore, employing regions of stability in dependence on electrolyte conditions, the phase diagrams of PEC and PEM systems are comparable to each other [29,30]. Most importantly for conductivity experiments and their interpretation, in PEC the content of small cations and anions is known, as it depends on the mixing ratio of the polyions. Furthermore, systems with mainly one type of counterion can be prepared, when excess salt is removed by dialysis. In this way conductivity data in dependence of the composition can be related to the conductivity contribution of a single type of charge carrier [31,32]. For this purpose solid PEC complexes have to be prepared by drying and pressing. The characterization of PEC is so far limited to techniques applied to soluble complexes, such as light scattering [33] or osmotic stress equilibration [34] Some early conductivity studies of dried PEC were reported in the 60’s of the last century [35], but such investigations are hindered by low dc conductivities of dry PEC and the necessity of applying sophisticated impedance spectroscopy devices for measuring the conductivity. In our research program, which is reviewed here, we perform impedance spectroscopic experiments on PEC as well as on PEM, employing the former as a model system with a known polyion/counterion stoichiometry. This contains the first systematic conductivity studies of PEC and the first frequencydependent conductivity spectra of PEM. The data base contains impedance spectra of different polyanion/polcation combinations in multilayers in dependence on humidity and impedance spectra of PEC as a function of composition and temperature [19,31,32,36]. Analyzing the data, we discuss the relevance of the contributions of different ionic species, such as the alkali cations, anions, or protons to the conductivity. We yield the conclusion that in hydrated PEM the conductivity is dominated by proton conduction, and in dry PEC the conductivity is governed by cation conduction, while the mobility of small anions is very low. In analogy to temperature, which activates ionic motion in PEC, increasing relative humidity yields an exponential increase of the conductivity and is therefore considered as an activation parameter. Furthermore, scaling concepts such as the time-temperature superposition principle are applied to PEC materials, yielding information about the temperature-dependence of the transport mechanism. And finally, modeling of different contributions to the spectra in the framework of concepts established in well-characterized inorganic ion conductors sheds more light on the transport processes in PEC and PEM.

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2. Materials and methods Experimental details on the preparation and on the measurement techniques are given elsewhere, see references [5,19,36] for the PEM materials and Ref. [31] for the PEC materials, respectively.

2.1 Polyelectrolyte complexes As a representative system for polyelectrolyte complexes we studied a series of complexes made of anionic poly(sodium 4-styrene sulfonate) (NaPSS) and cationic poly(diallyldimethyl ammoniumchloride) (PDADMAC), see the structures in Scheme 1. The complexes are denoted as x NaPSS · (1 − x) PDADMAC with the fraction of cationic monomers, x. The composition was varied between 0.35 ≤ x ≤ 0.70, in steps of 0.05. In addition, analogue complexes with LiPSS and CsPSS instead of NaPSS were studied for compositions x = 0.40, 0.50 and 0.60. Within the studied composition range, the equimolar ratio of PSS and PDADMAC with x = 0.50 represents a polyelectrolyte complex with only intrinsic charge compensation in the ideal case. However, in reality there will always be a very small amount of small ions present, as the charge compensation between the polycations and polyanions is never perfectly 1 : 1. For all other compositions, extrinsic charge compensation by small ions exists in addition to intrinsic charge compensation. In the PSS-rich complexes (x > 0.50), the extrinsic charge carriers are predominantly the Na+ ions, whereas in the PDADMAC-rich complexes (x < 0.50), it is the Cl− ions. All complexes were formed from solutions of reagent grade chemicals by precipitation, dialysis and freeze drying; the procedure is described in Ref. [31]. The preparation of CsPSS by ion exchange of freeze-dried NaPSS is explained in Ref. [36]. For PDADMAC-rich complexes with x < 0.50 we could distinguish two phases. One of them is a cotton-like phase similar to that of the PSS-rich compositions and the second one is a hard salt-like phase. The two-phase systems were milled and mixed before use. Small amounts of freeze-dried PEC samples were used for thermal analysis. Subsequent DSC measurements were performed between −73 ◦ C and 300 ◦ C with a heating rate of 10 ◦ C/min, see also Fig. 2 of Ref. [31]. During the first scan we observed a large endothermic peak with an onset temperature between 41 ◦ C and 60 ◦ C depending on composition. These are summarized in Ref. [31]. This large peak is absent in the following runs and can, therefore, be attributed to the evaporation of water which is still present after freeze-drying of the PEC. A much smaller endothermic peak, which was reproducible in all runs following the first one, can be explained by a glass transition of the PEC materials. More detailed information along with the Tg values determined from the DSC curves are given in Ref. [31]. The impedance measurements were performed on pre-dried, pressed, cylindrical samples. Gold electrodes were sputtered onto the top and bottom faces,

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Scheme 1. Polyelectrolyte structures employed for PEC or PEM formation.

respectively. Complex impedance spectra of the samples were measured in a frequency range of 10−2 Hz to 106 Hz using a Novocontrol α-S high resolution dielectric analyzer equipped with a Quatro cryosystem. During measurements, the samples were deposited in a chamber, where the temperature is controlled by a preheated flow of dry nitrogen. Impedance measurements were performed between −70 ◦ C and 300 ◦ C in steps of 10 ◦ C. To remove small amounts of remaining water, the samples were annealed at elevated temperatures for several hours before each measurement. More experimental details are given in Ref. [31].

2.2 Polyelectrolyte multilayers The following polyelectrolyte combinations were investigated: PAH/LiPSS, PAH/NaPSS, PAH/RbPSS, PAH/CsPSS, PDADMAC/NaPSS, and PAZ+ / PAZ− , respectively. The abbreviations stand for the following polycations (see

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also Scheme 1): poly(allylamine hydrochloride) (PAH), poly(diallyldimethyl ammonium chloride) (PDADMAC), cationic (poly(bis(3-amino-N,N,N-trimethyl-1-propanaminium iodide) phosphazene) (PAZ+ ) and for the following polyanions: poly(sodium 4-styrene sulfonate) (PSS) and poly(bis(lithium carboxylatophenoxy)phosphazene) (PAZ− ), respectively. The synthesis and characterization of the PAZ materials is described in Refs. [5,19]. Polyelectrolyte multilayers were prepared from aqueous solution of the polyions using the layer by layer technique developed by Decher and coworkers [7]. The number of monolayers n was always 100. The first layer was always made of poly(ethylene imine) (PEI) followed by layers of polyanions/polycations. The last layer was always formed by polyanions. For the impedance measurements, indium-tin oxide (ITO) coated glass substrates were used as substrates for the PEM films, the ITO layer acting as the bottom electrode in a sandwich geometry. The top electrode of the film was sputtered onto the multilayer using a mask. This provided parallel rows of circular gold electrodes with a diameter of 1 mm. Identical films without electrodes were made on oxidized Si wafers, such that the film thickness could be determined by ellipsometry [36]. Conductivity spectra were determined at T = (22 ± 2) ◦ C at various constant relative humidities (RH) employing an impedance analyzer (hp 4192 A, Hewlett-Packard) which covers a frequency range from 5 Hz to 13 MHz. All measurements were performed in a sealed glove box, where saturated salt solutions were applied for RH adjustment. The complex conductivity σˆ (ν) is obtained from the measured complex admittance Yˆ (ν) via σˆ (ν) = Yˆ (ν)d/A, where ν is the experimental frequency and A denotes the area of the gold electrodes. The sample thickness d was taken from ellipsometric measurements of a silica wafer with an identical PEM coating. More experimental details can be found in Refs. [19,36].

3. Conductivity spectra: overview 3.1 Spectra of PEC We present temperature-dependent conductivity spectra of dried xNaPSS · (1 − x)PDADMAC where x was varied between 0.35 and 0.70 in steps of 0.05. Figure 1 shows conductivity spectra for x = 0.40 and 0.60, respectively. We see similarities with, as well as deviations from the spectra of typical ion conducting bulk materials. It is obvious that the low-frequency part of each spectrum is determined by a frequency regime where the conductivity does not depend on frequency. This conductivity value of each isotherm can be identified with the dc conductivity which will be discussed in more detail in Sect. 4.1. All other compositions not displayed here show similar spectral shapes except for x = 0.70, the best conducting composition, where at the highest temperatures polarization affects the dc plateau.

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Fig. 1. Representative spectra of the real part of the conductivity of PEC with x = 0.40 (a) and x = 0.60 (b), respectively. The stars mark the onset frequencies defined as σ  (ν* ) = 2σdc .

With increasing temperature, T , the conductivity values increase and the onset of conductivity dispersion shifts to higher frequency. This effect of temperature on the conductivity spectra is well-known for other ion-conducting materials, too. In Fig. 1, we have used the common definition σ  (ν * ) = 2σ dc for the onset frequency, ν * , which characterizes the transition from the dc into the dispersive regime. In both PEC materials we find that the onset points are on a straight line, but their slope differs. For x = 0.40 the slope is smaller than one, for x = 0.60 it exceeds one. This is one of the marked differences between the spectra of PEC which are rich in PDADMAC and those which are rich in NaPSS and it will be discussed in more detail in Sect. 5.2. It is also visible in Fig. 1 that the conductivity increases with NaPSS content. At a given temperature, the conductivity for PEC with x = 0.60 is by more than two orders of magnitude higher than the conductivity of PEC with x = 0.40. If we compare Fig. 1a with Fig. 1b we see that roughly the same conductivity values which are obtained for x = 0.40 at 563 K are already measureable for x = 0.60 at 433 K. The difference between isothermal conductivities of different PEC is most pronounced in the dc regime, see also Sect. 4.1. Whereas at first sight the spectral shape in the dispersive regime of the conductivity spectra of both materials displayed in Fig. 1 seems to be in accordance with the conductivity spectra of other bulk materials, there are, however, differences. These differences are twofold. Depending on composition, PEC spectra differ from each other and all of them differ from those of most other ion-conducting bulk materials. The first finding is visible in Fig. 2a where we show two isotherms of PEC with x = 0.40 and 0.60, respectively, which have almost the same dc conductivity. Figure 2b displays the corresponding values

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Fig. 2. Spectral shape (a) of the real part of the conductivity and (b) of the real part of the permittivity.

of the real part of the permittivity ε which is connected to the imaginary part of the complex conductivity σ  via ε = σ  /(ωε 0 ). ω stands for the angular frequency which can be obtained from the experimental frequency by ω = 2πν, ε 0 is permittivity of free space. We find that the complex conductivities of the investigated PEC systems can be divided into two classes: those with x > 0.50 and those with x ≤ 0.50. In Fig. 2a it is obvious that for comparable dc conductivities the dispersive regime sets on earlier on the frequency scale for PEC with x > 0.50 than for the other PEC. In all materials with x > 0.50 we also observe a “shoulder” in the conductivity spectra, roughly between 0.1 kHz and 10 kHz. An analogue behaviour is hardly detectable for x ≤ 0.50. In the real part of the permittivity we detect higher values and ε decays more rapidly with increasing frequency for x > 0.50 than for x ≤ 0.50. The real and imaginary parts of the complex conductivity are interconnected via Kramers–Kronig relations [37,38]. This means that σ  (ν) and σ  (ν) (and therefore also ε (ν)) contain the same kind of information and can be transformed into each other, if the complete experimental spectrum is known. In this work, both the real part and the imaginary part of the complex conductivity have been experimentally determined, but in the following sections we will focus on the real part of the conductivity. An analysis of the exact shape of the conductivity spectra and its implication for the ion transport will be given in Sect. 6.

3.2 Spectra of PEM Impedance spectra of PEM were taken at different relative humidities, always at room temperature. We investigated a wide range of different polyanion/polycation combinations [19,36]. As an example the spectra of PAH/LiPSS

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Fig. 3. Impedance spectra of 100 layers of (a) PAH/LiPSS multilayers and (b) PAZ+ /PAZ− multilayers at RH between 17% and 95% [36].

are shown in Fig. 3a. A striking difference with respect to the spectra of PEC shown in Fig. 1 is immediately evident: Polarization effects are strongly pronounced, causing an additional regime at very low frequencies, where the conductivity is increasing with frequency. This polarization arises due to the low thickness of the layers (about 100 nm for 100 layers [36]), and is due to ionic charge carriers accumulating at the electrodes at low frequencies. In the spectra of Fig. 3 it is clearly evident that polarization becomes stronger with increasing dc conductivity, i.e. increasing relative humidity. In the intermediate frequency regime, a dc plateau can be identified, however, since polarization and the onset of dispersion are rather close together on the frequency scale, the plateau is less clearly evident as compared to the spectra of PEC. Finally, at high frequencies, a dispersive regime is observed. Thus, apart from the polarization, the regimes of PEM spectra are qualitatively similar in showing a dc and a dispersive regime, respectively. Increasing humidity leads to an increase of the conductivity. This is most pronounced in the dc regime, whereas – at least in the logarithmic representation – differences in the dispersive regime are less pronounced. The relevance of polarization effects is increasing with increasing dc conductivity, as expected. Apart from the polarization regime, thus, the influence of humidity on the spectra of PEM is very similar to the influence of temperature on the spectra of PEC. For different combinations of polycations and polyanions the sets of spectra taken at different humidity show similar properties. A second example is given in Fig. 3b, where multilayers are formed from cationic and anionic polyphophazene derivates [19]. Polyphosphazenes are currently under study in bulk salt-in-polymer electrolyte systems, since the polyphosphazene backbone is flexible and can provide a large mobility for Li ions coordinated to

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its side chains [5] This was the motivation for synthesizing ionically modified polyphosphazenes and forming multilayers thereof [19]. In Fig. 3b it is seen that at comparable RH values, the conductivities are 1 to 2 orders of magnitude larger than those of the PAH/LiPSS system. Whether or not this is an effect of the main chain flexibility, cannot be decided at the present stage, since the type and concentration of charge carriers in either system might be different, which can also lead to different conductivities. A detailed comparison of parameters characterizing the dc conductivity values, and in comparison to PEC, is following in the subsequent sections.

4. Analysis of the dc conductivities 4.1 Dc conductivities of PEC The dc conductivities of all PEC were extracted from Nyquist plots of the complex impedance by fitting parameters of a model equivalent circuit to the data. The equivalent circuit always consisted of a parallel connection of an Ohmic resistance and a constant phase element, see Ref. [31]. The same dc values were also obtained by identifying the conductivity values of the low-frequency plateau with the dc conductivity. We always performed two series of conductivity measurements. In the first series, the sample was cooled down to about −90 ◦ C. After that the temperature was increased up to 200 ◦ C in steps of 10 ◦ C. Then the samples were kept at 200 ◦ C for about 3 h and cooled down stepwise by 10 ◦ C to the lowest temperature at which the conductivity could still be determined. We refer to these measurements as “first measurement series”. In a second conductivity measurement series, the sample was first heated to 300 ◦ C and after a prolonged isothermal heat treatment of more than 6 h, the temperature dependence of the complex conductivity was measured by decreasing the temperature stepwise by 10 ◦ C. Arrhenius plots of the ionic conductivity of a PEC with x = 0.70 obtained from both first and second measurement series are shown in Fig. 4. The upper straight line shown in this figure corresponds to the conductivity before annealing the sample at high temperatures and it obeys the Arrhenius law. Strong deviations from Arrhenius behavior start above 25 ◦ C. This temperature is consistent with the DSC data obtained on the first heating and attributed to a loss of water. The loss of water continues until 200 ◦ C is reached. The conductivity observed during cooling down from 200 ◦ C is also Arrhenius-like; however, it is three orders of magnitude lower than the conductivity of the samples before drying. It is interesting to observe that in contrast to the huge discrepancy of the ionic conductivity values of “humid” and “dried” samples, the activation enthalpy of the conductivity is almost identical. The latter finding is in contrast to analogue PEC materials in which LiPSS or CsPSS is used instead of NaPSS. Here, the activation enthalpy is smaller

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Fig. 4. First (full circles) and second (open squares) measurement series of the dc conductivity as a function of reciprocal temperature for PEC with x = 0.70 [31].

in the first heating measurement series than in the first cooling down process. In following runs, the activation enthalpy then remains constant within experimental error. These results will be published in more detail in a forthcoming paper [39]. In the following we will focus on dc conductivities determined on dried PEC materials which were always obtained after annealing at 300 ◦ C. The results are presented in Fig. 5. The dc conductivities of all investigated PEC show Arrhenius behavior, see also Ref. [31]. It is also remarkable that in those PEC for which measurements could be performed at temperatures above and below the calorimetric glass transition temperature, there is no change in the temperature dependence of σ dc T when passing through the glass transition. This implies that PEC materials are “strong” glasses. In all other PEC of Fig. 5, where Tg is not indicated by an arrow, the presented data refer to the status above Tg . Figure 5 shows a strong dependence of both, the absolute conductivity values at a given temperature as well as the activation enthalpy derived of the slope from the presented lines. The values of the activation enthalpy ΔH dc of the dc conductivity as well as of the pre-exponential factors of the Arrhenius law: σ dc · T = A dc · exp[−ΔH dc/(k B T )] are summarized in Table III of Ref. [31]. Figure 6 shows the activation enthalpies along with the isothermal dc conductivity at 563 K for all investigated PEC as a function of composition. We see that from x = 0.30 up to roughly x = 0.55 the dc conductivity increases and the activation enthalpy decreases almost linearly. Above x = 0.55 the increase in NaPSS content is then accompanied by a much stronger increase of σ dc and decrease of ΔH dc, respectively. The strong increase of σ dc with x is reminiscent

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Fig. 5. Arrhenius plot of the ionic dc conductivities of various x NaPSS · (1 − x) PDADMAC dried PEC samples [31].

Fig. 6. Activation enthalpy (full circles, right Y -axis) and the dc conductivity (full squares, left Y -axis) measured at 290 ◦ C as a function of composition.

of inorganic glasses in which the dc conductivity increases with the number density of mobile ions in a power-law fashion [40–43]. The latter finding is attributed to a strong increase in ion mobility [40]. The dependence of the dc conductivities and the parameters derived thereof show that there are distinct differences between the ion dynamics in PDADMAC- and PSS-rich PEC, respectively. In both cases, however, the Arrhenius dependence of σ dc · T clearly shows that the ion dynamics in PEC materials is determined by thermally activated hopping processes of the mobile ions. The fact that the isothermal dc conductivity increases continuously with NaPSS content indicates that the chloride ions do not dominate the ion trans-

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port, even in PEC materials with an excess of polycations and thus Cl− as the most abundant mobile charge carrier. Otherwise, σ dc as a function of x should run through a minimum which is obviously not experimentally seen. The conductivity measured for PEC with x ≤ 0.50 could therefore be either due to residual Na+ ions or protons. The fact that the activation enthalpies in PEC with different types of small cations, viz. Li+ , Na+ , Cs+ , show different activation enthalpies in the dried state, points into the direction of transport by residual alkali ions rather than by protons. The question why the mobility of Na+ ions in PEC is very large compared to that of Cl− ions can partly be explained by the difference in size of the respective ions. However, since the difference in mobility appears to be orders of magnitude, it is likely that structural differences of the PEC play a large role as well: From the asymmetric behavior of the osmotic coefficient of hydrated PEC in dependence of mixing ratio, x, it was concluded that polyanion-rich PEC are homogeneous, while polycation-rich PEC undergo a micro-phase separation of neutral from anion-rich regions [34]. This is consistent with our observation of the appearance of the dried powder, see above. If there was no connectivity of anion-rich regions in a microphase separated material, the anion transport would be drastically decreased. In addition to cations or anions, protons or hydronium ions might contribute to the dc conductivity of the non-annealed PEC samples determined in the first heating run, see Fig. 4. The next section is dealing with the question of relevant charge carriers in hydrated polyelectrolyte systems, namely in PEM.

4.2 Dc conductivities of PEM From spectra such as those shown in Fig. 3, the dc conductivity can be extracted in different ways: One is to determine the dc values from Nyquist plots of the complex impedance by fitting an equivalent circuit consisting of an Ohmic resistance and a parallel constant phase element to the data. Another way is extrapolating the conductivity to low frequencies while neglecting the polarization effects according to a fitting procedure using a reference curve. The latter procedure was described earlier [19,44] and yields dc values which are in very good agreement with the Nyquist fit method, as we showed by a direct comparison [36]. Dc conductivities of different polyanion/polycation combinations are shown in Fig. 7. The conductivities are rather low, if compared to classical polymer electrolytes. The largest values are found for PDADMAC/PSS, whereas PAZ+ /PAZ− is less conductive and PAH/PSS has the lowest conductivity. The counter-cation employed in the PSS adsorption does not lead to significant changes of the conductivity. A rather low conductivity of PAH/PSS was already shown by Durstock and Rubner [16].

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Fig. 7. Logarithm of the dc conductivity of different polyelectrolyte multilayers at RH = 74% [36].

A difficulty in the interpretation of the conductivity values is the fact that the number density of the charge carriers is not known, and in different films different fractions of counterions of either type might be incorporated. Therefore the differences in σ dc showing up in Fig. 7 might include different contributions: (i) differences of the content of incorporated counterions, which act as charge carriers; (ii) differences of the type and size of relevant charge carriers, causing different charge carrier mobility; (iii) differences of the state of hydration, which are influencing the charge carrier mobility, or (iv) differences of the polyion network mobility. In the case of PDADMAC, issue (iii) can be expected to play a role and lead to comparatively large dc conductivities, since PDADMAC with its quaternary ammonium ion group is a very hygroscopic material and layers thereof can be expected to be more strongly hydrated compared to other polymers. This might lead to an enhanced conductivity. For the PAZ+ /PAZ− films the backbone flexibility is expected to be large (contribution (iv)), however, just from the data of Fig. 7 no separation of the different contributions influencing σ dc is possible. The question of the contribution of different ionic charge carriers, such as residual cations, residual anions, or protons from the hydration water, is aided by the results found for PEC and described in Sect. 4.1: In PEC the contribution of residual Na+ is dominant over that of Cl− [31]. This holds even in PEC with a large fraction of Cl− anions (and possibly only traces of Na+ ) incorporated. We can thus conclude that even though it might be mainly Cl− counterions that are incorporated into some PEM upon layer formation, they will not dominate the ion transport either. This conclusion holds for dry PEM, but with the assumption that the hydration does not enhance Cl− mobility much

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Fig. 8. Logarithm of the dc conductivity of multilayers of PAZ+ /PAZ− (circles), PAH/LiPSS (squares) and PDADMAC/LiPSS (triangles) as a function of relative humidity. The solid lines result from linear regression [36].

more than Na+ mobility, it holds for hydrated films as well. Thus, the conductivities in Fig. 7 are dominated by the cations or by protons [36]. In order to judge the contribution of the cations, we varied the type of counterion of the polyanion PSS. Figure 7 shows the same conductivity irrespective of the type of alkali counterion involved. This implies that either alkali ion incorporation into the film is negligible, or that ion density and mobility are identical for all alkali ions. However, as argued before, the only reasonable conclusion is that the conductivity is not dominated by incorporated counterions, but by protons or hydronium ions [36]. That being the case, the dependence of the dc conductivity on RH, or rather water content, is very interesting. The water content was not determined here, but in several publications it is described as increasing linearly or almost linearly with humidity [45,46]. Figure 8 shows the increase of log(σ dc ) with RH for some examples of multilayer systems. Remarkably, the increase of log(σ dc ) is linear over the whole range of RH, and this is valid for all PEM systems investigated [36]. Thus, there seems to be a general law of an exponential increase of the dc conductivity with the relative humidity. This is reminiscent of the exponential dependence on temperature found in thermally activated processes and we may conclude that the charge transport in PEM is a hydration-activated process. Though there is a general exponential law of the humidity-dependence of σ dc , not only the absolute values of σ dc , but also the slopes in Fig. 8 vary significantly for different polyion combinations. The slopes were extracted and are displayed in Fig. 9 for the different systems investigated [36]. Assuming that the exponential law of the humidity-dependence of σ dc is indeed a general feature, literature data can be included: In studies where dc conductivities were determined for at least two humidity values, the slope of an assumed exponen-

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Fig. 9. Slopes of log 10 (σdc ) vs. RH for different polyelectrolyte multilayer systems of our studies (full symbols) [36]. Open symbols: Slopes derived from interpolation of conductivities published by other authors [16,17,24], the open squares present lower limits of the slope, see text. Black frames: PEM made from the same polyion combinations.

tial dependence can be estimated from the published data points [16,17,24]. We note that the open squares represent not exact values, but lower limits of the slope, since in this case only an upper limit of the dry state conductivity was available [16]. The overview in Fig. 9 clearly shows that the slope values fall into different categories, emphasized by the black frames: Each combination of polyions has its typical value of a slope. The value can be interpreted as the sensitivity of the respective polymer matrix to hydration. As mentioned before, layers containing PDADMAC can be expected to be rather hygroscopic due to the hygroscopic nature of the PDADMAC component. Indeed, here we find the largest value of a; i.e. the conductivity is most sensitive to hydration. A similar or even larger sensitivity is only found for Nafion-containing PEM, see data point by Daiko [24]. Remarkable is also, that for all PAH/PSS layer systems, irrespective of the type of alkali counterion, and of preparation conditions, the slope is always the same, see open and filled squares in Fig. 9. Thus, only the polymer backbone and its hydration properties control the sensitivity of σ dc to humidity. This finding is consistent with the discussion above about the nature of the dominating charge carriers: The mobility of protons as dominant charge carriers in the conduction process will increase with the amount of water in the films, and the increase of mobility is steeper for a material with a stronger hydration behavior.

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5. Scaling properties 5.1 General aspects of scaling The conductivity spectra of many materials possess a T -independent shape and can, therefore, be superimposed to a so called “mastercurve” [47,48]. This is usually called “time-temperature-superposition-principle” (TTSP). The validity of the TTSP for the real part of the complex conductivity σ  /σ dc can be expressed by a function σ  /σ dc = F(ν/ν 0 ). ν 0 is an individual scaling parameter for each conductivity isotherm. The scaling function, F, is independent of temperature. An appropriate choice of ν 0 for each curve is necessary to superimpose spectra measured at different T to a master curve. A straightforward approach is to assign ν 0 to the onset frequency of the conductivity dispersion, ν * , which can be defined in various manners. In almost all studies the σ  (ν * ) = 2σ dc definition proposed by Kahnt is used [49]. The special case of Summerfield scaling [50,51] is fulfilled if σ dc T and ν * are proportional to each other. Connecting the onset frequencies in a plot of log(σ  T ) vs. log(ν) yields a straight line of slope 1. Summerfield scaling can be expressed by σ  (ν)/σ dc = F (ν/(σ dc T )). This scaling works for many ion-conducting materials, however it fails for systems like single cation glasses with high electronic polarizabilities [52,53] and mixed-alkali borate glasses [54,55]. Whereas for mixed alkali glasses the actual shape of the conductivity spectra changes with temperature [54,55], the shape remains constant for the other materials of refs. [52,53]. Instead, in the latter cases the simple proportionality between σ dc T and ν * is not valid anymore. From Fig. 1 of Ref. [53] published by Murugavel and Roling it is obvious that a straight line connecting the onset frequencies would have a slope that exceeds one. In other words, σ dc T increases more strongly with T than ν * does. One way to describe this type of deviation from Summerfield scaling is to include an additional scaling factor for the frequency scale that itself depends on temperature: σ  (ν)/σ dc = F (ν/(σ dc T ) f(T )). With f(T ) included, all spectra taken at different temperatures can then be scaled on a master curve. The scaling relation   σ  (ν) ν (1) =F Tα σ dc σ dc T had been reported on the basis of theoretical calculations by Baranovskii and Cordes [56]. In Refs. [52,53] Roling et al. showed that for their glasses f(T ) can be identified with T α , yielding the same scaling function. But whereas Baranovskii and Cordes [56] and also other authors doing simulation work [57, 58] reported on negative α-values, the experimentally determined α-values by Roling et al. were always positive [52,53]. A positive α-value means that the T -dependence of ν * is less pronounced than that of σ dc · T and it implies that the characteristic mean square displacement of the mobile ions at a time given by t * = 1/(2πν * ), r 2 (t * ), increases with temperature. There are two

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possible reasons for this [52,53]. Either the number density of mobile ions increases or the number of available pathways for the ions decreases with increasing temperature. In the latter case, the ions have to travel larger distances until they can finally leave their initial site, resulting in an increase of the characteristic mean square displacement with T . Although not outlined in the references cited before, one should keep in mind that a possible proportionality of the scaling factor f(T ) to T α as implied by the Baranovskii and Cordes scaling function can hardly be distinguished from an exponential type of dependence f(T ) ∝ exp(−1/T ), if the temperature range under investigation is small and not varying by decades. The latter proportionality would be consistent with an assumption that the formation of additional charge carriers contributing to the conductivity was thermally activated in an Arrhenius fashion. This could explain why the number density of mobile ions increases with temperature yielding a T -dependence of the conductivity spectra where the slope of the straight line connecting the onset frequencies is larger than one.

5.2 Scaling properties of PEC spectra We will now discuss the scaling properties of the PEC conductivity spectra with respect to their temperature dependence. On the one hand we will show that the time–temperature superposition principle (TTSP) is also valid for each PEC composition indicating that the shape of the conductivity spectra does not

Fig. 10. Temperature dependence of σdc · T (full symbols referring to left Y -axis) and ν* (open symbols referring to right Y -axis). Note that the left Y -axis is shifted by a factor 10−10 relative to the right Y -axis.

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Fig. 11. Baranovskii and Cordes scaling a) with α = 1.58 for x = 0.70 and b) with α = −1.68 for x = 0.35.

change with temperature for a given composition. In addition, we will discuss the scaling properties as a function of composition. Some of the results are already published in Ref. [32]. As marked in Fig. 1 by the star symbols, we have determined for each PEC composition the frequencies of the dispersion onset, ν * , using the relation: σ  (ν * ) = 2σ dc . The temperature dependence of the onset frequencies ν * for four compositions is shown in Fig. 10 (open symbols and right Y -axis), where it is compared with the respective temperature dependence of σ dc T (full symbols and left Y -axis). In each case we find the Arrhenius law being overall valid for the considered quantity. Very slight deviations are only seen at high temperatures. If Summerfield scaling with σ dc T ∝ ν * was valid, the straight lines of log(σ dc T ) and of log(ν * ) vs. 1/T of a given composition should be parallel. This is obviously not the case for PEC with x = 0.35 nor for those with x = 0.60 and 0.70. For the latter two PEC we find that log(σ dc · T ) decreases stronger with increasing 1/T than log(ν * ) does. This is also true for all other compositions with x > 0.50. For x = 0.50 the deviation in the 1/T -dependence between log(σ dc T ) and of log(ν * ) is not much pronounced, but quantitative analysis shows that log(σ dc T ) decreases a bit stronger with increasing 1/T than log(ν * ). An opposite trend is seen for the complex with x = 0.35. Here, the decrease of log(σ dc T ) with increasing 1/T is less pronounced than in log(ν * ). The temperature dependence of the onset frequencies ν * for all complex compositions can be described by the Baranovskii–Cordes scaling function which means that ν∗ ∝ σ dc TT −α = σ dc T 1−α , but the sign and the absolute values of α differ strongly with composition. Figure 11 shows conductivity spectra of PEC with x = 0.70 (a) and 0.35 (b) respectively, which have been scaled according to Baranovskii and Cordes,

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Fig. 12. α-exponent (full circles, right y-axis) and σdc (T ) at 563 K (solid squares left y-axis) as a function of PSS content x. The dashed lines are for eye guidance.

but with different α-values. The spectra for x = 0.70 can, indeed, be nicely superimposed to a master curve, see Fig. 11a); the time–temperature superposition principle is very well fulfilled. For x = 0.35 the superimposed spectra of Fig. 11b) show slight deviations at very high normalized frequencies, but the spectral shape is independent of T for several decades on the frequency scale, even far above the onset of dispersion. From these and other results not presented here, we conclude, that the TTSP is valid in the complete measured frequency/temperature range for x > 0.50 and in a limited frequency range for x < 0.50. However, σ dc T and ν * have distinct temperature dependences. The α-exponent values were obtained from linear regression of log(ν * /σ dc ) vs. log(T ) for all investigated PEC and are plotted in Fig. 12. We identify an increasing trend of α from −1.68 to 1.58 if x changes from 0.35 to 0.70. For α = 0, σ  (ν)/σ dc = F (ν/(σ dc T )T α ) is equivalent with Summerfield scaling. Considering the dashed line in Fig. 12, the α = 0 crossover occurs at x ≈ 0.47. Thus, a sample close to the 1 : 1 PSS/PDADMAC composition should obey best to Summerfield scaling. Here, α changes its sign from negative values for all PDADMAC-rich samples to positive values for all PSS-rich and the PEC with x = 0.50 samples. Positive α-values correspond with the composition regime where the dc conductivity increases very strongly with x, viz. the NaPSS-rich regime. Following the arguments at the beginning of chapter 5 a positive α-value implies that either the number density of mobile ions increases or the number of available pathways for the ions decreases with increasing temperature. This is the case for PSS-rich PEC with x ≥ 0.50. Like for the concentration dependence of the dc conductivity, we find again a similarity between PSS-rich PEC and some inorganic glasses where Roling et al. reported on positive α-exponent as well. The PDADAMAC-rich PEC behave differently. In analogy to the arguments given above, a negative α-value means that either the number density

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of mobile ions decreases or the number of available pathways for the ions increases with increasing temperature. Summerfield scaling of conductivity spectra retained for α = 0 can therefore be considered as a special case where neither the number density of mobile ions nor the number of available ion pathways depend on temperature, or both effects are present, but cancel.

6. Modeling Ion movements in disordered ion-conducting materials occur via hopping processes in which ions leave their sites and jump into neighboring vacant sites. Conductivity spectra probe the ion dynamics on different time scale, the latter being given by the inverse of the angular frequency ω [59–61]. Therefore, wide range conductivity spectra probe the transition from elementary steps of the ionic movement to macroscopic transport. According to linear response theory [62], the complex conductivity σ(ω) is proportional to the Fourier transˆ form of the current density autocorrelation function i(0) · i(t). If there is only one type of mobile charge carriers and if cross correlations between movements of different ions can be neglected (single particle approximation) the complex conductivity can be simply expressed in terms of the velocity autocorrelation function v(0) · v(t): ∞ N q2 v(0) · v(t) exp(−iωt) dt (2) σ(ω) = ˆ 3 V kB T 0

In Eq. (2), k B stands for the Boltzmann constant and T is the temperature. V is the volume of the sample, and N is the number of ions with charge q; t and ω stand for the time and the angular frequency, respectively. In fact, Monte Carlo simulations by Maass et al. [63] have shown that the overall shape of conductivity spectra is indeed well-described, if only correlations between jumps of a single ion are considered. The simplest approach is to assume that ion movements can be considered as random, see for example Ref. [61]. Here, the jump of an ion moving into a forward direction is only correlated to itself and the velocity autocorrelation function is proportional to a Dirac-Delta-function at t = 0. In this case, the complex conductivity obtained by Fourier transformation is constant at all frequencies; the real part of the conductivity shows no dispersion. σ  (ω) can be therefore identified with the dc conductivity at all frequencies. By contrast, conductivity spectra of most ion-conducting materials show that σ  (ω) varies with frequency. At low frequencies, the dc plateau is observed, but at higher frequencies σ  (ω) is found to increase with frequency. The corresponding velocity autocorrelation function consists of a Dirac-Delta function at t = 0, but also of a negative contribution which approaches zero at long times. The negative contribution indicates that ion movements are not random, but strongly influenced by backward correlations [61].

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There are many different models which describe the overall shape of the frequency-dependent conductivity which cannot all be summarized here. One successful approach uses computer simulations for treating the ion motions in static energy landscapes within a random barrier model [57,58,64] The random barrier model takes disorder into account by choosing an energy landscape with randomly distributed barriers of different heights. In most of these simulations, Coulomb interactions between the moving ions are neglected. In Ref. [57], however, Roling discusses the influence of Coulomb interactions on the scaling properties of conductivity spectra. Other successful models describing the overall shape of frequencydependent conductivity spectra are the counter-ion model and the dipolar model of Dieterich and co-workers [65–67]. Here, Monte Carlo simulations are used to describe the hopping motions of small ions. All Coulomb interactions between the mobile ions and their immobile counterions are taken into account. Disorder is introduced by randomly placing fixed counterions into the center of cubes forming a lattice. Whereas in the counterion model long range transport of the mobile cations is possible, the dipolar model only considers local motions of the mobile ions in the vicinity of their immobile neighboring counterions (dipols). Interactions between fluctuating dipols are taken into account. The MIGRATION model by Funke and co-workers [61,68,69] considers microscopic ion hopping in an energy landscape that changes with time. A distribution of barriers in the energy landscape is not taken into account. Instead, the model focuses on the time dependence of the energy landscape occurring when mobile ions leave their site. In an extended version of the MIGRATION model, only interacting dipoles are considered. The MIGRATION model has one big advantage over the other models for it yields a set of analytic equations with fitting parameters. These equations allow for calculating conductivity spectra which are in agreement with experimental spectra where the exact spectral shape differs slightly from system to system. We have, therefore, employed the MIGRATION model for performing fits to our spectra and will describe this model in some more detail. The acronym MIGRATION stands for MIsmatch Generated Relaxation for the Accommodation and Transport of IONs. The model considers in a detailed way backward correlations between the successive jumps of a mobile ion. These backward correlations give rise to the negative part in the velocity autocorrelation function and thereby to dispersion of the conductivity. The model is applicable for ion conductors such as polymers, inorganic glasses or crystals [61,69]. The central idea of this model is that because of their mutual repulsive Coulomb interactions equally charged mobile ions tend to stay apart from each other. If a mobile ion leaves its site by jumping into a vacant neighboring site, mismatch is created. The system then tends to reduce the mismatch, which can be either done by a correlated backward hop (“single particle route”) of

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the ion itself or by a rearrangement of its neighboring ions (“many particle route”). In the first case, the previous forward jump of the ion under consideration is unsuccessful, whereas in the second case the ion has successfully moved to a new site. Successful hops are the basis for long range ion transport. The rate of relaxation on the “many-particle route” is related to single-particle functions such as the velocity autocorrelation function from which the complex conductivity can be derived via Eq. (2). Local mismatch is, however, not only reduced by the rearrangement of the neighboring ions, but at the same time also progressively shielded. An empirical parameter K is introduced to quantify the time dependence of the shielding effect. More details are given in references [59,60,69]. The parameter K turns out to modify the shape of the conductivity spectra, increasing values of K resulting in a more gradual onset of the dispersion. In most single cation glasses and also in many crystals the value of K is found to be close to 2.0. The parameter K appears to be related to the effective number density of mobile ions. The smaller the number density, the higher is the value of K . Accordingly, materials with small concentrations of mobile cations are reproduced by model spectra with larger K values than those with high cation contents. Large values of K have also been reported for mixed cation glasses and for materials with a low dimensionality of the pathways for the ion transport [61,70].

6.1 Modeling of PEC spectra In the following we analyze the shape of the PEC conductivity spectra in order to shed light on the underlying ion dynamics. We will first focus on the NaPSSrich compositions with x > 0.50 where a phase separation can be excluded and we will try to analyze the ion transport mechanism on a microscopic scale by looking at the shape of the conductivity spectra. The shown spectra are representative for all compositions with x > 0.50, see also Fig. 2. Figure 13 shows one conductivity spectrum of 0.60 NaPSS · 0.40 PDADMAC taken at 393 K. The dashed line was obtained on the basis of the MIGRATION concept with the model parameter K = 2.4, see also Ref. [44]. Whereas the model curve describes the experimental spectrum very well at low and at high experimental frequencies, respectively, one sees variations in the intermediate frequency range. These deviations remain, even if the value of the parameter K is varied. The deviations between the MIGRATION curves which result from correlated-forward backward hopping motions of the ions and the measured curves, indicate that additional movements of charged particles or groups do contribute to the conductivity in the dispersive regime. The total spectrum displayed by the circles, can, however, be described very well, if an additional contribution to the conductivity is considered, see the line with small symbols in Fig. 13. This line was obtained by the following equation:

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Fig. 13. Experimental conductivity spectrum (open symbols) of PEC with x = 0.60 taken at 393 K. The dashed line shows a model curve from the MIGRATION concept (K = 2.4). The solid line results from a superposition of the MIGRATION curve and the spectrum represented by the line with small symbols which results from localized hopping, see text.   σ loc (ω) = σ loc (∞) · [1 + (ωt1 )−1 ]−q

with q > 1

(3)

the exponent being 1.7. The same type of equation has been reported for describing the high-frequency conductivity spectra of ion-conducting glasses [71–74]. The index “loc” characterizes localized motions. σ loc (∞) is a high-frequency-plateau value and t1 marks the crossover from the q-powerlaw dependence into the plateau σ loc (∞). In the literature spectra, the reported exponents were ranging from 1.1 to 1.3, being closer to 1 than to 2. Nevertheless we can transfer the concept of localized ionic motions proposed in Refs. [71,73–75] which is somewhere between the two scenarios of a Debye process on the one hand and localized hopping of interacting particles leading to a nearly-constant-loss-behavior (NCL) on the other hand. An exponent of 2 in Eq. (3) would correspond to a Debye-type process, where a non-interacting dipole moves locally. In such a Debye-case one can envisage an ion hopping in a double minimum potential which it cannot leave, even at long times. The environment is rigid, the potential energy landscape does therefore not change with time. The real part of the conductivity cor (ω) ∝ ω2 frequency dependence responding to a Debye process shows a σ Debye at sufficiently low frequencies and then turns into a high-frequency plateau regime. The other scenario which can be considered is that many dipoles strongly interact with each other. It was independently shown by Funke et al. within an extended version of the MIGRATION concept [61,69] and by Dieterich et al. within their dipolar version of the counterion model [65,66,76] that such processes lead to the well-known “nearly constant loss (NCL) behavior”. The NCL-behavior implies that the imaginary part of the permittivity becomes  (ω) ∝ ω depenalmost independent of frequency, which corresponds to a σ NCL

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Fig. 14. Experimental conductivity spectra (open symbols) of PEC with x = 0.60 mol % taken at different temperatures. The solid lines result from a fit described in the text.

dence in the real part of the conductivity. Conductivity spectra obtained on the basis of the two models given above, indeed show the proportionality of σ  ∝ ν over wide ranges in frequency. At very low-frequencies the model spectra bend over into a ω2 dependence, at high frequencies into a high-frequency plateau. As shown in Ref. [71] power-laws with q > 1 imply that the corresponding mean square displacement of the mobile ions will always merge into a plateau regime at sufficiently long times, indicating that the motion under consideration is strictly localized. In contrast to a pure Debye-type process, however, the environment of the considered ion is not completely frozen. If an ion performs a jump within its double minimum potential, neighboring ions will react to this hop and move very slightly, inducing a time-dependence of the potential energy. So there is some interaction between the ion hopping locally and its environment, but the interaction is not as pronounced as in the dipolar model of Dieterich [65,76] or in the extended version of the MIGRATION model of Funke [61,69,77]. The presence of a contribution as described by Eq. (3) in our PEC materials implies that localized motions of ions in not completely rigid environments have to be considered in PEC materials as well. Figure 14 shows different conductivity isotherms which can be all well described by a superposition of a MIGRATION-type curve and another curve resulting from Eq. (3). The MIGRATION curve represents ion transport which involves correlated forward-backward ion hopping sequences that lead to longrange transport at sufficiently long times. The other contribution characterized by the ωq frequency dependence which – in contrast to the PEM systems discussed in the following section – merges into a plateau regime within our frequency window, represents localized ionic movements. The fact that all parts of the conductivity isotherms (including the “shoulder”) follow the same scaling relation implies that both type of processes, potentially successful and localized hops probably involve the same kinds of ions, viz. the Na+ ions.

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Fig. 15. Spectra of PAZ+ /PAZ at different RH [36]. The symbols represent experimental spectra. The curved dashed lines were obtained on the basis of the MIGRATION concept. Superposition of these model spectra with an additional contribution represented by the straight line yields spectra which are in very good agreement with the experimental results (solid lines).

Spectra of x ≤ 0.50 can be fitted in a similar way. The component describing the localized motions of charged particles is still necessary to describe the spectra, but its contribution to the total spectra is much less pronounced than in the case of PSS-rich PEC, see also Fig. 2.

6.2 Modeling of PEM spectra This section deals with the shape of the PEM spectra which has similarities, but also differences from the spectra obtained for PEC. Figure 15 shows PEM spectra of PAZ+ /PAZ at different RH as symbols. The dotted lines are spectra whose shape is typical for conductivity spectra of many ion-conducting materials including polymers, glasses and crystals [61,68,69]. This spectral shape is not only found experimentally, but can be also very well reproduced by the MIGRATION concept using K = 2.0. From Fig. 15 it is obvious that the shape of the PEM-spectra agrees very well with the model spectra at sufficiently low frequencies (low, but well above the polarization, regime), but that there are strong deviations at higher frequencies. In the experimentally given frequency window, these deviations become more important the lower the relative humidity is. In order to shed light on the origin of the additional contribution to the conductivity, we determined the difference between our experimental and the MIGRATION spectra. It turns out that the extra contribution to the PEM conductivity spectra does not depend on humidity, but shows a power-law frequency dependence σ  (ν) ∝ ν q with an exponent q larger than 1, but smaller than 2. In PAZ+ /PAZ− PEM, the exponent is 1.6, in other system like e.g. PAH/PSS 1.7.

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We may conclude that the origin of the σ  (ν) ∝ ν q conductivity dependence with q > 1 originates from local jumps of the mobile ions in the vicinity of the counter charges localized on the polyions. This is reminiscent of the localized motions seen in the PEC systems, however, here we see no bending over into a high-frequency plateau. In addition, the localized movements are probed at higher frequencies than in the PEC case. Nevertheless, we can state, that also in PEM localized movements of molecules or ions play an important role. These movements could be due to localized movements of ions in not completely rigid environments as discussed in context with the PEC materials. Neighboring mobile ions and/or the polyions do react on ionic movements by slightly moving as well. In addition, it might be also possible that we are in fact observing a NCL-type process. NCL-type conductivity spectra show a transition from a ω2 dependence at low frequencies to a ω dependence. Between these two regimes, the slope of log(σ  ) vs. log(ω) can be between 2 and 1. We also cannot exclude that movements, e.g. rotations, of water molecules may contribute to the high-frequency component. But in any case, the high-frequency component is of localized type and does not contribute to the long-range transport and therefore to the dc conductivity. The higher the relative humidity is, the less these local motions contribute to the spectra within our frequency window. Due to the plasticizer effect of absorbed water in the PEM, the mobility of the ions – which are not only mobile on a localized scale, but also on a macroscopic scale – increases with RH. This classical MIGRATION type ionic hopping motion then dominates the PEM conductivity spectra. For the PAZ+ /PAZ− system, the weight of the σ  (ν) ∝ ν q contribution to the total spectra does not change significantly with RH within experimental error. The same is true for NaPSS/PDADMAC. In other PEM films there seems to be a very slight increase of the weight of the contribution of localized movements with RH. In all PEM systems, however, the conductivity contribution resulting from local motions only plays a minor role at high RH, but it is significant at low humidity.

7. Discussion This section focuses on the discussion of the results presented in the previous sections in a wider context. We arrive at conclusive statements on the mechanisms of ion transport in polyelectrolyte multilayers as well as in polyectrolyte complexes.

7.1 Discussion of PEC spectra From the Arrhenius dependence of the dc conductivity it is clear that the ion transport in PEC materials occurs via thermally activated hopping processes as in other solid ion-conducting materials. On the PSS-rich side the Na+ ions are

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governing the ion transport. We observe that σ dc increases (and the dc activation enthalpy decreases) strongly if the NaPSS content increases only slightly. The finding is in agreement with observations in several ion-conducting glasses, where the dc conductivity increases with the number density of mobile ions, NV , in a power-law-fashion. According to σ dc = NV μq, the power-law behavior σ dc ∝ NVγ therefore indicates that the mobility itself increases with a power law: μ ∝ NVγ −1 . In the literature, γ is typically in the order of 10, but depends on temperature. An increase of the ionic mobility is nicely explained with the framework of the dynamic structure model developed by Bunde, Ingram and Maass [40]. The strong increase of the ion mobility is linked to the fact that the number of available sites increases strongly with ion content. Whereas in dilute systems there are few not well connected ion sites and ions have to travel large distances before they find a new appropriate site, there is a connected system of pathways in materials with higher ion content. This interpretation seems to hold for those PEC materials with an excess of NaPSS as well. The concentration dependence of the dc conductivity and the parameters derived thereof lead to the conclusion that even in PEC with an excess of Cl− ions, the conductivity is not governed by these anions. From Figs. 6 and 12 we clearly see that the dc conductivity increases slightly with PSS content. If the Cl− ions were responsible for the ion transport one should observe an increase of σ dc with PDADMAC content, resulting in a minimum in σ dc at around x = 0.50 which is not visible. Instead, there are strong indications that the conductivity is always determined by the alkali cations. In the PDADMAC-rich complexes as well as in PEC with x = 0.50, there are residual Na+ ions which determine the low, but still measurable conductivity. Also the high activation enthalpies for σ dc T measured in the PEC with x ≤ 0.50 show that the content as well as the mobility of the mobile ions must be very low in these materials. The reason why additional Na+ ions could be incorporated into the PDADMAC-rich PEC (x ≤ 0.50) might be linked to different distances between the repeating charges on the polyions. The distance between the positive charges on the PDADMAC polycation is larger than between the negative charges on the PSS polyanion. This allows the formation of “pockets” in the structural arrangement of the polyanion in which Na+ ions can be incorporated. With the assumption that Na+ is the most mobile ionic species one can also explain the increase of the scaling factor α with PSS content. According to Roling’s computer simulations [57], α increases with increasing Coulomb interactions between the mobile ions and therefore also with their concentration. Let us first focus on the PSS-rich side where the Na+ ions will dominate the ionic conductivity. Cl− ions might also be present in the PSS-rich complexes, but their concentration and mobility will be much lower than the mobility of the smaller Na+ ions. As α is positive and increases with x, the Na+ ion number density increases and/or the number of available pathways decreases with temperature and the effect is more pronounced as the PSS content increases. If the number density of mobile ions was thermally activated, then more and

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more Na+ ions would become mobile with increasing T and thus contribute to the long-range transport. This argument is reminiscent of the “weak electrolyte model” put forward by Ravaine and Souquet [78] for explaining the ion transport in inorganic glasses. On the other hand, the number density of available pathways for the ion transport could decrease with temperature if the chain conformation of the polyelectrolyte network changed with temperature. Structural rearrangements and a higher local mobility of chain segments could block some pathways for the Na+ ions leading to an increase of the characteristic mean square displacement. If we turn to the PDADMAC-rich side of the complexes the situation is completely inverted. The scaling with negative α-values here implies that the number density of mobile ions will decrease and/or the number density of accessible pathways will increase with temperature. As it is unrealistic that a temperature increase will diminish the number of mobile ions we conclude that in PEC with x < 0.50, the number of available pathways for the mobile ions actually increases with T as predicted by theory. The effect is more pronounced as the PDADMAC content increases. The remaining question is why PDADMAC-rich and PSS-rich PEC behave differently concerning their temperature dependences of ion transport properties. The reason can be seen in structural differences between the two kinds of PEC-materials. The PSS-rich PEC which have to embed none or only very small amounts of large chloride ions can be considered as homogeneous materials with a high degree of connected pathways for the Na+ ion transport. On the other hand, the matrix of the PDADMAC-rich PEC has to host the large Cl− ions being here the dominant small counterions. This will lead to larger voids in the polyelectrolyte matrix where the Cl− ions reside. References [31,34] point out that there are indications for a phase separation in PEC with x < 0.50. Therefore, in PDADMAC-rich PEC Cl− -rich regions might exist along with regions of complexes which are almost free of chloride, and the pathways for the ion transport of the diluted Na+ ions in PDADMAC-rich PEC will be completely different than in PSS-rich PEC where Cl− will be of very minor importance. In PSS-rich PEC, Na+ ions might find a network of connected pathways, whereas in PDADMAC-rich PEC pathways might be more heterogeneously distributed. In conclusion, a consistent picture about the ion transport arises if we assume that in all PEC always Na+ ions govern the ion transport. In addition, we assume for PDADMAC-rich PEC that – similar to a diluted solution of a weak electrolyte – all Na+ ions are “dissociated” from their counter charges implying that they all contribute to the ion transport. As the number density of the Na+ is low in PDADMAC-rich PEC, the conductivity is very low. With increasing temperature, however, more pathways become accessible for the sodium ions. The higher the Na+ concentration in the PSS-rich samples becomes, the more we have to consider that the number density of mobile charge carriers is thermally activated. The release of more mobile Na+ with increasing T governs the temperature dependence of the onset of conductivity dispersion.

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If we consider the ion dynamics on different time scales by taking into account the fits to the experimental spectra we can envisage the following scenario. At very short times (corresponding to high frequencies in our spectra), the Na+ ions perform correlated forward backward hopping motions on small length scales. These movements are potentially successful at long times and lead to macroscopic transport. In addition to this transport, there are structural regions where ions can only move locally. This could be due to structural “pockets” formed by the polyions in which the ions hop around. It is also possible that a Na+ ion performs local hops around the oxygen atoms of the negatively charged −SO3 group. Charged polyelectrolyte segments might also perform local movements. The fact that localized motions seem to play a minor role in PDADMAC-rich PEC than in NaPSS-rich supports the idea that either Na+ and/or PSS-segments are involved in these movements.

7.2 Discussion of PEM spectra The knowledge of ion conduction mechanisms in PEC helps in interpreting the conduction mechanisms occurring in PEM, which have a less well defined stoichiometry. For example, identification of the type of dominant charge carrier in PEC, in combination with a variation of the alkali counterion species in PEM, helps to unambiguously ascribe ionic conductivity in hydrated PEM to proton conduction. There might be exceptions, for example specific polyelectrolyte combinations, which yield unusually highly extrinsically compensated PEM. There the counterion content and thus its contribution to the overall conductivity might be larger. One might also speculate about the regime of very dry PEM, where incorporated counterions of the polyelectrolyte components might dominate the transport, but for the humidity values RH ≥ 17% employed here, protons are dominating. Even for much lower humidity there is typically a large water content in PEM, for example 3.5 water molecules per pair of monomers at 0% humidity, as extracted from neutron reflectivity data [79,80], such that even in dry films it might be still protons which dominate the transport. In spite of some possible exceptions, the general property of hydrated PEM is thus that charge transport is governed by protons. The finding of proton conduction in PEM is not very surprising in view of the structural similarity of for example PSS, which is employed in many PEM systems, to NAFION, both carrying sulfate groups and hydrophobic moieties. One can speculate whether a local separation of hydrophobic and charged groups, as it is known in NAFION, might also occur in PEM. Due to the disordered structure of PEM, structural studies at this level of detail are difficult. It has, however, been proven by neutron reflectivity studies that – for suitable preparation conditions – hydrophobic interaction plays a major role in the layer-by-layer formation process [9,79]. It is therefore quite possible that hydrophilic pathways are existing in PEM, along which sites for protons are available. Such sites should then be closely related to water sites, and the con-

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duction process might follow the Grotthus mechanism. The size of water voids in fully hydrated PEM was determined to be about 1 nm in diameter [81]. Reducing the water content would then reduce the size of these water voids and reduce proton mobility as well. The exponential dependence of proton mobility on relative humidity, which we found, is indeed a very strong dependence. Over the whole range of humidity from 0 to 100% the water content in PEM varies only by about 30% of the total film weight [45,46]. Thus, the proton mobility increases over orders of magnitude, induced by only small variations of the water content. This is a reasonable finding, since indeed at low degrees of swelling immobile water was found, which does not exchange with hydration water [80]. The mobility of the protons will be closely connected to that of the hydration water. Open questions for further, more detailed studies are, whether the water content is the controlling parameter for conductivity, or in how far structural aspects on the molecular scale might play a role. The size and shape of water voids could have a significant effect on the connectivity of such hydrated pores, and thus on the long range mobility. Analyzing the spectra in terms of different contributions, which were extracted by modeling, see Sect. 6, we find close similarities of PEM and PEC again. Conductivity spectra of both species are well described by the MIGRATION model, only with different K values. Different K values can be due to a difference in the number of effectively mobile charge carriers. Thus, in hydrated PEM the effective charge carrier density appears to be larger (lower K value) than in dry PEC with extrinsic charge compensation. In PEC the charge carrier density is known, and for our compositions (0.30 < x < 0.70) it is less than 0.2 counterions per polyion charge. A proton density larger than that then has to be accommodated in a hydration shell of up to seven water molecules per polyion pair, a number found in neutron reflectivity for hydrated films [82]. Another reason for the difference in the K -value used for describing the PEC, and the PEM-spectra, respectively, could be lying in the connectivity/dimensionality of the pathways accessible for ion transport. Deviations from the MIGRATION model are quite different in PEM compared to those in PEC: Fast motions of the hydration water molecules are likely to give their contribution to the high frequency part of the spectra, in particular since water is a very polar molecule and can be expected to contribute to local polarizations. In PEM these water movements might even be coupled to the proton hopping motions.

8. Outlook In this review we described our previous work on the humidity dependence of PEM impedance spectra and the temperature dependence of PEC impedance spectra. PEC as a model system with a known stoichiometry of polyions

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and small ions could reveal interesting properties, in particular concerning the mobility of the cationic and anionic charge carriers. This helped to interpret the PEM spectra in terms of the relevance of different charge carriers, in spite of the unknown composition of the films. Remaining open questions are whether PEM show the same conductivity behavior in dependence of temperature, for example the same activation energies, or whether activation energies might further help to clarify the polycation/polyanion composition by comparison to PEC activation energies for varying compositions. For this purpose, temperature-dependent conductivity experiments of dry PEM and PEM equilibrated at a constant relative humidity are currently under way. Another interesting aspect is the crossover between cation-dominated conductivity and proton-dominated conductivity that should occur with increasing water content for PEC. Therefore, also humidity-dependent conductivity studies on PEC are currently being performed. The idea is to be able to separate the contributions of protons and other cations, and estimate the relative mobilities of either charge carrier. Such knowledge is relevant for potential applications of PEM either as proton or as Li conductor, in particular in how far conductivity contributions of protonic or cationic charge carriers can be tuned by the polyelectrolyte composition or by external parameters.

Acknowledgement We thank Y. Karatas and H.-D. Wiemhöfer (project A2, SFB 548) for providing the PAZ+ /PAZ− polymers and C. Hofmann for preparing PAZ+ /PAZ− multilayers. We also thank K. Funke (project C2, SFB 548) for many stimulating discussions over the past years. This work is the result of the research program of projects A6 and B6 within the collaborative research center “SFB 458”, funded by the Deutsche Forschungsgemeinschaft.

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Short and Medium Range Order in Ion-Conducting Glasses Studied by Modern Solid State NMR Techniques By Hellmut Eckert∗ Institut für Physikalische Chemie, WWU Münster, Corrensstr. 28–30, 48149 Münster, Germany (Received July 20, 2010; accepted in final form September 10, 2010)

Glass Structure / Solid State NMR Spectroscopy / Structure-Property Relations Glassy ionic conductors have attained considerable importance in the solid state battery technology field. To understand the influence of composition on ionic mobility and transport, information about the local and dynamic environments of the mobile ions in glasses is required. Solid state NMR spectroscopy is one of the most powerful tools for addressing these questions. Our research effort in this area comprises (1) the development of efficient techniques and strategies for the study of disordered materials, (2) the detailed application of optimized measurement and data analysis protocols to different glass systems under systematic compositional variation, and (3) the comprehensive interpretation of the results obtained in the overall literature context. This article attempts to convey a concise overview of our contribution to this field over the period from 2004 to 2010, to describe the most significant insights obtained and to place them into the overall field of Solid State Ionics.

1. Introduction Glassy ion conductors are promising candidates for applications as electrolyte materials for solid state batteries. Owing to their compositional variability, their materials properties can be tailored to specific application demands. For the design of optimized solid state electrolytes, fundamental research in this area concentrates on the inter-relationships between chemical composition, structural organization and ion dynamics. Structural information is also essential for the effort of modeling ionic motion in disordered solids at the atomic * E-mail: [email protected] Z. Phys. Chem. 224 (2010) 1591–1653 © by Oldenbourg Wissenschaftsverlag, München

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level. From the general relationship σ = z Fcu

(1)

the ionic conductivity σ is expected to increase as a function of the concentration of the mobile ions c in the material. In this expression z is the charge, F is Faraday’s constant, and u is the mobility of the ions. Experimental data confirm the σ ∼ c relationship, however, the slopes observed in various glass systems are found to differ substantially from each other [1]. This result suggests that different glass systems present substantial ion mobility differences thereby emphasizing the importance of their local environments. The key question as formulated by a distinguished colleague, namely, “where are the ions and where did they come from?” [2] concerns the local environments of the ions, their interactions with the network, and their overall spatial distribution. Such structural issues must be addressed when discussing any effects of the glass composition on the ionic conductivity, regarding the types and concentrations of both the network former and the mobile ion species, respectively. Compared to crystalline solids, glasses present a much more formidable challenge to structure elucidation. Owing to the lack of long-range periodicity in the glassy state, diffraction techniques are fairly powerless, and the structural concepts typically emerge from the joint interpretation of numerous complementary spectroscopic experiments. For a comprehensive picture glass structure must be discussed on different length scale domains: (1) shortrange order involving only the first atomic coordination spheres (distance region 0.15–0.3 nm), (2) second-nearest neighbor environments (0.3–0.5 nm), (3) nanostructure (0.5–3 nm), (4) mesostructure (3–500 nm) and, finally, microstructure (> 500 nm). Spectroscopic methods such as NMR, Raman, photoelectron and extended X-ray absorption fine structure (EXAFS) provide primarily local information at the atomic level. Thus, EXAFS can provide accurate bond distances and coordination numbers, and NMR spectra contain complementary information about bonding and site geometries, allowing quantitative identification of nearest neighbor atomic species and specific types of coordination polyhedra. While the utility of EXAFS rapidly deteriorates at distances beyond the first coordination sphere, NMR remains the most powerful approach for the study of structural issues on the 0.3 to 3 nm length scale (domains (2) and (3)), particularly in the sub-nanometer region commonly denoted as medium-range structure. The generally accepted structural model for ion-conducting oxide glasses is based on the Zachariasen model [3]. A covalent main group oxide (called network former), establishes an aperiodic framework based on corner-shared oxygen atoms. By incorporation of group- I or II oxides (called network modifiers) oxygen bridges are ruptured and the framework becomes a macro-polyanion, which balances the positive charges of the mobile alkaline or alkaline earth metal cations. Finally, intermediate oxides such as Al2 O 3 , MoO 3 , or TeO 2 ,

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which are not able to form glasses by themselves, can be incorporated in large amounts into multi-component systems and influence their physical properties. While the first coordination spheres of all of the above species are dominated by oxygen, the detailed coordination numbers and symmetries, as well as the types of oxygen atoms (bridging or non-bridging) present must be characterized. Furthermore, in mixed-anion glasses the first coordination sphere of the mobile cations may also contain halide, sulfide and nitride species, which requires detailed analysis by appropriate NMR methods. More sophisticated questions concern spatial proximities and interactions at larger distances, such as (1) the connectivity between different network former species, (2) the distance correlations involving network former and network modifier species, and (3) the overall spatial distribution of the network modifier ions. The most powerful NMR approach to such questions is measurement of internuclear magnetic dipole–dipole couplings, which have a straightforward relationship to inter-nuclear distances. Information is even available beyond the second coordination sphere, regarding, e.g. nanophase segregation and/or macroscopic phase separation effects and crystallization mechanisms. Within the frame of SFB 458, Ionic Motion in Disordered Systems – From Elementary Jumps to Macroscopic Transport – the goal of our project was to develop and optimize quantitative NMR measurement and data analysis techniques, for obtaining new insights into the structural issues described above. During the first half (2000–2004) of the total funding period the work focused on the refinement of advanced solid state NMR methodology and their adaptation to quantitative structural studies of glasses. Detailed examination protocols were developed, which were then applied to individual ion-conducting binary and ternary glass systems during the second half of the total funding period (2004–2009). The current review is organized as follows: after a brief introduction into the fundamental theoretical aspects (Sect. 2), Sect. 3 reviews the key experimental methods and their adaptation to structural studies of glasses, summarizing the state of knowledge prior to the last funding period. This review is essential for the understanding of the results presented in the subsequent Sects. 4–6, which are devoted to the experimental results obtained on different ion-conducting glass systems and their discussion in the appropriate literature context. Section 4 deals with the local environments of the mobile ions and their overall spatial distribution in simple binary network glasses. Section 5 summarizes our NMR results on the structural foundations of the mixed cation effect. Finally, Sect. 6 is concerned with the consequences of network former mixing, the incorporation of intermediate oxides and mixedanion effects upon structure/property correlations in more complex glasses. To validate certain NMR methods on model systems and to provide a suitable comparison with crystalline compounds, we also studied the structural and dynamic surroundings of the mobile ions and their relation to atomic mobility in a number of disordered crystalline ion conductors. These results will be detailed in a separate publication.

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2. Theoretical aspects 2.1 Basic principles of solid state NMR The fundamental theory of solid state NMR has been covered by number of excellent texts from different perspectives [4–7]. Solid state materials contain nuclei carrying spin angular momentum I and hence a magnetic moment μ. Both quantities are related by the expression μ = γI, where γ , the gyromagnetic ratio, is an isotope-specific nuclear constant. Being a microscopic particle, this magnetic moment is subject to the fundamental laws of orientational quantization, implying that one of the components (usually defined as the z-component) of the nuclear spin vector operator adopts only discrete multiples of Planck’s constant h: | I z | = mh

(2)

In this expression m, the orientational quantum number, can adopt values within the range {+I, I − 1, . . ., −I + 1, −I}, where I represents the nuclear spin quantum number. To detect nuclear magnetic moments, an external magnetic field is applied, represented by the magnetic flux density B 0 . The resulting Zeeman interaction, represented by the Zeeman Hamiltonian, H z = −μ z Bo

(3)

lifts the energetic equivalence of these orientational states and causes a splitting into 2I + 1 individual levels with energies E m = −mγh B o

(4)

By application of electromagnetic waves satisfying the Bohr condition ΔE = hω, allowed transitions between adjacent orientational states can be stimulated and observed spectroscopically. To a first approximation, the angular resonance frequency is given by ω = γB o

(5)

This resonance frequency is identical with the angular frequency, the so-called Larmor frequency ω L with which the nuclei precess in the applied magnetic field. Since the values of γ differ greatly for different kinds of nuclei, NMR spectroscopy is intrinsically element selective as, at a given magnetic field strength, different kinds of nuclei have different resonance and precession frequency ranges. With typical values of applied magnetic flux densities on the order of magnitude of 1 Tesla, the frequencies lie in the radio wave region (107 to 109 MHz). The precession frequencies are being measured using pulsed excitation, followed by Fourier transformation of the time domain response signal, the so-called free induction decay.

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2.2 Internal interactions in the solid state While the nuclear precession frequencies are usually dominated by the Zeeman interaction, they are additionally influenced by a number of internal interactions, whose parameters reflect the details of the local structural environment and whose effect on the energy levels can be calculated using standard perturbation theory. The total spin Hamiltonian relevant in solid state NMR can be written as H total = H z + H rf + H cs + H D + H J + H Q

(6)

where H z + H rf describe the interactions of the nuclear magnetic moments with the applied magnetic field and the oscillating magnetic field component associated with the externally applied radiofrequency waves. In contrast, H cs + H D + H J + H Q define the relevant Hamiltonians of distinct types of internal interactions, namely, (1) the magnetic interactions of the nuclei with the surrounding electrons (magnetic or chemical shielding), described by H cs , (2) the internuclear direct and indirect magnetic dipole–dipole interactions, described by H D and H J , respectively, and (3) interactions between the electric quadrupole moments of spin > 1/2 nuclei and the electrostatic field gradients surrounding these nuclei (quadrupolar interaction), described by H Q . In the solid state, all of these interactions are anisotropic and hence cause linebroadening in powdered samples. 2.2.1 Magnetic shielding Magnetic polarization effects induced in the electronic environment of the nuclei under the influence of B o modify the local magnetic fields experienced by the nuclei, hence influencing their precession frequencies. For a theoretical description of these effects, first-order perturbation theory suffices, and the secular Hamiltonian can be written as: ↔

H cs = γI σ B o

(7)

↔

In Eq. (7), σ is a second rank (3 × 3) tensor, which takes into account the anisotropy of the interaction. The tensor adopts diagonal form in a specific coordinate system, denoted as the principal axis system (PAS). The PAS can often be related to local symmetry. The diagonal parameters are usually characterized by the symbols σ 11 , σ 22 , and σ 33 , with the isotropic average being σ iso = 1/3(σ 33 + σ 22 + σ 11 ). Generally the convention |σ 33 − σ iso | ≥ |σ 11 − σ iso | ≥ |σ 22 − σ iso | is followed.

(8)

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For the case of axial symmetry (existence of an n ≥ 3-fold rotation axis) this relation reduces to:     1 2 (9) w L (θ) = γB 0 1 − σ iso − Δσ 3 cos θ − 1 3 Here Δσ = σ 33 − 1/2(σ 11 + σ 22 ) and the angles θ and φ specify the orientation of the PAS relative to the magnetic field direction. Note that for θ = 54.7◦ , the anisotropy vanishes, an effect that is exploited in magic-angle sample spinning. Magnetic shielding tensor values can nowadays be calculated with high precision using suitable ab-initio methods, incorporating electron correlation effects by density functional theory, and even relativistic corrections for heavy atom nuclei [8,9]. Experimentally accessible, however, are not absolute shielding values (which would require comparisons with bare nuclei) but rather chemical shifts measured relative to the precession frequency of a reference compound. δ iso = (ωsample − ωref )/ωref

(10)

2.2.2 Direct and indirect magnetic dipole–dipole coupling Nuclear precession frequencies are further influenced by magnetic dipole– dipole interactions, as the spins feel the magnetic moments of their neighbors. The effect is anisotropic, depending on the orientation of the inter-nuclear distance vector relative to B 0 . As a result, broad spectra are observed in powdered samples. There are two distinct mechanisms, (a) the direct through-space interaction, which merely reflects spatial proximity, and (b) the indirect interaction, where the coupling is transmitted via the polarization of bonding electrons. Both the direct and the indirect terms comprise a homonuclear and a heteronuclear contribution, respectively. The Hamiltonian of the direct through-space interaction is proportional to the inverse cube of the inter-nuclear distance, providing a straightforward connection to geometric structure. The Hamiltonian describing the magnetic dipole–dipole interaction for a homo- or heteronuclear two-spin system is given (in the limit of first-order perturbation theory) by the expressions  μ  γ 2 h 2  3 cos2 θ − 1    0 3 Iˆz2 − Iˆ2 H D homo = − (11) 3 4π r ij 2   μ  γ γ  3 cos2 θ − 1  0 I S 2 ˆz Sˆz . h I H D hetero = − (12) 4π r 3IS 2 The orientational dependence is governed by the term 3 cos2 θ − 1 where θ is the angle between the internuclear distance vector and the magnetic field direction. In systems characterized by multiple-spin interactions, the dipolar couplings are characterized by the average mean square of the local field (the

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second moment), which can be related to internuclear distance distributions by the van Vleck formulae [4] 3  μ 0 2 M2 homo = I(I + 1)γ 4 h 2 r ij−6 (13a) 5 4π i = j 4  μ 0 2 M2 hetero = S(S + 1)γ I2 γ S2 h 2 r −6 (13b) IS . 15 4π s The Hamiltonian describing the indirect magnetic dipole–dipole coupling is given by the tensor product ↔

H J = Iˆ J Sˆ

(14)

↔

where J is the indirect spin–spin coupling tensor. Its magnitude and anisotropy depend to a great extent on the symmetry of the electron distribution in the chemical bond, but also on the sizes of the magnetic moments involved. For a two-spin interaction, the tensor is generally axially symmetric, and can be split into an isotropic component Jiso and an anisotropic component ΔJ. The isotropic component produces peak splittings in MAS-NMR spectra. Heteronuclear spin–spin interactions yield a peak multiplicity of 2n I + 1, where I ↔ is the spin quantum number of the non-observed nuclei. In general, J depends not on distance, but on bonding properties only and provides direct evidence of bond-connectivity. 2.2.3 Nuclear electric quadrupolar interactions For nuclei with spin I > 1/2 the charge distribution is non-spherically symmetric. This asymmetry can be described by an electrical quadrupole moment superimposed upon a sphere containing the nuclear charge. Classical physics predict that such quadrupole moments can interact with inhomogeneous electric fields, i.e. electric field gradients q ij (EFGs) present at the nuclear site. The latter are generated internally by the asymmetric charge distributions associated with atomic coordination and chemical bonding effects. The EFG is a symmetric second-rank tensor, which can be diagonalized in a molecular axis system. The sum of the diagonal elements q xx + q yy + q zz vanishes (Laplace equation) so that the interaction can be described in terms of two parameters, the nuclear electric quadrupolar coupling constant C Q and the asymmetry parameter η = (q xx − q yy )/q zz

(15)

with 0 ≤ η ≤ 1. The quadrupolar interaction competes with the Zeeman interaction for spin alignment and the mathematical treatment is particularly straightforward if

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one interaction dominates. If H z H Q , the influence of the nuclear electric quadrupolar interaction can be calculated from first-order perturbation theory. The relevant interaction Hamiltonian takes the form  1 2 CQ 1 3 cos2 θ − 1 − η Q sin2 θ cos 2φ 3 Iˆz − I(I + 1) . H Q(1) = 2I(2I − 1) 2 2 (16) For each Zeeman state |m the corresponding energy corrections E (1) m are computed from the integrals m|H Q|m . As the energy corrections are proportional to m 2 , the energy level shifts of the | + 1/2 and the | − 1/2 states are identical at each orientation, and the width of the central | + 1/2 ↔ | − 1/2 transition is thus unaffected by the quadrupole interaction. In contrast, it follows from Eq. (16) that the frequencies of the | ± 1/2 ↔ | ± 3/2 transitions are orientationally dependent, giving rise to anisotropically broadened satellite transitions in powdered samples. If the condition H Q H z is not fulfilled but still H Q < H z , the perturbation approach has to be extended to second order. In this case, a more complicated orientational dependence causes anisotropic broadening even of the central transition. Highly characteristic lineshapes are observed, from which C Q and η can be extracted via lineshape simulation routines [10]. Theoretical ab-initio values are available from optimized molecular geometries or crystal structure data using programs such as the WIEN2k code [11]. Thus, the nuclear electric quadrupolar interaction parameters represent important structural validation criteria for unknown materials.

3. Experimental techniques and aspects of their application to glasses Considering all of the interactions discussed above, the total NMR Hamiltonian can be described by H total = H z + H rf + H csaniso + H csiso + H Dhomo + H Dhetero + H Jhomo, iso + H Jhomo, aniso + H Jhetero, iso + H Jhetero, aniso + H Q(1) + H Q(2) .

(17)

In the most general case, the NMR spectrum of a nuclear species in the solid state is influenced by all of these interactions simultaneously, making the analysis of the spectrum in terms of the relevant Hamiltonian parameters virtually impossible. However, a particular strength of solid state NMR lies in the fact that the Hamiltonian of Eq. (17) can be simplified, using special selective averaging techniques. Two such techniques, to be described in the following, include magic angle sample spinning and spin echo decay spectroscopy. In addition, homo- and heteronuclear decoupling methods can be used to eliminate the corresponding direct and indirect dipolar coupling terms. Furthermore, it is possible to combine these averaging techniques with special re-coupling

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methods that are designed to re-introduce specific parts of the spin Hamiltonian. By applying such selective averaging and/or re-coupling techniques, one can look selectively at individual parts of the spin Hamiltonian. Using the potential of multidimensional spectroscopy it is further possible to separate different spin interactions into two (or more) different time (and frequency) domains and to correlate them with each other. In many cases such one-, two- or multidimensional methods are applied in combination with special coherence transfer schemes that rely on magnetic dipole–dipole interactions or dynamic effects (exchange or relaxation) for correlating spins with each other. The design of new pulse sequences is greatly aided by the availability of program packages that calculate the behavior of the spins under the influence of the external and internal interaction Hamiltonians using standard time-dependent perturbation theory. Especially noteworthy is the program package SIMPSON, which enables the simulation of complex NMR experiments under the precise experimental conditions used at the spectrometer [12]. This freeware has proven invaluable in the analysis of complex experiments and profoundly influenced NMR research in many laboratories. The following sections describe a number of useful individual experiments as well as some theoretical and practical aspects of their application to glassy materials.

3.1 Magic-angle sample spinning Isotropic motion of the nuclei in solution affords the simplest form of selective averaging. If the correlation time describing this motion is short compared to the inverse frequency describing the spectral dispersion due to anisotropic interactions, all of the latter are being averaged out, and the NMR Hamiltonian simplifies to H tot = H csiso + H Jhomo, iso + H Jhetero, iso .

(18)

The success of high-resolution liquid state NMR spectroscopy as a structural tool in chemistry is based on this principle. A rather similar situation can be accomplished in polycrystalline solids by rotating the sample about an axis that is inclined by an angle of 54◦ 44 relative to the magnetic field direction (magic angle sample spinning) [13]. This manipulation replaces the actual orientation θ by an orientational average over the rotor period, which simply corresponds to the orientational angle of the rotation axis. As for θ = 54.7◦ the term 3 cos2 θ − 1 equals zero, all of the anisotropic interactions whose frequency dispersions scale with this term are canceled out, provided that the spinning frequency ωr is sufficiently large. In this case, the NMR Hamiltonian simplifies to the expression: H tot = H csiso + H Jhomo, iso + H Jhetero, iso + H Q (1) + H Q (2) .

(19)

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In this expression the terms H Q (1) and H Q (2) denote the relevant parts of the quadrupolar Hamiltonians in first and second order, which are modulated by MAS. Specifically, H Q (1) results in spinning sideband manifolds, whose envelope reflects the shape of the outer Zeeman transitions, while H Q (2) will produce structured lineshapes for the central transition as mentioned above. The center of gravity of the signal is comprised both of the isotropic chemical shift contribution and a second-order quadrupolar shift, the magnitude of which decreases with increasing field strength. Both these contributions can be separated by systematic studies of the field dependence or by the multiple-quantum NMR experiment (see below). While the MAS-NMR technique leads to highly resolved spectra with linewidths on the order of tens of Hz in many crystalline compounds, in the glassy state the continuous variations of local environments often produce wide isotropic chemical shift distributions, which limit the spectroscopic resolution. Nevertheless, different types of local coordination environments can often be resolved in simple model glasses, such as the three- and four-coordinate boron species (B(3) and B(4) ), Al species with different coordination numbers (Al(4) , Al(5) and Al(6) ), as well as the silicate or phosphate species with different numbers of non-bridging oxygen species (Q (n) , where 4 ≥ n ≥ 0). Furthermore, the isotropic chemical shift measured by MAS-NMR is frequently sensitive to the second-nearest neighbor environments.

3.2 Multiple-quantum (MQ)-MAS-NMR For quadrupolar nuclei affected by second-order quadrupolar perturbations the resolution can be improved by combining MAS-NMR with multiple quantum excitation. As shown by Frydman et al. [14], the second-order quadrupolar broadening can be eliminated by correlating the evolution of a |m ↔ | − m

multiple quantum transition with the central |1/2 ↔ | − 1/2 transition during the course of a two-dimensional experiment, with corresponding coherence pathway selection by appropriate phase cycling. For sensitivity reasons, usually the triple-quantum coherence involving the |3/2 ↔ | − 3/2 Zeeman states is selected, even for nuclei with I > 3/2. Figure 1 shows a typical pulse sequence, along with a coherence level diagram. Triple-quantum coherence is excited (among other coherences) by the first short, intense radiofrequency pulse. The subsequent evolution during an incremented time period t1 is terminated by a second intense pulse, which generates zero-quantum coherence. Then a weak detection pulse produces transverse magnetization (single-quantum coherence), which is acquired during the detection period t2 . Double Fourier Transformation with respect to both time domains t1 and t2 , followed by appropriate data manipulation generates a 2-D spectrum, from which the isotropic chemical shift δiso cs and the isotropic quadrupolar shift δ Q can be evaluated. The centers of gravity of the signals observed in the isotropic (F1-) and the anisotropic (F2-) dimensions of

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Fig. 1. Timing and coherence level diagram of a triple-quantum NMR experiment.

this experiment, δ F1 and δ F2 are given by: δ F1 = (34δiso cs − 60δ Q )/9 δ F2 = δiso cs + 3δ Q

(20a) (20b)

where   δ Q = −(SOQE)2 / 30ν 2L [2I(2I − 1)]2 .

(20c)

In this expression ν L is the nuclear Larmor frequency, and the parameter SOQE, the so-called second order quadrupolar effect is given by the relation: SOQE = C Q (1 + η2Q /3)1/2 .

(20d)

Thus, the analysis of the centers of gravity in both dimensions of a TQMAS experiment affords the isotropic chemical shift δiso cs and the parameter SOQE.

3.3 Spin echo decay spectroscopy Another selective averaging technique involves a time reversal scheme, caused by the application of 180◦ pulses (Hahn spin echo), see Fig. 2. Detailed theoretical analysis shows that the 180◦ inversion pulses can reverse the effects of all internal Hamiltonians displaying a linear dependence on the operator Iz . Thus, neither the magnetic shielding nor the heteronuclear dipole–dipole interactions have any influence on the amplitude of the spin echo at time 2t1 . On the other hand, fluctuations or alterations of precession rates tend to reduce the spin echo amplitude. Specifically, homonuclear dipole–dipole and nuclear electric

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Fig. 2. Timing diagram of the spin echo decay method. The spin echo amplitude is measured as a function of total dipolar evolution time 2t1 .

quadrupole couplings, which scale bilinear with the operator Iz , are not being refocused [15]. With respect to its dependence on the evolution time 2t1 the total internal Hamiltonian of Eq. (17) effectively reduces to: H tot = H Dhomo + H Jhomo, iso + H Jhomo, aniso + H Q(1) + H Q(2) .

(21)

As a result, for a spin-1/2 system a plot of normalized echo intensity I/I o as a function of evolution time 2t1 is only affected by homonuclear dipole– dipole interactions. For a multi-spin system, a Gaussian decay is expected (and observed),

M2 homo I (2t1 ) 2 (22) = exp − (2t1 ) I0 2 affording a reliable measurement of the homonuclear van Vleck second moment M2homo . Within the frame of SFB 458, the spin echo technique has been adapted and applied for examining the spatial distribution (homogeneous, random or clustered) of sodium ions in a variety of model glasses. The selective measurement of homonuclear 23 Na-23 Na dipole–dipole couplings entails certain complications arising from the quadrupolar character (I = 3/2) of these nuclei. Theoretical analysis shows that spin echo decay spectroscopy can only yield reliable dipolar coupling information if the π-pulses are applied entirely selectively to the central |1/2 ↔ | − 1/2 transition [16]. If the resonance frequencies between the coupled nuclei are sufficiently similar to allow for spin-exchange the

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Fig. 3. Experimental and calculated M 2 (23 Na-23 Na) values characterizing the 23 Na-23 Na magnetic dipolar interactions in crystalline sodium salts. Squares: data from Ref. [17] (7.04 T); circles: data from Ref. [47] (9.4 T).

dipolar second moment is given by:  μ 2 0 M2 = 0.9562 γ 4h2 r ij−6 . 4π j

(23)

The validity of Eq. (23) was confirmed experimentally for homonuclear 23 Na23 Na dipole–dipole interactions in crystalline solids, for which the M2 values are readily calculable from the known crystal structures [17]. As shown in Fig. 3, the experimental M2 values usually deviate by less than 20% from the theoretically expected ones, making this technique useful for quantifying cation distributions in glasses. For many of the crystalline compounds, and in the case of all Na-containing glasses we have studied, however, the experimental data deviate significantly from the Gaussian behavior suggested by Eq. (22). This behavior can be attributed to the presence of a wide distribution of dipolar second moments, such that at longer evolution times the less strongly coupled spins contribute more strongly to the spin echo amplitude. For this reason, one usually restricts the dipolar analysis to the initial decay regime. In practice a data range of 2t1 < 200 μs is a good compromise between minimizing this systematic error and ensuring a sufficiently widely spaced number of data points. If the deviations are particularly large, M2 values can also be determined as a function of data range, and extrapolated to the origin [17]. Finally, in glasses sub-ambient temperatures (T < 200 K) are generally required to suppress any dynamic contributions to the 23 Na spin echo

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decays. The latter arise from nuclear electric quadrupolar coupling fluctuations caused by sodium ion motion on the micro-to-millisecond timescale and produce exponentially shaped decay curves. Unfortunately, the spin echo decay method is not straightforwardly transferrable to other akali ion nuclei such as 7 Li and 133 Cs. These isotopes have moderately small electric quadrupole moments causing first order quadrupolar splittings that are comparable to the radio frequency excitation window. As a result, no selective irradiation of the central transition is generally possible, and the quadrupolar coupling contributes to the spin echo decay. An elegant way to circumvent this problem is to measure the heteronuclear dipole–dipole coupling between 7 Li and 6 Li instead, in a sample enriched with 6 Li (typically by 90%), using the spin echo double resonance (SEDOR) technique described in the next section.

3.4 Spin echo double resonance (SEDOR) Besides affording the selective measurement of homonuclear magnetic dipole– dipole interactions, spin echo decay spectroscopy can also be used for the selective measurement of heteronuclear dipole–dipole interactions between two different spin species. In this case a 180◦ recoupling pulse is applied to the non-observed spins during the dipolar evolution period [18]. The corresponding pulse sequence, termed SEDOR (spin echo double resonance), compares the spin echo intensity of the observed nuclei as a function of dipolar evo-

Fig. 4. 7 Li{6 Li} SEDOR on 6 Li enriched (95%) lithium carbonate. The Hahn spin echo amplitude of the residual 7 Li nuclei is measured as a function of dipolar evolution time 2t1 in the absence (solid symbols) and the presence (open symbols) of the 6 Li recoupling pulses. The solid curve is calculated with Eq. (24) using the M 2 value calculated with Eq. (13b) from the known internuclear distances in lithium carbonate. The dashed curve represents the best fit to the experimental data. From Ref. [19].

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Fig. 5. 7 Li{6 Li} SEDOR measured on (Li2 O)0.1 -(SiO2 )0.9 glass (top) and (Li2 O)0.1 -(B2 O3 )0.9 glass (bottom). Both glasses have a 6 Li/7 Li isotope ratio of 9 : 1. The Hahn spin echo amplitude of the residual 7 Li nuclei is measured as a function of dipolar evolution time 2t1 in the absence (solid symbols, fitted by the dashed curve) and the presence (open symbols) of the 6 Li recoupling pulses. The dotted curves are fits to the experimental data using Eq. (24). From Ref. [19].

lution time (a) in the absence and (b) in the presence of these recoupling pulses. Experiment (a) produces a decay F(2t1 )/F0 , which is largely governed by homonuclear dipole–dipole interactions, while experiment (b) results in an accelerated decay reflecting the contribution from the heteronuclear dipole– dipole interaction. For multi-spin systems, a Gaussian decay is expected, F(2t1 ) I(2t1 ) = exp −(2t12 M2hetero ) I0 F0

(24)

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the rate of which is given by the heteronuclear second moment M2hetero . As illustrated by Fig. 4 for 90% 6 Li-enriched Li2 CO3 , the experimental data closely match the SEDOR curve predicted by Eq. (24), using the M2 values calculated from the internuclear Li−Li distances of its crystal structure. Within the frame of SFB 458, 7 Li{6 Li} SEDOR has been used to study the spatial distribution of lithium ions in single and mixed alkali glasses [19]. Figure 5 compares the experimental data for (a) a silicate and (b) a borate glass with similar lithium concentrations. Note the substantial difference in the corresponding SEDOR curves, indicating that the dipolar interactions in the silicate glass are significantly stronger than in the borate glass having roughly the same lithium content (see Sect. 4.4).

3.5 Rotational echo double resonance (REDOR) An even more powerful experiment, termed rotational echo double resonance (REDOR) is able to provide site-selective dipolar coupling information under the high-resolution conditions afforded by magic angle spinning. As illustrated in Fig. 6, the dipolar coupling constant oscillates according to the term sin ω r t and is averaged out over the rotor cycle. However, if we invert the sign of the dipolar Hamiltonian by applying a π-pulse to the non-observed I-spins, this average is non-zero; the interaction is re-coupled. Figures 7 and 8 show two typical pulse sequences used for such purposes [20,21]. In both sequences, the re-coupling is accomplished by 180◦ pulse trains. In Fig. 7 these pulse trains are applied to the I-spins, while the

Fig. 6. Principle of the REDOR experiment. Under MAS conditions the heteronuclear dipolar Hamiltonian oscillates sinusoidally with rotor orientation, leading to cancellation of HD upon completion of the rotor cycle (right, top). Sign inversion created by a π-pulse applied to the non-observed I spins interferes with this cancellation (right, bottom).

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Fig. 7. Timing and coherence level diagrams of the REDOR pulse sequence of Schaefer and Gullion [20] preferred if the detected spin species S is quadrupolar.

S-spin signal is detected by a rotor-synchronized Hahn spin echo sequence. In Fig. 8 the pulse trains are applied to the observed-S spins, while the I-spins are subjected only to one central π pulse in the middle of the rotor period. In both cases one measures the normalized difference signal ΔS/S o = (S o − S)/S o in the absence (intensity S o ) and the presence (intensity S) of the recoupling pulses. A REDOR curve is then generated by plotting ΔS/S o as a function of dipolar evolution time NTr , the duration of one rotor period multiplied by the number of rotor cycles. For isolated two-spin-1/2 pairs, this REDOR curve has universal character and can be directly used to extract the internuclear distance [20–22]. In contrast, for larger spin-clusters the REDOR curve depends on the detailed shape and distance geometry of the spin system [23]. In glasses, one generally expects a distribution of spin geometries and magnetic dipole coupling strengths. As previously shown, these problems can be circumvented by limiting the REDOR data analysis to the initial curvature, where ΔS/S o < 0.2 [24,25]. In this limit of short dipolar evolution times, the REDOR curve is found to be

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Fig. 8. Timing and coherence level diagrams of the REDOR pulse sequence of Garbow and Gullion [21] preferred if the detected spin species S = 1/2 and dephasing in the dipolar field of quadrupolar nuclei I > 1/2 is studied.

geometry-independent and can be approximated by:   I 1 1 ΔS 2 = (2m) (NTr )2 M2 2 S0 2I + 1 m=−I π (I + 1)I

(25)

where the average van-Vleck second moment Eq. (13b) can be extracted from a simple parabolic fit of the experimental data. Wide distributions of dipolar coupling strengths manifest themselves in strong deviations from the parabolic shape of this curve. If I > 1/2 nuclei are involved, a number of complications enter which must be accounted for by appropriate procedures. In principle one can distinguish three cases: (a) S > 1/2, I = 1/2, (b) S = 1/2, I > 1/2, and (c) S > 1/2, I > 1/2. Case (a) is handled well by the pulse sequence of Fig. 7 for either twoor multispin systems. Provided the length of the π(S) pulse is well-defined (in the limits either of entirely non-selective or entirely selective excitation of the central transition) any effect of nuclear electric quadrupolar coupling that is

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Fig. 9. Effect of the nuclear electric quadrupolar coupling constant C Q on the universal S{I} REDOR curves of a two-spin system. Simulations assume I = 3/2, S = 1/2, ν(I) 1 = 55 kHz and νr = 15 kHz, and an arbitrarily chosen asymmetry parameter (η = 0.5) of the EFG tensor. For each value of C Q the correct π(I ) pulse length was determined by SIMPSON simulation and used in the calculation of the REDOR curve. Open and closed symbols represent results calculated for different magnitudes of the dipolar coupling constant, proving that the curves are indeed universal. Dotted line: theoretical behavior predicted by Eq. (25). Solid line: theoretical behavior predicted by Eq. (25) in the limit of entirely selective excitation of the central coherence. From Ref. [26].

present will have an equal influence on intensities S o and S, resulting in overall cancellation. For cases (b) and (c) expression (25) remains valid in the limit of zero or very weak nuclear electric quadrupolar interactions (coupling constants C Q < 50 kHz) affecting the I spins. For larger C Q -values, the anisotropic broadening of the |1/2 ↔ |3/2 “satellite transitions” produces large resonance offsets, which reduce the efficiency of the π pulses to cause population inversion. The decisive parameter in this regard is the ratio ν 1 /ν Q , where ν 1 is the nutation frequency of the I spins, which is governed by the amplitude of the rf pulses, and ν Q is the quadrupolar frequency. For sufficiently strong quadrupolar interactions, as ν 1 /ν Q approaches zero, only the central |1/2 ↔ | − 1/2 coherences will be affected. In this limiting case only those S-spins that are coupled to I-nuclei in Zeeman states with |m I | = 1/2 are expected to yield a REDOR response, and the pulse sequence of Fig. 8 is the method of choice. Figure 9 shows a typical set of S = 1/2{I = 3/2} REDOR curves calculated for systematic variation of the nuclear electric quadrupolar coupling constant C Q of the spin species I [26]. Across the entire range of C Q -values, the REDOR curves calculated for dipolar coupling constants of different magnitudes (D = 200 Hz, and D = 600 Hz, open, and filled symbols, respectively) are found to be super-imposable, indicating that they represent universal curves [26].

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Fig. 10. Procedure for quantitative analysis of REDOR data on S{I > 1/2} nuclei (see text). From Ref. [26].

Figure 9 shows that for small C Q values, the two-component behavior predicted by Eq. (25) is readily apparent. The REDOR curve consists of two segments: the shallow part belongs to those S-spins that are coupled to the I-spins in the | ± 1/2 Zeeman states, while the steeper part belongs to those S-spins that are coupled to the I-spins in the | ± 3/2 Zeeman states. With increasing magnitude of C Q , the REDOR response is successively attenuated, reflecting the fact that the spins in the outer | ± 3/2 Zeeman states are less and less affected by the π(I) pulses. We can then introduce a phenomenological efficiency factor f 1 (0 < f 1 < 1) which takes into account the extent to which the dipolar coupling of S spins to I nuclei in their outer Zeeman states influences the REDOR curve, leading to: 1 ΔS = (2 + 18 f 1 )(NTr )2 M2 . S0 15π 2

(26)

The whole data analysis procedure can then be summarized as follows (see Fig. 10) [26]. Based on a nuclear electric quadrupolar coupling constant known from experiment, a REDOR curve is computed, using the precise experimental conditions of the REDOR experiment done. This simulation curve is fitted to a parabola, described by Eq. (26), resulting in the appropriate f 1 value, which is then applicable for the analysis of the experimental data set. Using the f 1 value determined in this fashion, the experimental data set is fitted to Eq. (26), resulting in an experimental M2 value characterizing the heteronuclear dipolar interaction. This value can then be compared with a second moment calculation from the van Vleck formula for testing hypothetical structural scenarios.

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Fig. 11. 11 B{31 P} REDOR results on (a) crystalline BPO4 and (b) a silver borophosphate glass of composition 0.5(Ag2 O)-[0.5(B2 O3 )0.5 (P2 O5 )0.5 ]. Red and blue curves show the data for threeand four-coordinated boron atoms. From Ref. [27].

Case (c) (S > 1/2, I > 1/2) remains seemingly intractable for quantitative applications. The chief problem in applying the pulse sequence of Fig. 8 arises from the nuclear electric quadrupolar interactions affecting the S spins, which interfere with the ability of the π-pulse trains to refocus the transverse magnetization. In contrast, the pulse sequence of Fig. 7 is able to produce experimental REDOR curves, which are, however, difficult to interpret quantitatively as they

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Fig. 12. Two-dimensional homonuclear J-resolved spectroscopy. Following the preparation by a π/2 pulse, evolution takes place under the influence of homonuclear J-coupling during the evolution time t1 . Chemical shift and heteronuclear J-coupling evolution are refocused by the central π pulse. The evolution is stopped by the second π/2 pulse, while the third π/2 pulse is used for detection. During the acquisition period t2 the regular MAS Hamiltonian (Eq. 19) is effective.

show a pronounced dependence on spinning frequency [26]. As a best-effort solution, one may resort to sample-to-sample comparisons on a relative basis and the use of empirical calibration procedures with closely related model compounds run under identical experimental conditions in this case. Validation of the above REDOR approaches on crystalline model compounds is desirable for any of the cases (a) through (c). Figure 11a shows experimental 11 B{31 P} REDOR data on the crystalline model compound BPO 4 . The data are found to be in excellent agreement with the simulated curve, where the dephasing of the 11 B nuclei in the dipolar field of four 31 P nuclei, arranged in a regular tetrahedral geometry at the distance of 277 pm has been calculated. Figure 11b shows a useful application of 11 B{31 P} REDOR to a silver borophosphate glass [28]. In this glass, both three- and fourfold boron atoms are found in the network, and can be well-differentiated by 11 B MAS-NMR. However, as illustrated in Fig. 11, only the four-coordinated boron atoms show a sizeable REDOR effect, indicating a next-nearest neighbor relationship. In contrast no 11 B-O-31 P connectivities are present for the trigonal boron species.

3.6 Homonuclear J-resolved spectroscopy The technique of homonuclear J-resolved spectroscopy combines the two types of selective averaging afforded by MAS and the Hahn spin echo [28,29], in a two-dimensional experiment. In glassy solids, the MAS-NMR peak width is often dominated by distributions of isotropic chemical shifts, which makes it generally impossible to resolve peak splitting produced by indirect spin–spin interactions in directly acquired spectra. This problem is solved by conducting a Hahn spin echo in the indirect (t1 ) dimension of a two-dimensional NMR experiment (see Fig. 12). At the end of each evolution period t1 the inhomogeneous broadening is refocused, so that the amplitude and phase of the signal are exclusively governed by the strength of the isotropic J-coupling. Thus, the corresponding signal observed after Fourier-transforming the data acquired along the time domain t1 allows a selective measurement of the homonuclear J-interaction.

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3.7 Crosspolarization Cross-polarization uses the magnetic dipole–dipole couplings for the transfer of polarization between two spin systems I and S [30]. In general, transverse spin magnetization of the abundant nuclei (species I) is created by a 90◦ pulse, and spin-locked by applying a strong radiofrequency field along the direction of the magnetization in the rotating frame. The I-spins are thereby prepared in a non-equilibrium state of very low spin temperature. Simultaneously, a second radio frequency field is applied to the dipolar-coupled S-spins. If the I- and S-precession frequencies in the rotating frame are equal, i.e. if the Hartmann– Hahn matching condition γI B1I = γS B1S [31] applies, the I-spin system can relax towards thermal equilibrium, by transferring some of its polarization to the S-spin system. This cross-relaxation pathway utilizes the dipolar interactions and hence requires spatial proximity and a reasonable extent of rigidity on the NMR timescale. Thus, if we can obtain an S-signal via cross-polarization from an I-spin reservoir we have proven the spatial proximity of both spin species. Within the frame of SFB 458 we have used this pulse sequence to examine the spatial proximity between 23 Na and 7 Li in mixed alkali glasses. Unlike the common case of organic solids and polymers studied by 13 C{1 H} CPMAS NMR, several complications need to be considered in this case. First of all, inorganic glasses generally present weakly-coupled spin systems, in which the Hartmann–Hahn matching conditions are modulated by the process of sample spinning [32,33]. In general, two different types of matching conditions can be realized, given by: ν eff (I ) − ν eff (S) = nν r

(27a)

ν eff (I ) + ν eff (S) = nν r .

(27b)

or

Here ν eff (I) and ν eff (S) represent the effective precession frequencies of spins I and S (which are determined by the chosen radiofrequency amplitudes), ν r denotes the MAS spinning frequency and n can take integer values of 1 or 2. Equation (27a) represents polarization transfer between the two spins I and S via an energy conserving flip–flop process (zero-quantum CP) whereas Eq. (27b) corresponds to a polarization transfer associated with a non-energy conserving flip–flip or flop–flop process where the energy is balanced by the mechanical rotation of the sample (double-quantum CP). The second complication to be considered arises from the fact that both the 7 Li and the 23 Na nuclei are quadrupolar nuclei. The quadrupolar interaction not only influences the nutation frequencies of these spins, but it can also interfere seriously with the ability of spin-locking their transverse magnetizations because of spin-state mixing effects [34]. Thus, the challenge of such CP/MAS experiments consists in finding a Hartmann–Hahn matching condition with rf amplitudes that enable

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Fig. 13. Pulse sequence used for cross-polarization. The rf field applied to the I nuclei during the contact time is phase shifted by 90◦ relative to the corresponding preparation pulse.

Fig. 14. Intensity of 7 Li{23 Na}CP/MAS spectra in LiNaSO4 as a function of rf-field amplitude at constant contact time of 5 ms, νr = 14 kHz. The 7 Li nutation frequency was kept constant at 1.8 kHz while that of 23 Na is systematically varied, as expressed by the chosen attenuation (in dB) of the rf source. Maxima a and c correspond to zero-quantum, maxima b and d to doublequantum CP conditions. From Ref. [35].

both nuclear species to be effectively spin-locked simultaneously. Figure 14 shows a detailed 7 Li{23 Na} CPMAS optimization study for the model compound LiNaSO 4 where the 23 Na radio frequency amplitude is systematically varied while that of 7 Li is held constant (1.8 kHz). It clearly illustrates that good Hartmann–Hahn conditions can be realized, even though the conditions have to be chosen carefully [35].

4. Local environment and spatial distribution of the mobile ions in model glasses The local environment of the mobile ions in ion-conducting glasses is one of the key issues related to structure/property relations in these and other solid electrolyte materials. Basic questions of interest are

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1. the average ion-oxygen distance, 2. the average number of oxygen atoms defining the first coordination sphere, (coordination number), 3. the geometric arrangement of these oxygens (coordination symmetry), and 4. the average number of anionic species (non-bridging oxygen atoms) in the first coordination sphere. The last aspect is also related to the spatial ionic distribution, as multiple negative charges must be compensated locally by the appropriate number of positive charges. Thus, a large number of anionic species in the first coordination sphere of the ions automatically implies strong ion clustering. Within SFB 458 we have studied these questions for a range of binary alkali oxide glass systems, using complementary solid state NMR experiments. The observables considered include 1. NMR chemical shifts (6,7 Li, 23 Na, and 133 Cs), using MAS and MQMAS NMR, 2. Magnetic dipole–dipole interactions between the nuclear isotopes associated with the network modifier ions (23 Na-23 Na and 7 Li-6 Li), using spin echo decay or SEDOR spectroscopies, and 3. Magnetic dipole–dipole interactions between the nuclear isotopes associated with the network modifier ions (7 Li, 23 Na etc.) and those associated with the network former species (29 Si, 11 B, 31 P etc.) using appropriate REDOR techniques.

4.1 Alkali silicate glasses Alkali silicate glasses have been studied in considerable detail both on the basis of MD simulations and experiments. In the region of relatively low alkali oxide contents (< 33 mol. %) these studies have lent strong support to the “modified

Fig. 15. Schematic illustrating lithium clustering in lithium silicate glasses, bringing multiple Liions into proximity of multiple Q (3) units.

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Fig. 16. Site-resolved 29 Si{7 Li}REDOR data on a lithium silicate glass. Spin-lattice relaxation times were shortened by doping with 0.1 mol % MnO.

random network model” (MRN) in which the modifier cations are viewed to be concentrated in cluster regions and coordinated primarily by non-bridging oxygen atoms [36]. The percolation of such clusters may lead to the formation of “ion conducting channels” at higher concentrations. The general results of these studies are consistent with the tendencies of alkali silicate glasses to phase separate [37]. Within SFB 458 we have explored the local environment in binary lithium and sodium silicate glasses by a range of complementary singleand double resonance NMR techniques. As illustrated in Fig. 15, clustering of the cations also implies clustering of the charge-compensating anions, which are associated with the Q (n) sites (n < 4), because only in this way excess local charges can be avoided [38,39]. As a consequence of clustering, the 29 Si-7 Li dipolar interactions are significantly stronger for the anionic Q (3) sites than for the neutral Q (4) sites. We have confirmed this prediction using 29 Si{7 Li}REDOR data of lithium silicate glasses containing less than 20 mol. % Li2 O (see Fig. 16) [40]. While the Q (3) units show a substantial REDOR effect, the signal of the Q (4) units is essentially unaffected by 7 Li irradiation, indicating that bridging oxygen atoms connecting between two Q (4) units make no contribution to the local Li environment [40]. Figure 17 shows the compositional dependence of the corresponding dipolar second moments for a series of lithium silicate glasses, illustrating the large difference in 7 Li-Q (3) and 7 Li-Q (4) dipolar coupling strengths

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Fig. 17. Dependence of M 2 (29 Si-7 Li) measured the Q (3) and the Q (4) units in binary lithium silicate glasses as a function of lithium oxide content. From Ref. [40].

Fig. 18. The scaled pair correlation function f Si,Li (r) for different concentrations for Q (3) units (left) and Q (4) units (right). From Ref. [40].

over a wide range of compositions. We conclude from these NMR results that the Q (4) species reside in spatial regions that are largely cation-depleted, consistent with the known tendency of lithium silicate glasses to phase-separate.

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Fig. 19. Dependence of M 2 (6 Li-7 Li) [106 rad2 s−2 ] on atomic concentration in binary lithium silicate glasses. The line corresponds to the prediction of the model described in the text.

Figure 18 shows Q (3) -Li and Q (4) -Li pair distribution functions f Si,Li (r) extracted from complementary MD simulations. We can combine these results to reveal new information about the structural organization of the lithiumrich nanophase. Based on the average Q (3) -Li distances determined from the maxima in these curves, we can estimate the number of Li nearest neighbors to a Q (3) silicon unit from the experimental M2 values. For example, taking r Si,Li = 318 pm as the relevant average Q (3) -Li closest distance in the x = 0.1 glass, each closest lithium neighbor would – on average – produce a contribution of 3.3 × 106 s−2 to M2 (29 Si{7 Li}). Based on the experimental M2 -values near 11–12 × 106 rad2 s−2 and considering that these values also include minor contributions from more remote 7 Li spins, we can conclude that the Q (3) units are surrounded by three closest lithium ions. Independent confirmation for the model illustrated in Fig. 15 comes from a study of the dipole–dipole interactions among the lithium spins, studied by 7 Li{6 Li} SEDOR [19]. Figure 19 summarizes the M2 (7 Li{6 Li}) values in a series of lithium silicate glasses as a function of lithium number concentration. The data not only reveal the typical signature of cation clustering, but the limiting value of M2 (7 Li{6 Li}) near 10 × 106 rad2 s−2 measured at low lithium contents is quantitatively consistent with the presence of four lithium ions in the second coordination sphere at an average Li···Li distance of 273 pm [40].

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Conceptually similar results have been obtained for sodium silicate glasses [41]. Again, the combined analysis of 23 Na spin echo decay spectroscopy and 29 Si{23 Na} REDOR experiments indicates that the sodium arrangement is highly nonstatistical, manifesting itself in significantly enhanced values of the corresponding dipolar second moments.

4.2 Alkali phosphate glasses In a similar vein as the work described above, the REDOR technique has been applied to characterize the network former/network modifier correlations in alkali phosphate glasses. Figure 20 shows experimental 31 P{23 Na} REDOR curves for two samples containing 60% and 35% sodium oxide, respectively [26]. Clearly, the magnitude of the 23 Na dipolar field present at the 31 P site increases with increasing number of non-bridging oxygen atoms. Since the overall local charge requiring cationic compensation increases from zero to two in the sequence Q (3) → Q (2) → Q (1) the observed experimental result is not unexpected. Furthermore, the M2 (31 P{23 Na}) values for the Q (2) sites increase significantly with increasing sodium content in the glass. Also the neutral Q (3) site displays dipolar coupling of significant strength to 23 Na nuclei, even though no charge compensation is needed here. This result suggests that the non-bridging oxygen atoms of the Q (3) units contribute significantly to the first coordination spheres of the Na+ ions. This result is consistent with a statistical alkali ion distribution and argues against cation clustering or even phase separation effects. Figure 21 summarizes the results obtained from systematic site-resolved 31 P{23 Na}REDOR measurements on sodium ultraphosphate glasses [43]. For all the glasses, the M2 (31 P-23 Na) values are significantly larger for the Q (2) units than for the Q (3) units  and increase (as expected) with increasing cation content. Because M2 ∝ r −6 , we can test different atomic arrangement scenarios against the experimental data. In this regard, useful guidance comes from the concentration dependence of M2 (x): M2 (x) ∝ x 0 is expected for phosphorus sites in a sodium environment independent of x (cluster formation); M2 (x) ∝ x 1 points towards a statistical distribution scenario, while M2 (x) ∝ x 2 corresponds to a maximization of the 31 P–23 Na distances in an ordered distribution scenario. For the simulation of these simple model distributions, the Na and P atoms were placed on the positions of a CsCl lattice and no assumptions about the locations of the oxygen atoms were made. In the ordered (“homogeneous”) model, starting from x = 0.5, the lattice constant of the model structure was adjusted to match the corresponding sodium number density of the glass considered. In the statistical model, a fixed lattice constant was used, and the sodium number density of the glass was matched by removing the sodium atoms from this lattice at random (“decimated lattice model”). Figure 21a compares the experimental

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Fig. 20. Site resolved 31 P{23 Na} REDOR results on two (Na2 O) x -(P2 O5 )1−x glasses using the sequence of Fig. 8. Top: x = 0.35; bottom x = 0.60. Heteronuclear M 2 values are calculated using the procedure summarized in Fig. 10 on the basis of experimentally determined f 1 values. From Ref. [26].

M2 (31 P{23 Na}) values, separately for the Q (2) and the Q (3) units, for both scenarios. For either unit type, the values predicted for the statistical model deviate substantially from the experimental data. The ordered model appears

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Fig. 21. (a) Dependence of M 2 (31 P{23 Na}) on sodium ion concentration in sodium ultraphosphate glasses. Separate values are plotted for the Q (3) and the Q (2) units. Dashed line illustrates the compositional dependence expected for a statistical distribution, while the solid curve shows the predicted behavior for a homogeneous model. From Ref. [43]. (b) Compositional dependence of M 2 (31 P{23 Na}) evaluated separately for the Q (2) and the Q (3) sites. The various curves correspond to the model described in the text for the Q (2) units, using different Q(2) ···Na distances. From Ref. [43].

to describe the compositional trend for the M2 (Q (3) ) values, but not for the M2 (Q (2) ) values. As expected, for x = 0.5, there is good agreement between the M2 (31 P{23 Na}) values of sodium metaphosphate glass and crystalline Na3 P3 O9 .

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Fig. 22. Dependence of M 2 (23 Na-23 Na) on Na ion concentration in sodium ultraphosphate glasses as measured in Ref. [44]. Data are compared with two different decimated lattice models as described in the text.

In the limit of low x values, the distance between the anionic Q (2) units and the charge-balancing sodium cations can be estimated by assuming that M2 (31 P{23 Na}) is mainly determined by the dipole–dipole interaction within an isolated 31 P-23 Na spin pair. Additionally, the interaction of the Q (2) unit with the more remote 23 Na spins can be estimated by embedding this Q (2) Na+ pair inside the homogeneous model mentioned above. The resulting model prediction is shown in Fig. 21b for various assumed Q (2) -Na distances. The best agreement is found for d(31 P– 23 Na) = 330 pm, corresponding rather well with typical Na−P distances in crystalline sodium phosphates. Based on these considerations, we conclude that the sodium ions tend to be isolated at low concentrations. Independent information regarding the sodium distribution comes from 23 Na spin echo decay spectroscopy data previously published in Ref. [44]. Figure 22 reproduces these data and compares them with two statistical distribution scenarios. The dotted curve shows the predicted behavior for a decimated CsCl lattice model, in which Na and P occupy the cation and the anion sites, respectively. For this model, the lattice constant (which corresponds to the closest Na−Na distance) is 400 pm, yielding a closest P−Na distance of 346 pm in good agreement with the 31 P{23 Na} REDOR result. Figure 22 illustrates that this model gives reasonable agreement with the experimental M2 values, particularly at low sodium contents (dotted line). At higher sodium contents (x > 0.30), the data are better matched by the dashed line, which is calculated for a decimated lattice model based on Na3 P 3 O 9 . Altogether, these results indicate a random spatial distribution of the Na+ ions in sodium ultraphosphate glasses.

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Fig. 23. Site resolved 11 B{23 Na} REDOR results on a sodium borate glass recorded with the pulse sequence of Fig. 7.

Fig. 24. Dependence of M 2 (11 B{23 Na}) on sodium number density in sodium borate glasses. The curves portray the compositional dependences expected from simulations based on a statistical (solid curves) and a homogeneous (dotted curve) distribution scenario of sodium relative to boron. From Ref. [47].

4.3 Alkali borate glasses Alkali borate glasses differ fundamentally from silicate and phosphate glasses with regard to the network modification process. Rather than creating nonbridging oxygen atoms, the oxide ions contributed by the network modifier

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Fig. 25. Dependence of M 2 (23 Na-23 Na) on sodium number density in sodium borate glasses. The curves portray the compositional dependences expected from simulations based on a statistical (solid line) and a homogeneous (dotted curve) distribution scenario. For further details see text. From Ref. [47].

component are used to convert the neutral trigonal planar BO 3/2 groups (B(3) units) into anionic four-coordinate BO−4/2 (B(4) ) species. As the 11 B resonances of both structural units are well resolved by MAS, it is again possible to prove lithium or sodium cation clustering by site resolved REDOR experiments. Figure 23 shows a typical 11 B{23 Na} REDOR curve obtained in a sodium borate glass. Clearly, the extent of dephasing is rather similar for both types of boron structural units [45,46]. This result indicates that the B(3) and B(4) units are in intimate contact, that oxygen atoms within B(3) -O-B(4) bridges contribute to the first coordination sphere of the sodium ions and that there is no cation clustering. For the B(3) groups the M2 (11 B (3) {23 Na}) values increase linearly with concentration (Fig. 24), consistent with a random sodium distribution in the glasses. Very similar behavior is observed for the B(4) units, except in the limit of very low concentrations, where the corresponding M2 (11 B (4) {23 Na}) values deviate from the linear correlation and approach a constant “baseline” value of M2 = 4.2 ± 0.5 × 106 rad2 s−2 . This finding is immediately plausible if one bears in mind that the minimization of local site energies requires appropriate anionic compensation of the positive Na+ charge. For this reason each Na+ must have at least one anionic BO−4/2 unit in its vicinity. Since the results discussed above indicate that the Na+ ions are statistically distributed, the “baseline” value of M2 (11 B (4) {23 Na}) = 4.2 ± 0.5 × 106 rad2 s−2 thus reflects the dipolar interaction within an isolated spin pair. Using Eq. (13b) we calculate an 11 B-23 Na internuclear distance of 316 ± 6 pm. This value is comparable to the range of BNa distances (from 285 pm to 320 pm) in various sodium borate model compounds. Regarding the dipolar interactions between the 11 B (3) species and the 23 Na spins, we can consider the linear dependence

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of M2 (11 B{23 Na}) on sodium content to reflect the increasing probability with which a BO 3/2 group participates in the local coordination sphere of a sodium ion. Complementary information on the spatial distribution of the sodium ions is available from 23 Na spin echo decay experiments. In contrast to the situation in silicate glasses, the M2 (23 Na-23 Na) values increase linearly with atomic concentration, consistent with a random distribution. We can discuss the spatial distribution of the sodium ions in these glasses using a simple decimated lattice-model, with the face centered cubic structure of NaCl as a starting point: The cation sites of this lattice are occupied by the sodium ions, whereas the anion sites are filled by the boron atoms. The lattice constant is chosen such that the Na···B distance is 316 pm, as determined independently from the REDOR result. This corresponds to a minimum Na···Na distance of 440 pm [45,46]. For full Na occupancy, the overall Na number density of this model is 1.59 × 1028 m−3 . To account for the lower sodium number densities of the actual glasses, a corresponding fraction of sodium atoms are removed at random from their respective lattice positions. Subsequently, average values of M2 (11 B{23 Na}) and M2 (23 Na-23 Na) are calculated, by averaging over a large number of randomly chosen sodium sites and taking into consideration that each B(4) unit must be next to at least one Na+ ion. As illustrated in Figs. 24 und 25 this simple model can predict the compositional evolutions of both the M2 (11 B{23 Na}) and the M2 (23 Na-23 Na) values in very good agreement with experimental data. To demonstrate the potential of NMR to differentiate against alternative distribution models, similar calculations were carried out for a homogeneous distribution scenario, in which the lower sodium number density of the glasses is taken into account by keeping the lattice fully occupied, but by simply increasing the lattice constant [46]. This model maximizes the minimum possible sodium–sodium and sodium–boron distances at each composition, thereby producing the minimum possible values of M2 (23 Na-23 Na) and M2 (23 Na-11 B). As illustrated by Fig. 24, it yields much lower M2 values than observed and produces a parabolic dependence of the corresponding M2 values on composition, at variance with the experimental results. It is only in the limit of relatively high sodium contents (x = 0.30), that the M2 (23 Na-23 Na) data seem to cross over towards the values predicted by the homogeneous distribution scenario. Apparently, the Coulombic Na+ -Na+ repulsions result in a spatial distribution of the sodium ions that differs subtly from that predicted by a purely statistical scenario, particularly in glasses with higher sodium contents. In contrast, no such effect is noticed for the distribution of sodium relative to boron.

4.4 Discussion The detailed experimental results outlined in the previous sections now facilitate a comprehensive discussion of structural trends observed in binary oxidic network former glasses. Strikingly different results between these glass

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Fig. 26. (a) Dipolar second moment M 2 (6 Li-7 Li) measured via 7 Li{6 Li} SEDOR and (b) 6 Li isotropic chemical shifts in lithium silicate and lithium borate glasses as a function of ion number density. From Ref. [19].

systems become evident, when we present their NMR observables jointly as a function of cation number density. For example, Fig. 26a compares the dependence of M2 (6 Li-7 Li) on the lithium content in the lithium silicate and

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borate glass systems, revealing the fundamental difference in their spatial arrangements discussed above. Interestingly, the analogous compositional dependence is observed for the isotropic 6 Li or 7 Li chemical shifts (see Fig. 26b). In the silicate glasses the signals are shifted to significantly higher resonance frequencies indicating a greater degree of Li−O bond covalency in this system. This behavior is expected because of the strong Li−O···Li electrostatic interactions within the lithium clusters. An analogous correlation can be found in the comparison of the various sodium-containg glasses [41–43,45–47]. Figure 27a summarizes the M2 (23 Na23 Na) values as a function of number density, again emphasizing the rather different behavior of sodium silicate glasses as opposed to all the other glassforming systems: a more or less constant value of M2 = 4.5 × 106 rad2 /s2 is observed at Na concentrations below 1.5 × 1028 m−3 , indicating significant clustering of the sodium ions. In contrast, M2 values are much smaller in sodium borate, phosphate, tellurite and germanate based glass systems, and scale linearly with sodium number density, consistent with a statistical spatial distribution. At Na concentrations higher than 1.5 × 1028 m−3 the M2 values for the silicate glasses converge with this line, indicating that the sodium distribution can also be considered as random in this composition range. Figure 27b shows an analogous compositional trend for the 23 Na isotropic chemical shifts. While the magnitudes of the chemical shifts depend on the type of network former observed, we observe close to linear dependences on cation content for all those glass systems, where no clustering is detected by spin echo NMR. Only the data of the sodium silicate glasses stand out, as they are essentially independent on cation concentration over a wide composition range. This trend again reveals the close connection between the local environment and the clustering tendency of the alkaline ions in glasses. In most oxide glass networks the anionic charge is dispersed over several atoms: for example, the two non-bridging oxygens in metaphosphate glasses, the four boron–oxygen bonds associated with the anionic B(4) units in borate glasses, or similar bridging oxygen structures presumed to be present in the germanate and tellurite systems. In contrast, the Q (3) units present in silicate glasses have their anionic charges fully localized on the non-bridging oxygen atoms, producing a much stronger ionic potential experienced by the cations. As a result, multiple Li+ -NBO and Na+ -NBO interactions become energetically more favorable at the local level, resulting in clustering and the well-known tendencies of alkali silicate glasses to phase-separate. The inhomogeneous distributions of alkaline ions in silicate glasses are consistent with the proposal of ion-conducting channels, maintaining the possibility of long-range ion transport even at relatively low cation contents. In contrast, this possibility does not exist, if the mobile ion distribution is statistical, as this would result in very large average jump distances at low cation contents. On this basis it is now possible to understand why at a given number density the ionic conductivities of alkali silicate are significantly larger than those of the corresponding alkali borate glasses, as reported in [1].

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Fig. 27. (a) Dipolar second moment values M 2 (23 Na-23 Na) and (b) 23 Na isotropic chemical shifts as a function of atomic number density in various ion conducting oxide glass systems.

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5. Mixed-alkali effects in oxide glasses Glasses containing two different types of network modifier cations have gained considerable interest because of their unusual properties concerning the ionic mobility, namely the so called mixed-alkali-effect (MAE) [48]. In these glasses a dramatic decrease of ionic conductivity is observed, if one type of the mobile cations is replaced by a homologue at a constant overall cation content. Early interpretations attributed this effect to certain anomalies regarding the relative distribution of the two types of ions with respect to each other and rather contradictory ideas were proposed, ranging from the formation of “unlikecation pairs” as a result of an interaction preference, all the way to phase segregation as a result of mutual repulsion [49]. An important breakthrough was achieved by the dynamic structure model proposed of Maass, Ingram and Bunde [50] introducing the important concept of site mismatch between the two unlike cations. On the basis of EXAFS results the mobile ions are known to possess their own distinct environments in the glass structure leading to preferred diffusion pathways for each type of ion. As a consequence the path connectivities, which depend only on the relative cation content, determine the mobility of both cation types and their respective contributions to ionic conductivity. This model is supported by molecular dynamics simulations [51] as well as tracer diffusion experiments [52]. Most importantly, within the frame of the dynamic structure model and its subsequent refinements, the MAE can be understood as a natural consequence of a statistical distribution of the unlike cations relative to each other. In this connection, important experimental confirmation has come from solid state NMR for a number of mixed lithium sodium silicate [53–55], thiogermanate [56] and borate glasses [57]: dipolar second moments determined from 23 Na spin echo decays, and 7 Li{6 Li} or 23 Na{6 Li} SEDOR data consistently point towards such random relative cation distributions.

5.1 Site modifications and secondary mismatch effects Despite the success of the dynamic structure model in explaining the phenomenology of the MAE rather well, some questions in the context of the mixed alkali effect are still not fully resolved. In particular the idea of individual, non-interfering pathways for the cations cannot explain the very strong immobilization effects especially in the dilute foreign ion substitution limit [58]. In addition to these experiments trends of the 23 Na chemical shift in mixed alkali borate glasses [57,59–61] clearly show that the local environment of the sodium ions changes continuously upon substitution of sodium with a homologue, a finding that also supports the idea of site modification. As an example, Fig. 28 shows some recent data on sodium rubidium borate glasses [61]. With increasing extent of substitution of Rb for Na, the isotropic chemical shift moves towards higher frequencies, indicating an increase in bond covalency.

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Fig. 28. 23 Na isotropic chemical shifts in sodium borate and sodium rubidium borate glasses as a function of molar sodium oxide concentration. From Ref. [61].

The opposite trend (shift towards lower frequencies) is observed upon replacing sodium by the smaller lithium cation [57]. Detailed quantum mechanical chemical shift computations [62] supported by an analysis of chemical shift data obtained on sodium silicates [63] suggest that these chemical shift trends can be interpreted as follows [57]: the sodium sites shrink upon substitution with a larger cation whereas they expand upon substitution with a smaller cation (i.e. lithium). Inspection of a large body of chemical shift trends in various mixed-alkali glasses suggests that these structural readjustments are rather universal [59–74]: substitution of an alkali ion A by larger homologue B always leads to a slight compression of the A site and an expansion of the B site, compared to the situation in the corresponding single-alkali glass. Based on this evidence for site modification, we have suggested an additional (“secondary”) mismatch effect as a fundamental principle underlying the MAE, which seems particularly relevant in the dilute foreign ion limit (see Fig. 29, bottom): each foreign ion B acts not only by interrupting transfer paths between the well-matched A sites by mere occupancy, but also by distorting regular A sites close to B ions into A sites via specific cationcation interactions. As a result, each B ion has the effect of immobilizing multiple A ions in its immediate vicinity. To provide further proof of this concept, those lithium ions located in the immediate vicinity of sodium were selectively detected by 7 Li{23 Na} crosspolarization MAS-NMR experiments, conducted on ((Li2 O)0.9 (Na2 O)0.1 )0.3 (B2 O3 )0.7 and ((Li2 O)0.95 (Na2 O)0.05 )0.3 (B2 O3 )0.7 glasses. This study was done with detailed attention to the methodological aspects of cross-polarization between two quadrupolar nuclei in the weak coupling limit at described in

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Fig. 29. Site mismatch models explaining the mixed-alkali effect in glasses. Top: primary mismatch caused by substitution of the homologous cation species at typical jump distances. Bottom: Secondary mismatch caused by the presence of like-cation sites that have been altered owing to their proximity of a homologous cation species.

Sect. 3.7 [35]. As an important result of this study, the lithium ions in the immediate vicinity of 23 Na detected by CPMAS were found to have the same chemical shift (within experimental error) as the average of all of the lithium ions (detected by standard MAS). Apparently, the isotropic chemical shift distribution in these glasses is dominated by other sources of dispersion (distribution of interatomic distances, bond angles and coordination numbers). Interesting new information is available, however, from a comparison of the static spectra (recorded under non-spinning conditions) obtained via crosspolarization from 23 Na nuclei. As illustrated in Fig. 30, these spectra can be simulated with two overlapping components: a narrow, motionally narrowed component, which is emphasized in the spectrum detected under standard conditions, and a broad, static component, which dominates the spectrum under CP conditions. The width of this latter component is essentially temperature independent, thus supporting the idea that the mobility of lithium sites in the immediate proximity to sodium ions is significantly reduced as compared to the remainder of the lithium population. This fact provides rather strong dir-

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Fig. 30. Static 7 Li-NMR (top) and {23 Na}7 Li CP spectra (bottom) for glass with composition ((Li2 O) 0.9 (Na2 O) 0.1 )) 0.3 (B2 O3 ) 0.7 , measured at three different temperatures. Experimental data are compared with simulations on the basis of two lineshape contributions as displayed in the figure. From Ref. [35].

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Fig. 31. Dependence of M 2 (23 Na-23 Na) on sodium number density in sodium rubidium borate glasses. Data on binary sodium borate glasses are included. From Ref. [61].

ect evidence for the immobilization effect observed for the lithium ions in the immediate vicinity of sodium species, the so-called “secondary mismatch effect”.

5.2 Like-cation preferences While both the continuous compositional trends in the isotropic chemical shifts as well as most of the SEDOR data available on mixed alkali glasses strongly support intimate mixing and random distributions of the two types of cations relative to each other, certain deviations from a strictly statistical model are evident in various glasses, including examples from borate [61], phosphate [70] and silicate systems [75]. An important example is given in Fig. 31, which summarizes the dependence of M2 (23 Na-23 Na) on sodium number density in sodium rubidium borate glasses [61]. While for the binary sodium borate reference system, the concentration dependence of M2 is strictly linear (as expected for a statistical model), the mixed alkali system presents clear deviations from this behavior, suggesting a subtle like cation preference. The degree to which this phenomenon occurs is an interesting question that can be pursued on the basis of the M2 data shown in Fig. 31. We can define a “like-cation preference parameter”   full   / M2 − M2single (28) μ = Mmixed − Msingle 2 2 where M2mixed and M2single represent the experimental data measured on the mixed-alkali glass and the single alkali glass having the same sodium number density, while M2full refers to the maximum value expected for a completely

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Fig. 32. Dependence of M 2 (23 Na-23 Na) on cation composition in K x Na1−x PO3 glasses. Square: M 2 for crystalline NaPO3 (phase A, polyphosphate chains). Triangle: M 2 for crystalline (NaPO3 )3 (cyclic phosphate phase). Straight line: Calculated linear dependence with composition for a random distribution of 23 Na in the glasses. From Ref. [70].

segregated glass having the same total alkali/boron ratio. In the simplified terms of Eq. (28), 0 ≤ μ ≤ 1, where the limit μ = 0 corresponds to statistical mixing, while the limit μ = 1 describes full like-cation segregation. Preliminary experimental μ values near 0.6 (for y = 0.2) and 0.2 (for y = 0.3) suggest that like-cation preferences increase with decreasing total ion content. Most recently, similar like-cation segregation effects have been observed in M2 (23 Na-23 Na) measurements on mixed-alkali K-Na-metaphosphate glasses, see Fig. 32 [70].

6. Mixed network former effects in oxide glasses The large majority of technically relevant glasses are based on more than one network former species. The combination of several network formers usually offers the possibility of fine-tuning physical property combinations to special technological demands, and in certain cases new physical properties arise, as a result of a specific interaction of the two network former species. Similar mixed-network former effects can result from the incorporation of intermediate oxides into the glass network if the latter adopts, at least partially, a network former role. With regard to the structural aspects of such glasses, and the development of potential structure/property correlations two principal questions arise. The first concerns the quantitative connectivity distribution in the network. Do the two network former species interact preferentially, leading to an increase in the fraction of heteroatomic connectivities or even to the formation of new structural units not present in the single-network former glasses? Or do

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they repel each other, leading to domain segregation or even phase-separated glasses? Or is the distribution merely statistical? A second issue concerns the distribution of negative charge in the framework, reflecting the competition of both network formers about the modifier cation. Is there a selective attraction of cations by one of the network former components or is a model of proportional sharing the more appropriate description? To address the questions posed above, we have developed an integrated spectroscopic approach including a combination of multinuclear solid state NMR methodology, Raman spectroscopy, and X-ray photoelectron spectroscopy. The latter technique is particularly powerful as the O-1s binding energy, which is sensitive to the types and bonding partners of the oxygen atoms, provides extremely specific information about the connectivity distribution, which is of great assistance for the interpretation of the NMR results. The present review will focus on three topics: (1) the structure/property correlations in sodium borophosphate glasses at constant cation content, (2) the structural consequences of alloying NaPO 3 glasses with intermediate oxides, and (3) mixed anion effects in ion-conducting oxyfluoride glasses.

6.1 Structure/property correlations in sodium borophosphate glasses A well-known example for the mixed network former effect, is the sodium borophosphate glass system. In the ion-conducting glass system (Na2 O)0.4 [(P2 O5 )1−x (B2 O3 )x ]0.6 the successive replacement of the network former species phosphorus oxide by boron oxide at constant alkali content leads to a dramatic, non-linear increase of the ionic conductivities (see Fig. 33), suggesting profound structural changes [76,77]. Figure 34 summarizes the 31 P and 11 B MAS-NMR spectra measured on these glasses [78]. Spectral assignments are made in terms of the local P(n) mB and B(n) mP units, where n is the number of bridging oxygen atoms and m is the number of heteroatomic bridges for each network former type. These assignments are based on appropriate mass and charge balance constraints [78] and are partially assisted by 31 P{11 B} and 11 B{31 P} REDOR experiments [27,79–81]. The detailed phosphate and borate speciations as extracted from these spectra, are summarized in Fig. 35. The structure of the pure sodium phosphate glass is constructed from neutral branching groups (P(3) 0B units) and anionic chain-type metaphosphate (P(2) ) species. Upon substitution of the network former P 2 O 5 by 0B B2 O 3 , the main change is the successive replacement of anionic P(2) units by tetrahedral B(4) groups. This holds up to a substitution level of x = 0.4, above which the structural transformations are more complex and also involve doubly charged phosphate end group (P(1) ) and neutral three-coordinated B(3) species. Comparing Figs. 33 and 35 we note that the concentration of the anionic B(4) units and the ionic conductivities vary with composition in a virtually identical fashion, suggesting a direct correlation between ionic conductivity and local structure. The data summarized in Fig. 35 can also be used to calculate the

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Fig. 33. Compositional dependence of the activation energies (top) and of the dc ionic conductivities at 298 K in the glass system (Na2 O) 0.4 [(P2 O5 )1−x (B2 O3 ) x ] 0.6 .

Fig. 34. 31 P and 11 B MAS-NMR spectra and proposed assignments of glasses in the system (Na2 O) 0.4 [(P2 O5 )1−x (B2 O3 ) x ] 0.6 .

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(n) Fig. 35. Concentration of the various P(n) mB and B mP network former species in the glass system (Na2 O) 0.4 [(P2 O5 ) 1−y (B2 O3 ) y ] 0.6 .

number of B-O-B, P-O-P, and P-O-B linkages, in order to assess the connectivity distribution in the network. Figure 36 illustrates that the heteroatomic bonds are greatly enhanced in this system, suggesting an energetically favorable interaction between the two different network formers. In this vein, 11 B{31 P} REDOR results conducted in this glass system [79–81] indicate that B(4) 4P and (3) B(4) units tend to avoid the 3P units have particular stability, whereas trigonal B proximity of phosphate (see also Fig. 11b). The speciation equilibria involving the various types of borate and phosphate units can be expressed schematically by considering an equilibrium of the type P(2) + B(3) ⇔ P(3) + B(4) .

(29)

As a result of the favorable P-O-B interaction in the second coordination sphere the right side of Eq. (29) is favored, resulting in an enhanced degree of network connectivity. This connectivity can be expressed in terms of the total number of bridging oxygen units per network former species, which can be calculated from the data in Fig. 35. Figure 37a shows that this number increases steeply up to x = 0.4, remains more or less constant up to x = 0.9 and then decreases steeply again for x = 1 (the binary sodium borate glass). Essentially the same compositional dependence is observed for the value of the glass transition temperature Tg (Fig. 37b), indicating a close relationship between this parameter and the average degree of network connectivity.

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Fig. 36. Bridging oxygen distribution derived for sodium borophosphate glasses glasses along the composition line (Na2 O) 0.4 [(B2 O3 ) x (P2 O5 ) 1−x ] 0.6 .

Fig. 37. Average number of bridging oxygen per network modifier (top) and glass transition temperature (bottom) for sodium borophosphate glasses glasses along the composition line (Na2 O) 0.4 [(B2 O3 ) x (P2 O5 ) 1−x ] 0.6 as a function of substitution level.

The data of Fig. 35 can also be discussed from the viewpoint of a competition of the two network formers phosphorus oxide and boron oxide for the network modifier species, i.e. addressing the question of how many sodium

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Fig. 38. Concentrations of sodium ions interacting with anionic B(4) units (solid symbols) and of sodium ions interacting with anionic P(2) units (open symbols) as a function of substitution level x. The lines correspond to a scenario involving a complete transfer of ionic charge from the phosphate (dashed line) to the borate network (solid line) according to Eq. (29).

ions in the glass interact with the anionic B(4) groups and how many are bound by the anionic P(2) units, respectively. These numbers are plotted in Fig. 38 as a function of substitution level x. Note that up to x = 0.4 the experimental data correspond to a scenario involving a complete charge transfer from the phosphate to the borate network, as described by Eq. (29), whereas at higher substitution levels, the data indicate a more equal sharing of the network modifier between both network former species. Again, the comparison with Fig. 33 suggests that the value of the ionic conductivity measured in these glasses reflects the distribution of anionic charges in the network, suggesting that the presence of four-coordinate borate species in the network is particularly favorable for ionic transport. As a reason we may consider that in the B(4) units the charge is more effectively delocalized over four bridging oxygen atoms than in the P(2) units, where the charge is distributed between the two non-bridging oxygen atoms. The larger extent of charge dispersal effected by the B(4) units results in less deep Coulombic traps for the mobile Na+ ions, enhancing their ionic mobilities. Similar preferences for heteroatomic linkages were detected by comprehensive spectroscopic studies of the sodium and silver borophosphate glass system along other composition lines [27,79–82], in sodium aluminophosphate glasses [83] and in sodium aluminoborate glasses [84–86]. In contrast, preferences for homo-atomic connectivities were detected by dipolar NMR spectroscopic methods in borosilicate [87] and boron silicon nitride glass systems [88].

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6.2 Mixed network former effects in NaPO3 -based glass systems containing intermediate oxides Intermediate oxides such as Al2 O 3 , Sc2 O 3 , MoO 3 , WO 3 , and TeO 2 can be incorporated into glassy NaPO 3 , over wide composition ranges, producing strong effects on their physical properties. As an example, Fig. 39 shows the compositional dependence of the glass transition temperature for two representative systems. Alloying NaPO 3 glass with MoO 3 glasses produces a well-defined maximum in the glass transition temperature near the equimolar composition whereas the effect of TeO 2 is only minor in comparison. The structural organization of (NaPO3 )1−x (MoO3 )x and (NaPO3 )1−x (WO3 )x glasses has been discussed on the basis of extensive multinuclear NMR and Raman spectroscopic investigations [80,89,90]. Similar structural trends have been found, which will be discussed here at the example of the (NaPO3 )1−x (MoO3 )x system [90]. Figure 40 summarizes their solid state 31 P MAS-NMR spectra. The addition of MoO 3 to NaPO 3 glass successively replaces the regular Q (2) site near −20 ppm by a new well-defined resonance near −8 ppm, which dominates the spectrum at x = 0.30; at higher MoO 3 contents the spectra are rather poorly resolved. The structural interpretation of these data is greatly facilitated by J-resolved spectra, examples of which are shown in Fig. 41. For the −20 ppm signal these spectra show the expected triplet consistent with the two P-O-P linkages present in the P(2) sites. Furthermore, both a doublet component (near −8 ppm), and a singlet component (near −3 ppm) can be identified. The relative signal areas lead to the conclusion that for each MoO 3 unit added to the glass approximately two P(2) 0Mo units are transformed into P(2) 1Mo groups. In principle the process of P-O-Mo interlinking operative in (2) this composition region could produce both P(2) 1Mo and P 2Mo units. However, the J-resolved spectra actually give no evidence for the formation of P(2) 2Mo units for x < 0.30. This result is consistent with a simple “binary” structural transformation model, in which the MoO 3 component first converts all the P(2) 0Mo (2) polyphosphate chains into P(2) 1Mo units before the latter are converted to P 2Mo units (or other structures without P-O-P linkages) at higher MoO 3 contents. Figure 42 illustrates that this binary model agrees well with the experimental data, particularly for x < 0.30. Larger deviations from this model are observed at higher x-values suggesting more complex structural transformation processes. In this concentration range the Raman spectra give strong evidence of Mo-O-Mo linkages [80]. Finally, one of the questions yet to be answered concerns the possibility of some charge transfer between the phosphorus and molybdenum species. In this regard, two Raman scattering peaks observed near 910 and 940 cm−1 do suggest the formation of non-bridging oxygen atoms attached to molybdenum [90]. However, neither 23 Na or 95 Mo NMR nor other spectroscopic methods have been able to provide clear evidence for or against a redistribution of anionic species in this glass system.

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Fig. 39. Dependence of Tg on the compositional parameter x in (NaPO3 ) 1−x (TeO2 ) x , and (NaPO3 ) 1−x (MoO3 ) x glasses.

Fig. 40. 31 P MAS-NMR spectra of (NaPO3 )1−x (MoO3 ) x glasses. The percentages of MoO3 are indicated in the figure. From Ref. [90].

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Fig. 41. 31 P-J-resolved spectra of two representative (NaPO3 ) 1−x (MoO3 ) x glasses. The individual J-coupling multiplets are resolved in the second dimension. From Ref. [90].

In view of Fig. 39 it is instructive to compare the above-discussed changes in phosphate network structure with those obtained in the (NaPO3 )1−x (TeO2 )x glass system [91]. The tellurium atoms in these glasses are presumed to be present in four-coordination, corresponding to Te(4) sites able to form to four Te-O-Te or Te-O-P linkages. The onset of Te-O-P linking should be detectable by 31 P MAS NMR, however, as in the NaPO 3 -MoO 3 system, only one new spectroscopic feature is resolved (Fig. 43), namely a signal near 7–12 ppm, the location of which changes with composition. We attribute this feature to (2) overlapping resonances of both the P(2) 1Te and the P 2Te units. While the resolution cannot be further improved by J-resolved spectroscopy in this case, important new information is available by combining the 31 P NMR data analysis with complementary X-ray photoelectron spectroscopy (XPS) results. Figure 44 shows the O-1s spectra as a function of composition. Four lineshape contributions are distinguishable, corresponding to P-O-P, P-O-Te, and Te-O-Te linkages, as well as non-bridging oxygen atoms. While the quantitative populations of the three different types of bridging oxygen atoms are found to be relatively close to those predicted by a statistical model, they do consistently suggest a certain tendency towards preferred homoatomic linkages (see Fig. 45). This information can now be combined with the peak area analysis of

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Fig. 42. Experimental P(2) mMo species distributions obtained experimentally in (NaPO 3 ) 1−x (MoO 3 ) x glasses. Dashed lines include the predictions from the binary model described in the text. From Ref. [90].

Fig. 43. 31 P MAS-NMR spectra of (NaPO3 ) 1−x (TeO2 ) x glasses.

the NMR spectra in Fig. 43. On the one hand, the summed concentration of the (2) P(2) 1Te and the P 2Te units is given by   (2)   B(1 − x) = P(2) 1Te + P 2Te ,

(30)

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Fig. 44. O-1s XP spectra of (NaPO3 ) 1−x (TeO2 ) x glasses and crystalline model compounds. The bottom part shows a typical deconvolution into the four types of oxygen atoms present in this system.

where B is the fractional area of the new spectral component observed in the 7–12 ppm range in the 31 P NMR spectra. On the other hand, the concentration of the bridging oxygen atoms contained in a P-O-Te linkage is given by    (2)  A(3 − x) = P(2) 1Te + 2 P 2Te ,

(31)

where A is the fractional area of the O-1s signal component attributed to the heteroatomic P-O-Te linkages and 3 − x is the total oxygen concentration.

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Fig. 45. Population distribution of P-O-P-, P-O-Te-, and Te-O-Te-bridging oxygen species in (NaPO3 ) 1−x (TeO2 ) x glasses as extracted from the O-1s XP spectra. Solid curves are the predicted concentrations based on a statistical connectivity scenario.

31 Fig. 46. P(2) P MAS-NMR lineshape analysis mTe species distribution obtained by combining the with the bridging oxygen distribution obtained from the XPS data of (NaPO3 ) 1−x (TeO2 ) x glasses. Solid curves show the prediction of a statistical connectivity distribution model.

Expressions (30) and (31) represent two equations with two unknowns, from (2) which the individual concentrations of P(2) 1Te and P 2Te can be extracted. Figure 46 summarizes the results. Consistent with the XP-spectroscopic data the concentrations of the P(2) 0Te units are somewhat larger than predicted from the statistical model. In particular the P(2) 2Te units are under-represented. All of these results suggest that – in contrast to the strong preferred interaction between the glass

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constituents NaPO 3 and MoO 3 , NaPO 3 -TeO 2 glasses are organized in a more random fashion, with even some indication of a preference of homoatomic linkage formation. Presumably, these structural differences are related to the markedly different compositional dependences of the glass transition temperatures shown in Fig. 39. Finally, the preponderance of evidence argues against any charge transfer occurring between the phosphate and the tellurite networks. First of all, the XP-spectra shown in Fig. 44 give no evidence for Te-bonded non-bridging oxygen atoms (such as present in sodium tellurite, Na2 TeO 3 ). Secondly, both the Na-1s and the Te-3d binding energies measured by XPS in the present glass system are markedly different from those obtained in binary sodium tellurite glasses [80,91]. Finally, Raman spectra of these glasses give no evidence of phosphate repolymerization. All of these results indicate that the TeO 2 glass component merely functions as a neutral diluent species, forming TeO 4/2 linkages connecting to each other and P(2) phosphate species.

6.3 Mixed anion effects in ion conducting glasses As discussed in Sect. 6.1, mixed network former effects imply the possibility that different charge compensating anions contribute to the local environment of the mobile cations in systems where network modifier sharing occurs, such as in borophosphate glasses. Similar mixed-anion effects arise if the network forming oxide is mixed with various nitride, sulphide, or halide

Fig. 47. 7 Li{19 F} SEDOR results on 48B2 O3 -10PbO-28LiF-6Li2 O glass. The black solid curve is a polynomial fit to the data while the red solid curve is a fit to Eq. (32), with C = 0.32 and M 2 = 1240 × 106 rad2 /s2 . From Ref. [97].

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Fig. 48. Spectral comparisons of the Fourier Transforms of I(2t1 ) (red) and F(2t1 ) − I(2t1 ) (black) for three different dipolar evolution times 2t1 in 48B2 O3 -10PbO-28LiF-6Li2 O glass. The differences in the corresponding linewidths are evident.

species. Examples of this effect include ion conducting oxynitride [92], oxysulfide [93] and oxyhalide glasses [94–96]. In particular for the latter systems, the presence of lithium halide additives are known to enhance the conductivity of oxide-based glasses considerably. In general, the favourable influence upon the ion conductivity is attributed to the mixing of oxidic and halide

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Fig. 49. 19 F rotor-synchronized MAS-NMR experiments undertaken on 48B2 O3 -10PbO-28LiF6Li2 O glass. (a) Stack plot of the Fourier transforms as a function of dipolar evolution time (listed on right). Spinning sidebands are indicated by asterisks. (b) Plot of normalized intensity as a function of dipolar evolution time. Note the significantly faster decay of the −198 ppm compared to the −137 ppm resonance.

(sulphide, nitride) anions in the first coordination sphere of the cations, leading to increased disorder in the corresponding local energy landscapes. In principle, direct proof for and a possible quantification of such mixed anion environments could come from REDOR experiments in which the magnetic dipole–dipole interactions between the cation nuclei (7 Li or 23 Na) and those nuclei belonging to the different types of anions are detected. Unfortunately, in oxysulfide, oxynitride and in most oxyhalide glasses such ex-

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Fig. 50. 19 F{7 Li} and 19 F{11 B} REDOR experiments on 48B2 O3 -10PbO-28LiF-6Li2 O glass, leading to the peak assignments discussed in the text: Reference signal S0 and dephased signals S obtained after a fixed dipolar evolution time of 80 μs are shown. Spinning sidebands are indicated by asterisks.

periments have been hampered by the lack of suitable NMR active isotopes. Recently, however, the proof of concept was given for a glass of composition 48B2 O 3 -10PbO-28LiF-6Li 2 O [97]. As shown in Fig. 47 the 7 Li{19 F} SEDOR decay observed for this glass is clearly bimodal, as represented by the equation   I(2t1 ) F(2t1 )  C + (1 − C) exp −0.5M2 (2t1 )2 = I(0) F(0)

(32)

with C = 0.32 ± 0.03. This result indicates that about 1/3 of the lithium ions in this glass do not have fluoride ions in their first coordination spheres. In the SEDOR experiment, the contribution of such lithium ions is emphasized by the signal I(2t1 ), while the F-bonded lithium ions are emphasized in the SEDOR difference signal F(2t1 ) − I(2t1 ). Figure 48 shows that the static lineshapes obtained by Fourier-transforming the echoes recorded at dipolar evolution times in the 100–200 μs region of the SEDOR experiment are quite distinct. The signal of the F-bonded lithium ions is significantly broader than that of the nonF bonded lithium ions, as expected from the heteronuclear 19 F-7 Li magnetic dipole–dipole interactions. The 19 F MAS-NMR spectra shown in Fig. 49 give some complementary insight into the fluorine inventory of this glass. Two signals are detected, at −198 ppm and at −137 ppm vs. CFCl3 , respectively. Based on 19 F{7 Li} REDOR experiments the −198 ppm signal is attributed to Li-bonded fluoride species (Fig. 50, left). In contrast, the −137 ppm resonance is attributed to fluoride atoms attached to the boron species in the network as also suggested by its more rapid 19 F{11 B} REDOR decay (Fig. 50, right). Figure 49 also indicates that both types of fluoride species differ substantially with respect to their spin– spin relaxation times. The rapid spin echo decay observed for the Li-bonded

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F species suggests that many of the lithium ions are coordinated to more than just one fluoride ion, resulting in 19 F-19 F dipolar interactions of considerable magnitude.

Acknowledgement Support by the Deutsche Forschungsgemeinschaft via the SFB 458 Ionic Motion in Disordered Materials – From Elementary Jumps to Macroscopic Transport to the project Structural and Dynamic Environments of the Mobile Ions in Disordered Materials was essential for the work reported here. Special thanks are due to Professor Klaus Funke for his leadership in initiating this SFB and for guiding it with inspiration and wisdom for its ten-year duration. This review summarizes previously published contributions from numerous co-workers, doctoral students and postdoctoral associates. Special thanks are due to Dr. Carla Araujo (2006), Dr. Christian Brinkmann (2003), Dr. Susanne Causemann (2010), Dr. Rashmi Ramesh Deshpande (2009), Dipl.-Chem. Heinz Deters (2008), Dr. Stefan Elbers (2004), Dr. Jan-Dirk Epping (2004), Dr. Sandra Faske (2007), Dr. Barbara Koch (2010), Dr. Daniel Mohr (2010), Dr. Christine Mönster (2007), Dr. Stefan Puls (2005), Dr. Devidas Raskar (2008), Dr. Eva Ratai (2000), Dr. Matthias Rinke (2009), Dr. Wenzel Strojek (2006), Dr. Thorsten Torbrügge (2005), Dr. Jan-Henning Trill (2002), Dr. Julia Vannahme (2007), Dr. Ulrike Voigt (2004), Dr. Michael Witschas (2000), and Dr. Dominika Zielniok (2008), whose doctoral theses were conducted on various topics of this review and whose salaries were (at least in part) directly supported by the SFB. Scientific contributions made by past and previous postdoctoral coworkers Dr. Jin Jun Ren, Dr. J.C.C. Chan and Dr. L. Zhang and by group members PD Dr. Leo van Wüllen and Dr. Karin Meise-Gresch are also most gratefully acknowledged. We are grateful for the many collaborations with other project leaders of the SFB, most notably Prof. Dr. Bernt Krebs, Prof. Dr. Rainer Pöttgen, Prof. Dr. Tom Nilges, Prof. Dr. Andrew Putnis, Prof. Dr. Michael Vogel, PD Dr. Cornelia Cramer, Prof. Dr. Helmut Mehrer, Prof. Dr. Monika Schönhoff, Prof. Dr. Andreas Heuer and Professor Dr. Klaus Funke. Further support in the form of personal fellowships awarded by the NRW Graduate School of Chemistry and the Fonds der Chemischen Industrie to a number of doctoral students is gratefully acknowledged.

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Nonlinear DC and Dispersive Conductivity of Ion Conducting Glasses and Glass Ceramics By Halgard Staesche and Bernhard Roling∗ Fachbereich Chemie, Philipps-Universitaet Marburg, Hans-Meerwein-Str., 35032 Marburg, Germany (Received June 30, 2010; accepted in revised form September 20, 2010)

Ionic Motion / Nonlinear Ionic Conductivity / Glasses / Glass Ceramics Frequency-dependent third-order conductivity spectra σ3 (ν) of various ion conducting glasses and glass ceramics were obtained by applying sinusoidal electric fields with high amplitudes and by analysing the resulting higher-harmonic currents. In the DC conductivity regime, the third-order conductivity σ3, dc was found to be positive for all materials and at all temperatures. From the ratio of the third-order conductivity to the low-field conductivity, σ3, dc /σ1, dc , apparent jump distances were calculated. These apparent jump distances are much larger than jump distances between neighbouring sites in the glasses and decrease with increasing temperature. In (Li 2 O) 1−x · (Na 2 O) x · Al 2 O3 · (SiO2 ) 4 glasses, the mixed alkali effect leads to a minimum in the apparent jump distance, while partial crystallisation of Li2 O · Al 2 O3 · (SiO2 ) 2 glasses leads to an increase of the apparent jump distance. In the dispersive regime, the third-order conductivity σ3 (ν) of all glasses and glass ceramics is negative and exhibits an approximate power-law dependence, however with a larger exponent than the dispersive low-field conductivity σ1 (ν). For a given material, the third-order conductivity spectra σ3 (ν) obey the time–temperature superposition principle and can be superimposed by using the Summerfield scaling. Remarkably, the shift between the σ3 (ν) master curves of different materials is much stronger than the shift between the σ1 (ν) master curves. In order to rationalize this effect, we calculate the nonlinear dispersive hopping conductivity in a double minimum potential approximation.

1. Introduction Solid ionic conductors are an important class of materials and have been used in the form of thin films in different electrochemical devices, such as microbatteries, fuel cells and electrochromic windows [1–6]. Thin films combine a reduced mechanical stiffness, important for improved processibility, with reduced electrical resistance, important for improved power output of the device. In thin films, the application of even small voltages generates large electric fields, which in turn lead to nonlinear ion transport. * Corresponding author. E-mail: [email protected] Z. Phys. Chem. 224 (2010) 1655–1676 © by Oldenbourg Wissenschaftsverlag, München

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Electronically conducting glass films can also exhibit nonlinear electrical properties. For instance, phase change materials, such as chalcogenide glasses, are being used as non-volatile memory devices [7]. In these materials, local heating induced by an electric field transforms the highly resistive amorphous material into a low-resistive crystalline state. Generally, the ionic conductivity of solid electrolytes increases with increasing electric field strength [8–10]. Thus, for both basic science and application purposes, a better understanding of the processes governing nonlinear ionic conductivity is highly desirable. From an application point of view, the nonlinearity of the ion conductivity leads to a film resistance which is lower than expected from the linear ion conductivity. From a basic science point of view, the field dependence contains valuable information about the fundamental mechanism of the ion transport in solids and the nature of the potential landscape of the mobile ions [11–13]. Among the different classes of electrolytes, alkali ion conducting glasses exhibit a high electrochemical stability and a transference number of unity for the alkali ion conduction. These advantages allow the usage of glassy electrolyte films with thicknesses in the sub-micron range in lithium microbatteries [14]. Since lithium batteries can supply a voltage up to 5 V, the electric field in a 100 nm film would be as high as 500 kV/cm, which is clearly beyond the linear ion transport regime. In the 1950s to 1980s, a number of nonlinear dc conductivity studies have been carried out on different ion conducting glasses [8–10]. In these studies, dc electric fields, E dc , were applied, and the resulting dc current densities, jdc , were determined. Generally it was found that the current density increases with the field in a superlinear fashion. However, applying dc electric fields has two main drawbacks: Firstly, it is difficult to determine the influence of charge transfer resistances at the electrolyte/electrode interfaces on the measured currents. Secondly, little information is obtained about Joule heating effects. Fundamentally, it is important to differentiate between Joule heating effects and an intrinsic field dependence of the ion transport. This is more directly achievable by using ac electric fields. In the weak nonlinear regime, the dependence of the current density j(E) on the electrical field E can be described by the following power series: j(E) = σ 1 E + σ 3 E 3 + σ 5 E 5 + ...

(1)

At sufficiently low fields, contributions of the second and higher terms on the right hand side are negligible. Thus, σ 1 denotes the low-field conductivity, whereas σ 3 , σ 5 etc. are higher-order conductivity coefficients. We note that for isotropic materials, j(E) is an odd function, leading exclusively to odd terms in Eq. (1). In dc experiments it was found that the effective conductivity σ eff = j/E increases with the field [8,15,16]. The question arising is whether the increase is due to Joule heating effects or due to intrinsic field-

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dependent transport effects. Based on Eq. (1), application of a sinusoidal field E = E 0 sin (2πνt) will give rise to the following expression for the current density being in phase with the electric field, j  : j  = σ 1 E 0 sin (2πνt) + σ 3 E 30 sin3 (2πνt) + σ 5 E 50 sin5 (2πνt) + ... 3 1 = σ 1 (ν)E 0 sin (2πνt) + σ 3 (ν)E 30 sin (2πνt) − σ 3 (3ν) E 30 sin (3 · 2πνt) 4 4 10  5  5 5 + σ 5 (ν)E 0 sin (2πνt) − σ 5 (3ν) E 0 sin (3 · 2πνt) 16 16 1  5 + σ 5 (5ν) E 0 sin (5 · 2πνt) + ... (2) 16 An equivalent equation can be derived for the current density being out of phase with the field, j  [17]. On the basis of Eq. (2), one can see that an intrinsic field dependence of the ionic conductivity leads to higher harmonic contributions to the current density. Joule heating increases the low-field conductivity σ 1 (ν) and produces also higher harmonic currents at 3ν. However, these 3ν currents are proportional to (E 0 )2 [18], so that they can be clearly distinguished from the higher harmonic current term in Eq. (2). Consequently, the higher harmonic currents can be directly used to obtain the higher-order conductivity coefficients σ 3 (ν), σ 5 (ν) etc. A first step to interpret nonlinear conductivity spectra is to consider a simple regular hopping model with a distance a between adjacent sites. For this model, the dependence of the dc current density jdc on the dc electric field E dc is given by [19]:   qaE dc jdc ∝ sinh (3) 2k B T Here, q denotes the charge of the mobile ions, while k B and T are Boltzmann’s constant and the temperature, respectively. A Taylor expansion of Eq. (3) leads to  2 σ 3, dc 1 qaE dc = (4) σ 1, dc 6 2k B T This implies that within the framework of this model it is possible to obtain the jump distance from the ratio of the third-order conductivity to the low-field conductivity. However, real ionic conductors are more complex, exhibiting a distribution of jump distances, site energies and potential barriers. In this case, one can define an apparent jump distance a app via:  24σ 3, dc (k B T )2 (5) a app = σ 1, dc q 2

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Thus, the apparent jump distance is a phenomenological quantity, which is not expected to be identical to the distances between neighbouring sites in real ionic conductors. Indeed, nonlinear dc conductivity measurements on different ion conducting glasses and polymers yielded values for a app in the range of 15–50 Å [20–22], i.e. much larger values than found in molecular dynamics simulations for the nearest-neighbour hopping distances (typically 1.5–3 Å [23]). In this paper, we extract the third-order dc conductivities of various glasses and glass ceramics from their third-order spectra σ 3 (ν), and we analyse the temperature and composition dependence of the apparent jump distance a app . Moreover, we consider in detail the dispersive part of the σ 3 (ν) spectra and draw comparisons to the dispersive part of the low-field spectra σ 1 (ν). In the literature, σ 1 (ν) spectra have been analysed using different formalisms. In the dc regime and over the first decades of the dispersive regime, a Jonscher power law is a reasonable approximation [24,25]:    p1  ν  (6) σ 1 (ν) = σ 1, dc 1 + ∗ ν1 Here, ν ∗1 denotes a characteristic frequency, while the exponent p 1 is often in a range between 0.6 and 0.7. The dc conductivity reflects long-range diffusion of the ions, while the dispersive conductivity at higher frequencies reflects short-time subdiffusive motions of the ions [25,26]. Furthermore, it was found that the low-field conductivity spectra of many solid electrolytes obey the time–temperature superposition principle. For a given material, the σ 1 (ν) spectra obtained at different temperatures can be superimposed by using the Summerfield [25–28]. In this scaling  scaling  formalism, σ 1 (ν)/σ 1, dc is plotted vs. ν/ σ 1, dc T . Superposition of the spectra indicates that the ion transport mechanism is independent of temperature. In particular, the applicability of Summerfield scaling implies that the mean square displacement  the ion at thecrossover from subdiffusive to diffusive  of   dynamics, r 2 cr ≡ r2 t ∗ = 1/ 2πν ∗1 is independent of temperature [25,26]. On the other hand, r 2 cr does depend on the composition of the ion conductor. For instance, in the case of x Na2 O·(1 − x) GeO 2 glasses it was found that the rms displacement r 2  cr decreases from about 10 Å to about 0.6 Å when the sodium oxide content x was increased from 0.005 to 0.4 [26]. This huge difference in the spatial extent of the subdiffusive ion dynamics points to huge differences in the potential landscape of the mobile ions. In this paper, we test the applicability of the Jonscher power law and of the Summerfield scaling to the third-order conductivity spectra σ 3 (ν) of various glasses and glass ceramics. We show that a modified version of the Jonscher power law can be used to fit the σ 3 (ν) spectra of single alkali glasses, while it results in poor fits for mixed alkali glasses and partially crystallised glasses. For a given glass or glass ceramic, the Summerfield scaling can be used to super-

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impose the σ 3 (ν) isotherms. Remarkably, the σ 3 (ν) master curves of different materials exhibit a strong shift with respect to each other on the scaled frequency axis. These shifts are much stronger than the shifts observed between the σ 1 (ν) master curves of the same materials. In order to rationalize this effect, we calculate the nonlinear dispersive conductivity due to ion hopping in a double-well potential approximation.

2. Experimental For the nonlinear conductivity measurements, the following alkali aluminosilicates and sodium germanates were prepared: (Li2 O) 1−x ·(Na2 O) x ·Al2 O 3 · (SiO 2 )4 glasses with x = 0.1, 0.2 and 0.5 (LNASx); Li 2 O·Al2 O 3 ·(SiO 2 )2 glass and glass ceramics with 6% and 13% crystallinity (LAS, amorphous, “am”, and x% crystallinity) and (Na2 O) x ·(GeO 2 )1−x glasses (NGx, with x = 0.15, 0.25 and 0.30). For the chemical formulas, the compositional parameter x is given in the range from 0 and 1, whereas for the abbreviations, x was multiplied by 100. The glasses were prepared by the melt quenching technique as described elsewhere [13]. Crystallisation of the LAS samples was carried out according to Ref. [29]. The glass samples were cut into slices with an Accutom 5 (Struers). The desired thickness of the samples was achieved by lapping with a PM5 (Logitech). This way, thicknesses ranging from 80 μm to 200 μm were obtained. For the sample holder, several requirements have to be met. The materials holding the sample must exhibit a high electrical resistance, also at high voltages. The electrodes should be non-blocking and should have an electrical resistance many orders of magnitudes lower than the glass sample, so that the applied voltage drops exclusively in the sample. This can be achieved with the sample holder shown in Fig. 1, which consists of a quartz glass tube (conductivity < 1016 S/cm) in a quartz glass container. The sample is attached to the tube using a high-voltage resistant Araldite glue (Huntsman). Both the container and tube are filled with a saturated salt solution, which acts as a nonblocking electrode. The salt and solvent used depend on the temperature range covered in the experiment. At low temperatures, sodium or lithium nitrate dissolved in a mixture of propylene carbonate and propylene glycol was employed, whereas for high temperatures, sodium chloride was dissolved in dry glycerol. Contact with the high-voltage measurement system was achieved with platinum wires dipping into the salt solution. Impedance measurements were carried out with a Novocontrol Alpha-AK high performance impedance analyser equipped with a high voltage booster and a high-voltage interface. With the high voltage booster, ac voltages with a maximum amplitude of 2000 V (1414 V rms) can be applied. In Fig. 2 we show the principle setup of the high voltage system HVB4000. The alpha analyser internally generates an ac voltage in the range of 0 V to 1.92 peak voltage

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Fig. 1. Schematic illustration of the experimental setup for nonlinear conductivity spectroscopy on thin glass samples [30].

which is routed to the GEN socket of the HVB, increased by a factor of 1.05, and then routed to the Trek 623 V input socket. After amplification by a factor of 1000, the high voltage signal is transferred back to the HVB 4000 via a 750 kΩ resistance which limits the maximum current to about 4 mA. The signal passes an internal high voltage activation safety switch. The signal is then split and relayed both to the sample and a voltage divider, where the signal is reduced by a factor of 1000 and transmitted back to the alpha input voltage channel V1. The current response of the sample is sensed by the HVB 4000 current-in socket, converted into a voltage and transmitted to the alpha input voltage channel V2. The sample temperature was controlled by the Novocontrol Quatro cryosystem.

3. Results and discussion 3.1 Third-order conductivity spectra For obtaining third-order conductivity spectra, we consider the current density at 3ν in Eq. (2). Rearranging the relevant terms yields: 5 −4 j  (3ν) = σ 3 (3ν) E 20 + σ 5 (3ν) E 40 + ... E0 4

(7)

Thus, plotting the data as a graph of −4 j  (3ν)/E 0 vs. E 20 gives either rise to a straight line or, if the influence of the fifth-order term 54 σ 5 (3ν) (E 0 )4 term is

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Fig. 2. Schematic of the Novocontrol Alpha-AK high performance impedance analyser in combination with the HVB4000 and Trek 623b high voltage amplifier [31].

not negligible, a curved line. In the latter case, we derive σ 3 (ν) from the linear coefficient of a second-order fit. In Fig. 3, we show typical isotherms of the low-field conductivity σ 1 (ν) and of the third-order conductivity σ 3 (ν) for the LNAS20 glass. Both spectra are characterised by a dc conductivity plateau regime at low frequencies and a dispersive regime at higher frequencies.

3.2 Apparent jump distances In Fig. 4 we show typical Nyquist plots of the complex impedance coefficients Zˆ 1 (ν) and Zˆ 3 (3ν) of the LNAS0 glass sample [17], with Zˆ 1 (ν) = 1/Yˆ1 (ν) and

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Fig. 3. Low-field conductivity spectra σ1 (ν) and third-order conductivity spectra σ3 (ν) of the mixed alkali glass LNAS20 at different temperatures; the applied field was 113 kV/cm; empty symbols denote negative values and filled symbols positive values, respectively.

Fig. 4. Nyquist representation of impedance coefficients (a) Zˆ 1 (ν), and (b) Zˆ 3 (ν); (c) conductivity spectra of the LNAS0 glass sample (applied field 71 kV/cm, temperature 263 K); same frequency ranges are represented by red and blue circles and lines, respectively [17].

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Fig. 5. Apparent jump distances of various sodium germanate glasses; lines are drawn to guide the eye.

Zˆ 3 (3ν) = 1/Yˆ3 (3ν), respectively. The complex admittances themselves are related to the complex conductivity coefficients via [17]: A 3A σˆ 1 (ν) + σˆ 3 (ν)(E 0 )2 d 4d A (8) Yˆ3 (3ν) = − σˆ 3 (3ν) (E 0 )2 4d where A and d are the surface area and the thickness of the sample, respectively. The low-field impedance plot exhibits the usual semicircle in one quadrant of the complex plane, characteristic for ion transport in the bulk. The thirdorder impedance plot, on the other hand, covers three quadrants. Thus, Zˆ 3 (3ν) exhibits two changes in sign. At higher frequencies, there is a change in the sign of the real part, Z 3 (3ν), which reflects the change in sign of σ 3 (ν) as shown in Fig. 3. This feature will be discussed in Sect. 3.3. At lower frequencies, the imaginary part Z 3 (3ν) changes its sign, reflecting a crossover from bulk ion transport to ion blocking effects at the solid/liquid interface [17]. The value of the real part Z 3 (3ν) at this low-frequency crossover was taken as dc value Z 3, dc and was used for calculating σ 3, dc . Apparent jump distances for the various glasses were then derived using Eq. (5). In Figs. 5–7, we show results for the sodium germanate glasses, the mixed lithium-sodium aluminosilicate glasses and for the partially crystallised aluminosilicate glasses. All glasses and glass ceramics studied here show a decrease of the apparent jump distances with increasing temperature. Yˆ1 (ν) =

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Fig. 6. Apparent jump distances of lithium-sodium aluminosilicate glasses; lines are drawn to guide the eye.

Fig. 7. Apparent jump distances of LAS glass and glass ceramics; the lines are drawn to guide the eye.

We note that in a recent publication, we described a distinct feature of the apparent jump distances of a NG15 glass [12], namely an increase of a app with increasing temperature. However, this finding could not be confirmed on application of more stringent experimental conditions, in particular a reduced water content of the liquid electrolyte. Thus, the feature described in Ref. [12] was

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most likely due to a slow dissolution of the glass sample by the attack of water being present in the hygroscopic glycerol. As shown in Fig. 5, the apparent jump distances of the NGx glasses do not show a clear compositional trend. In the temperature range of the experiments, the NG30 glass exhibits the highest a app values, while the NG15 shows the steepest decline of a app with increasing temperature. Although considerable progress has recently been made in the theoretical understanding of the mixed alkali effects in glasses [32,33], the origin of the effect still eludes full explanation. The mixed alkali effect manifests, for instance, as a strong drop of the low-field dc conductivity, when one type of alkali ion is partially replaced by a dissimilar type of alkali ion, while keeping the total alkali content constant [34]. Here, we show results for the apparent jump distances in an aluminosilicate glass system with varying contents of lithium and sodium oxide, see Fig. 6. We find that replacing a small amount of lithium by sodium has little influence. However, in the 1 : 1 mixture, the a app values are considerably lower than in the pure single alkali glasses. Thus, the mixed alkali effect manifests in a drop of the apparent jump distance. Up to a degree of crystallisation of 43%, partial crystallisation of the LAS glass leads to an increase of the conductivity of the sample, followed by a decrease in conductivity [35]. It is believed that the increase is due to the existence of fast ion conduction pathways at the interface between the amorphous and the crystalline phase. The question is whether this has an effect on a app . Figure 7 demonstrates that this is indeed the case. Partial crystallisation of the glass by 6% and 13% leads to a significant increase of the apparent jump distance. As already mentioned, the apparent jump distance is a phenomenological quantity which cannot, in a simple fashion, be related to length scales of the ion transport, such as hopping distances between neighbouring sites. In the past few years, considerable progress has been made in the theoretical description of nonlinear ion transport in disordered materials [12,36,37]. In this context, we point to the theoretical article by Friedrich, Heuer and coworkers in this special issue [38]. Nevertheless, the physical meaning of the apparent jump distances is still not well understood. Therefore in this article, we limit ourselves to the description of experimental trends in the apparent jump distances.

3.3 Jonscher power law and Summerfield scaling Typical conductivity spectra of the low-field conductivity σ 1 (ν) and of the third-order conductivity σ 3 (ν) are shown in Fig. 3. The low-field conductivity σ 1 (ν) is positive throughout the whole frequency range. In contrast, σ 3 (ν) displays a change in sign in a frequency range close to the onset of dispersion in σ 1 (ν). The values of σ 3 (ν) are positive in the dc regime and negative in the dispersive regime. Consequently, in the log–log representation shown in Fig. 3 we have plotted the modulus of σ 3 (ν). Analysis of the apparent slopes

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Fig. 8. Higher-order conductivity spectra σ3 (ν) of: (a) the LAS, am glass at 273 K and (b) LAS, 13% cryst. glass ceramic at 253 K; filled symbols correspond to positive values and empty symbols to negative values, respectively; the solid line denote fits with the Jonscher-type power law expression (8). d log|σ  (ν)| in the dispersive regime shows that p 3 = d log3ν ≈ 0.85–0.95 is considerably larger than the apparent slopes in the dispersive part of the σ 1 (ν) spectra, d log(σ 1 (ν)) p 1 = d log(ν) . Consequently, we tested whether it is possible to fit the σ 3 (ν) spectra with a modified version of the Jonscher power law:    p3  ν (9) σ 3 (ν) = σ 3, dc 1 + ∗ ν3

with an exponent p 3 . While it was possible to fit the σ 3 (ν) spectra of all single alkali glasses, the spectra of partially crystallised LAS glasses and of mixed alkali glasses are not well fitted by Eq. (9), see Fig. 8. Furthermore, we analysed the scaling properties of the third-order conductivity spectra σ 3 (ν). In Fig. 9 we display, as an example, the Summerfield scaling applied to σ 3 (ν) isotherms of a LNAS20 sample. In analogy to the well-known Summerfield scaling for σ 1 (ν), we normalised the σ 3 (ν) axis by σ 3, dc and the frequency axis by σ 3, dc T . Within the experimental error, the σ 3 (ν) isotherms superimpose. Note that the scatter of the data close to the change in sign is caused by the small modulus of σ 3 (ν). The σ 3 (ν) isotherms of the other glasses and glass ceramics could also be superimposed by means of the Summerfield scaling. The resulting master curves of the various samples are summarised in Figs. 10–12. The Summerfield master curves of the sodium germanate glasses shown in Fig. 10 show a considerable shift with respect to each other in the dispersive regime. In comparison, the shifts between the Summerfield master curves of the low-field conductivity are much smaller. This implies that the NG15 exhibits the strongest nonlinearity of the dispersive conductivity, while the NG25 glass exhibits the weakest nonlinearity.

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Fig. 9. Summerfield plot of the third-order conductivity spectra σ3 (ν) of the LNAS20 glass.

Fig. 10. Summerfield plots of (a) the low-field conductivity spectra σ1 (ν) and (b) the third-order conductivity spectra σ3 (ν) of sodium germanate glasses; values at high frequencies are negative and those at low frequencies positive; lines are linear fits to the data in the dispersive regime.

The σ 3 (ν) master curves of the mixed alkali LNAS glass system are shown in Fig. 11. Here, the single alkali glass containing exclusively Na+ ions exhibits the weakest nonlinearity in the dispersive regime, while the 1 : 1 mixed alkali glass exhibits the strongest nonlinearity. Overall, the data indicate that mixed alkali glasses are characterised by a stronger nonlinearity of the dispersive conductivity than single alkali glasses. The Summerfield master curves of the LAS glass and glass ceramics are shown in Fig. 12. We observe that partial crystallisation leads to a weakening of the nonlinearity of the dispersive conductivity. In order to rationalize the observed features in the frequency dependence of σ 3 (ν), we address now two questions: (i) Why is σ 3 (ν) negative in the dispersive regime? (ii) What can be the reason for different strengths of the nonlinear effect in the dispersive conductivity of different materials? In this context, we

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Fig. 11. Summerfield plots of (a) the low-field conductivity spectra σ1 (ν) and (b) the third-order conductivity spectra σ3 (ν) of the mixed alkali glasses LNASx; values at high frequencies are negative and those at low frequencies positive; lines are linear fits to the data in the dispersive regime.

Fig. 12. Summerfield plots of (a) the low-field conductivity spectra σ1 (ν) and (b) the higherorder conductivity spectra σ3 (ν) of the LAS glass ceramics (amorphous and partially crystallised); values at high frequencies are negative and those at low frequencies positive; lines are linear fits to the data in the dispersive regime.

note that negative σ 3 (ν) values in the dispersive regime have already been observed in computer simulations [12]. However, a physical explanation has not yet been given. As a starting point, we consider ion hopping in a disordered potential landscape. On short time scales, the ions carry out hops between neighbouring sites. Forward–backward correlations lead to subdiffusive dynamics and to a dispersion of the conductivity [39,40]. When the time window is small enough, the ion dynamics can be described by a double minimum potential (DMP) approx-

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Fig. 13. Hopping dynamics of an ion in the presence of a field (a) in a symmetric double minimum potential and (b) in an asymmetric double minimum potential; Γ+ is the hopping rate in the direction of the field, and Γ− the hopping rate in the opposite direction; E a is the barrier height, ΔV the site energy difference, and a denotes the hopping distance.

imation, as sketched in Fig. 13. The underlying assumption is that the ions hop essentially between two neighbouring sites. If these sites have the same energy, the DMP is called “symmetric” (Fig. 13a), otherwise it is called “asymmetric” (Fig. 13b). From these considerations, it is evident that the DMP approximation is limited to short time scales and cannot be used to describe the transition from subdiffusive to diffusive motion on long time scales. In the following, we will calculate the nonlinear ion currents in the time domain under the influence of a constant electric field. We would like to stress that in the nonlinear case, there is no general relation between this time-domain current and the frequency-domain current under the influence of sinusoidal electric fields. Therefore, it will not be possible to draw a quantitative comparison between the time-domain DMP calculations and our experimental results in the frequency-domain. However, we are of the opinion that the time-domain calculations are helpful to understand qualitative aspects, such as the positive or negative sign of σ 3 (ν) and the strength of the nonlinear effects. As will be discussed later, this is supported by a good qualitative agreement between our time-domain results for a symmetric DMP and a frequency-domain calculation of the nonlinear dielectric relaxation of dipolar molecules [41]. Within the DMP approximation, two different time regimes can be distinguished. On very short time scales, the ion dynamics are characterised by few uncorrelated hops. In the frequency domain, this corresponds to the high frequency plateau regime [39,40], which is not accessible in our experiments. On longer time scales, the ion current in the DMP decreases until a constant electrical polarisation is reached. In the frequency domain, this leads to a dispersive conductivity. Consequently, we will distinguish between the “very-short-time limit” and the “longer-time limit” of the DMP approximation. The “longer-time limit” should not be confused with the transition to diffusive motion on time scales when the ions possess a long-range mobility within the potential landscape.

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We start with the symmetrical DMP with identical energies of ion sites A and B. At time 0, the probability to find an ion at either site is equal. On application of an electric field the time-dependent probability to find an ion in site B at time t, WB (t), is described by: dWB (t) = Γ+ − (Γ+ + Γ− ) WB (t) dt

(10)

with Γ+ (t) = Γ exp(u) and Γ− (t) = Γ exp(−u) denoting the hopping rate of the ion in and against the direction of the field, respectively. The parameter u . Γ is the hopping rate in the absence of a field, q is the is given by: u = 2kqaE BT charge of the ion, a is the hopping distance between the two sites, and k B and T have the usual meaning. An exponential decay function with a relaxation time τ can be used to solve the differential equation: WB (t) = C + D exp(−t/τ)

(11)

Combining Eqs. (10) and (11) yields the following expressions for the parameters C and D and for the relaxation time τ: Γ+ , D = 0.5 − C = − tanh(u) Γ+ + Γ− 1/τ = Γ+ + Γ− = 2Γ cosh(u)

C=

(12)

Upon application of very high fields, the long-time probability to find an ion at site B approaches unity and the average dipole moment approaches 12 qa. Therefore, the polarisation in the limit of longer times can be expressed as: 1 a NV q [W (t → ∞) − W (t = 0)] 3 2 1 a 1 = NV q (−D) = NV qa tanh(u) 3 2 6

P∞ =

(13)

On the other hand, the current density in the very-short-time limit is given by:   dWB (t) 1 a a 1 1 = NV q (−D) j0 = NV q lim 3 2 t→0 dt 3 2 τ 1 = NV qaΓ sinh(u) (14) 3 The interesting point now is the behaviour of the current densities in the nonlinear case, i.e. when a large field is applied. On very short time scales, the current density is dominated by a sinh function. The Taylor expansion of sinh(u) = u + 1 3 u + ... shows that at high fields, the current density increases superlinearly 6 with increasing field. On the other hand, the longer-time polarisation P∞ increases sublinearly with increasing field according to tanh (u) = u − 13 u 3 + ....

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Since the polarisation is obtained by temporal integration over j(t), a superlinear increase of the very-short-time current and a sublinear increase of the longer-time polarization with increasing field imply a faster temporal decay of j(t) at high fields. This is directly seen from the field dependence of the inverse relaxation time, 1/τ ∝ cosh (u) = 1 + 12 u 2 + .... Thus, on time scales t  τ, the time-dependent current density can be written as:     1 1 3 j (t) = NV qaΓ u + u + exp (−2Γt) exp −Γu 2 t (15) 3 6   For t  τ, the term exp −Γu 2 t overcompensates the 16 u 3 term, leading to a drop of the longer-time current density with increasing field. This is illustrated in a plot of the time-dependent effective conductivity σ eff (t) ≡ Ej(t)dc in Fig. 14a. Remarkably, the same type of nonlinear dynamic behaviour was found by D´ejardin and Kalmykov [41] in a frequency-domain calculation of the nonlinear dielectric relaxation of rigid dipolar molecules, see Fig. 15. The reorientational motion of dipolar molecules bears strong analogies to localised hopping motions of ions, since both types of motion lead to fluctuating dipole moments, which are influenced by external electric fields. The low-field dielectric loss ε1 (ν) caused by the random reorientation of dipolar molecules is characterised by a Debye peak. The third-order dielectric loss ε3 (ν) is positive at high frequencies and crosses over into negative values at low frequencies. The positive values of ε3 (ν) at high frequencies have the same physical origin as our superlinear increase of j(t) with increasing field. Namely, a strong electric field pushes the orientation of the dipoles and the movements of ions, respectively, in the direction of the field. This leads to a faster establishment of the longer-time polarisation, resulting in a sublinear increase of the longertime current with increasing field (for both rigid dipolar molecules and ions in DMP). In the frequency domain, this manifests in negative values for ε3 (ν) and σ 3 (ν), respectively. Next, we consider the asymmetric DMP as sketched in Fig. 14b. The asymmetry is characterised by the site energy difference ΔV . Now, we have to differentiate between two cases: (i) ΔV > 0, i.e. site A has a lower potential minimum than site B and (ii) ΔV < 0, where site B has the lower potential minimum.  In case (i) ΔV > 0, we define the jump rates as Γ+ (t) = Γ exp − kΔV BT × exp(u) and Γ− (t) = Γ exp (−u). Following the same procedure as for the symmetric case we obtain the following expression for the inverse relaxation time 1/τ:

   ΔV 1/τ = Γ+ + Γ− = Γ exp − exp(u) + exp(−u) ≈ Γ exp(−u) kB T (16)

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Fig. 14. Schematic illustration of the time-dependent effective conductivity in (a) a symmetric double-well potential at different field strengths; (b) a strongly asymmetric double-well potential; the dashed lines represent the effective conductivity at low fields, while the solid lines represent the effective conductivity at higher fields.

The approximation in Eq. (16) holds for strong asymmetries ΔV  k B T . In this case, the longer-time polarisation P∞ and the very-short-time current density j0 are given by:   1 NV qa exp (−ΔV/k B T ) exp (2u) − 1 6 1 j0 = NV qaΓ exp (−ΔV/k B T ) sinh(u) 3

P∞ =

(17)

Equations (16) and (17) imply that both the very-short-time current and the longer-time polarisation increase with increasing field and that the decay of j(t) becomes slower with increasing field. Thus, in contrast to the symmetric DMP, the current density j(t) increases superlinearly with increasing field on all time scales, as illustrated in Fig. 14b. Furthermore, it is important to note that the

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Fig. 15. Low-field and third-order dielectric loss spectra for the reorientation of rigid dipolar molecules in a strong electric field [41]. Both ε1 and ε3 were normalised by their maximum values to allow for better comparison.

factor exp (−ΔV/k B T ) implies smaller current densities and polarisations than for the symmetric DMP: In case (ii) ΔV < 0, we define the jump rates as Γ+ (t) = Γ exp(u) and | Γ− (t) = Γ exp − |ΔV exp (−u). This results in: kB T

   |ΔV | exp(−u) ≈ Γ exp(u) 1/τ = Γ+ + Γ− = Γ exp(u) + exp − kB T   1 P∞ = NV qa exp (− |ΔV | /k B T ) 1 − exp (−2u) 6 1 j0 = NV qaΓ exp(− |ΔV | /k B T) sinh(u) (18) 3 The very-short-time current is identical to case (i), however in contrast to case (i), the exponential decay of j(t) becomes faster with increasing field, see Fig. 14b. This results in a decrease of the longer-time current density j(t  τ) with increasing field. In a disordered potential landscape, we expect a broad distribution of asymmetries. Therefore, we have to average over (a) positive and negative ΔV values and over (b) a broad range of |ΔV | values. Let us first consider average (a). From Eqs. (16) and (18) it is evident that the major contribution to the longer-time current density j(t  τ) arises from the DMP with ΔV > 0. Thus, in the case of strong asymmetries, the longer-time current density increases superlinearly with increasing field. For average (b), it is important to

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realize that the current density j(t) is proportional to exp (− |ΔV | /k B T ), i.e. the major contribution to the overall current arises from DMPs with small |ΔV | values. These DMPs are characterised by a sublinear increase of j(t  τ) with increasing field. Thus, we expect the more symmetric DMPs to govern the nonlinearity of the dispersive conductivity. In the DMP approximation, this is the origin of negative σ 3 (ν) values in the dispersive regime. The DMPs with large |ΔV | values give positive, but rather small contributions to σ 3 (ν) in the dispersive regime. Consequently, an increasing width of the |ΔV | distribution should lead to less negative σ 3 (ν) values. In summary, we state that in the framework of the DMP approximation, the asymmetry of the potential landscape plays an important role for the nonlinearity of the dispersive conductivity. For a better comparison with experimental results, frequency-domain calculations for asymmetric DMPs should be carried out. As seen in the work by D´ejardin and Kalmykov [41] this requires considerable more effort than calculations in the time domain. Furthermore, it will be important to check the range of applicability of the DMP approximation by comparing our results to computer simulations of the nonlinear ion conductivity in disordered potential landscapes.

4. Conclusions We have measured third-order conductivity spectra of various ion conducting glasses and glass ceramics in a frequency range from 10 mHz to 1 kHz. The third-order spectra σ 3 (ν) were obtained by analysing higher harmonics in the current density after applying ac electric fields with large amplitude. From the dc values of the low field conductivity σ 1, dc and of the third-order conductivity σ 3, dc , we have calculated apparent jump distances a app . The apparent jump distances of the sodium germanate and of the mixed alkali and partially crystallised aluminosilicate glasses are in a range from 20–40 Å and decrease with increasing temperature. While the mixed alkali effect manifests in a minimum of a app , partial crystallisation leads to an increase of a app . For obtaining a better understanding of the physical meaning of a app , more theoretical work is required, see the article by Friedrich, Heuer and coworkers in this special issue [38]. The frequency-dependent σ 3 (ν) spectra are characterised by a change in sign, from positive values in the dc regime to negative values in the disper

sive regime. In the dispersive regime, the modulus σ 3 (ν) shows an approxid log|σ  (ν)| mate power-law behaviour, however with a slope p 3 = d log3ν ≈ 0.85–0.95 significantly larger than found for the low-field conductivity σ 1 (ν). For all glasses and glass ceramics, both the low-field spectra σ 1 (ν) and the third-order spectra σ 3 (ν) obey the time–temperature superposition principle. For a given material, master curves of σ 3 (ν) can be obtained by applying the Summerfield scaling. The resulting σ 3 (ν) master curves of different materials exhibit

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remarkably strong shifts with respect to each other on the scaled frequency axis. These experimental findings were rationalised by considering ion hopping in a double minimum potential (DMP) approximation. In the framework of this approximation, the strength of the nonlinear effects is strongly influenced by the asymmetry of the DMPs.

Acknowledgement Financial support of this work by the German Science Foundation (DFG) in the framework of the SFB 458 and of the Individual Grants Programme is gratefully acknowledged. Furthermore, we would like to thank Andreas Heuer, Rudolf Friedrich and Sevi Murugavel for valuable discussions.

References 1. K. Noda, T. Yasuda, and Y. Nishi, Electrochim. Acta 50 (2004) 243. 2. G. B. Appetecchi, J. Hossoun, B. Scrosati, F. Croce, F. Cassel, and M. Salomon, J. Power Sources 124 (2003) 246. 3. L. C. D. Jonghe, C. P. Jacobson, and S. J. Visco, Annu. Rev. Mater. Res. 33 (2003) 169. 4. J. M. Tarascon and M. Armand, Nature 414 (2001) 359. 5. E. P. Murray and S. A. Barnett, Nature 400 (1999) 649. 6. F. Croce, G. B. Appetecci, L. Persi, and B. Scrosati, Nature 394 (1998) 456. 7. L. Lacaita and D. J. Wouters, Phys. Status Solidi (a) 205 (2008) 2281. 8. J. M. Hyde and M. Tomozawa, Phys. Chem. Glasses 27 (1986) 147. 9. J. L. Barton, J. Non-Cryst. Solids 203 (1996) 280. 10. J. O. Isard, J. Non-Cryst. Solids 202 (1996) 137. 11. A. Heuer, S. Murugavel, and B. Roling, Phys. Rev. B 72 (2005) 174304. 12. B. Roling, S. Murugavel, A. Heuer, L. Lühning, R. Friedrich, and S. Röthel, Phys. Chem. Chem. Phys. 10 (2008) 4211. 13. H. Staesche and B. Roling, Phys. Rev. B 82 (2010) 134202. 14. P. Vinatier and Y. Hamon, in Charge Transport in Disordered Solids, edited by S. Baranovski (John Wiley & Sons, Ltd., 2006). 15. J. Vermeer, Physica 22 (1956) 1257. 16. J. P. Lacharme and J. O. Isard, J. Non-Cryst. Solids 27 (1978) 381. 17. H. Staesche, S. Murugavel, and B. Roling, Z. Phys. Chem. 223 (2009) 1229–1238. 18. C. B. Crauste-Thibierge, F. Ladieu, D. L’Hote, G. Biroli, and J.-P. Bouchaud: arXiv:1002.0498v2. 19. N. F. Mott and E. A. Davis, Electronic Processes in Non-Crystalline Materials, Clarendon, London (1979). 20. Y. Tajitsu, J. Mater. Sci. 31 (1996) 2081. 21. Y. Tajitsu, J. Electrost. 43 (1998) 203. 22. L. Zagar and E. Papanilolau, Glastech. Ber. 42 (1969) 37. 23. A. Heuer, M. Kunow, M. Vogel, and R. D. Banhatti, Phys. Chem. Chem. Phys. 4 (2002) 3185. 24. K. Jonscher, Nature 267 (1977) 673. 25. J. C. Dyre, P. Maass, B. Roling, and D. L. Sidebottom, Rep. Prog. Phys. 72 (2009) 046501. 26. B. Roling, C. Martiny, and S. Bruckner, Phys. Rev. B 63 (2001) 9.

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S. Summerfield, Philos. Mag. B 52 (1985) 9. S. Summerfield and P. N. Butcher, J. Non-Cryst. Solids 77/78, (1985) 135. R. M. Biefeld, G. E. Pike, and R. T. Johnson, Phys. Rev. B 15, 5912 (1977). S. Murugavel and B. Roling, J. Non-Cryst. Solids 351 (2005) 2819. User’s Manual of the HVB 4000, Issue 2/2008 Rev. 1.0 by Novocontrol Technologies. P. Maass and R. Peibst, J. Non.-Cryst. Solids 352 (2006) 5178. R. Peibst, S. Schott, and P. Maass, Phys. Rev. Lett. 95 (2005) 115901. J. O. Isard, J. Non-Cryst. Solids 1 (1968) 235. B. Roling and S. Murugavel, Z. Phys. Chem. 219 (2005) 23. M. Einax, M. Koerner, P. Maass, and A. Nitzan, Phys. Chem. Chem. Phys. 12 (2010) 645. M. Kunow and A. Heuer, J. Chem. Phys. 124 (2006) 214703/1. S. Röthel, R. Friedrich, L. Lühning, and A. Heuer, Z. Phys. Chem. 224 (2010) this issue. K. Funke, Defect Diffus. Forum 143 (1997) 1243. K. Funke, C. Cramer, and B. Roling, Glass Sci. Techn. 73 (2000) 244. J. L. D´ejardin and Yu. P. Kalmykov, Phys. Rew. E 61 (2000) 1211.

Dynamics of Network Formers and Modifiers in Mixed Cation Silicate Glasses By K. Sunder, M. Grofmeier, R. Staskunaite, and H. Bracht∗ Institut für Materialphysik, Wilhelm-Klemm-Str. 10, 48149 Münster, Germany (Received July 7, 2010; accepted in revised form September 23, 2010)

Alkali / Alkaline-Earth / Silicate / Glasses / Network Formers / Network Modifiers / Self-Diffusion / Conductivity Conductivity measurements and diffusion studies with radioactive and stable isotopes were performed to investigate the diffusion of both network modifiers (alkali (A) and alkaline-earth (M) ions) and network formers (oxygen (O) and silicon (Si)) in various mixed-cation silicate glasses below the respective glass transition temperatures. The dynamics of the network modifiers A and M were studied in bulk glasses prepared from the melt and in sol–gel derived glass films. The diffusion anneals were performed under air and under reducing conditions in forming gas. No impact of the glass preparation on the diffusion of the network modifiers was found. However, the diffusion of the alkaline-earth ions is affected by additional defects formed under reducing conditions. The dynamics of the network formers O and Si was investigated against the type of alkali and alkaline-earth ions, their composition ratio, and the number of non-bridging oxygens. We found that Si diffusion is directly related to the rigidity of the glass. Combining the data on O diffusion with previous results on the diffusion of alkaline-earth ions M in glasses of the same composition, we conclude that the diffusion of O is likely assisted by M via the formation of O−M pairs.

1. Introduction Amorphous silicon dioxide doped with alkali and alkaline-earth oxides and even pure amorphous silicon dioxide are used for various applications not only in form of bulk glasses, but also as glass films. Bulk glasses are widely used as window glasses, solid electrolytes [1], and for sealing of nuclear waste [2]. Important applications of thin glass layers concern anti-reflection coatings of lenses [3], corrosion protection [4], and surface hardening [5]. Pure amorphous layers of silicon dioxide with only a few nanometer in thickness are used as insulating oxide in metal–oxide–semiconductor (MOS) structures that form one basic element of microelectronic devices [6]. Structures of amorphous silicon * Corresponding author. E-mail: [email protected] Z. Phys. Chem. 224 (2010) 1677–1705 © by Oldenbourg Wissenschaftsverlag, München

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dioxide layers with embedded nanoparticles are considered as a new class of flash memory devices [7] and solar cells [8]. In order to meet the requirements associated with the various applications of glasses, the specific functional properties of the material must be controlled. That is, glasses for solid electrolytes should possess high conductivities, whereas e.g. silica layers in MOS devices should be highly insulating. The properties of glasses that are important for almost all glass applications are the transport properties of the glass constituents. For solid electrolytes the mobility of ions determines the macroscopic conductivity, whereas both mobile ions and immobile charged defects affect the macroscopic capacitance–voltage characteristic of MOS structures. Engineering materials with improved properties requires an in-depth understanding of the type of defects, their interaction and atomic mobility. This, in particular, holds for crystalline materials but to some degree also for strongly bond oxide glasses, whose local structure resembles their crystalline phase. In this work we report experimental studies on the dynamics of all constituents of mixed alkali alkaline-earth silicate glasses in order to uncover collective jump processes. The mobilities of alkali and alkaline-earth ions both in bulk and thin film glasses are determined by means of conductivity measurements and radiotracer diffusion experiments, respectively. These studies demonstrate that the radii ratio of the alkali and alkaline-earth ions strongly affects the activation enthalpy of the divalent cations. The impact of alkali ions on the mobility of alkaline-earth ions holds for bulk glasses prepared from the melt and for glass films derived with the sol–gel technique. However, differences in oxygen diffusion in bulk and thin film glasses are observed and ascribed to oxygen related defects formed during sol–gel processing. Silicon diffusion studies were performed by means of isotopically enriched sol–gel glasses deposited as thin films on bulk silicate glasses. The activation enthalpy of oxygen and silicon diffusion as function of the glass composition in comparison to the results on the mobility of the divalent cations led us to propose the formation of pairs between oxygen and alkaline-earth ions that, in particular, contribute to the diffusion of oxygen in silicate glasses. In Sect. 2.1 experimental details on the diffusion of the network modifiers, i.e. of monovalent and divalent cations, and in Sect. 2.2 of the network formers oxygen (O) and silicon (Si) in mixed alkali (A) alkaline-earth (M) silicate glasses are described. The results on the dynamics of the network modifiers A and M and the network formers O and Si are presented and discussed in Sects. 3.1 and 3.2, respectively.

2. Experimental 2.1 Diffusion of network modifiers Bulk glasses were prepared from dried powders of Aerosil (99,8%) and the carbonates of alkali and alkaline-earth elements (> 99%) [9,10]. The material of

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the desired composition was melted in a platinum crucible at temperatures between 1770 and 1870 K for 5 to 6 h. The melts were casted into steel molds pre-heated at 570 K to obtain cylindrical bulk glasses of 30 mm height and 10 mm diameter. The glasses were annealed 50 K below their respective glasstransition temperatures Tg for 2 h and cooled down to room temperature with a rate of 1 K min−1 . The cylindrical bulk glasses were cut into discs of 1.8 mm thickness. One surface of each sample was mechanically grinded with sandpaper of 15 μm grain size, followed by a four-step polishing procedure with diamond suspensions of decreasing grain size from 9 to 1/4 μm to achieve a scratch- and pit-free surface. Two thin film glasses of the composition Na2 O · 2CaO · 4SiO2 and K 2 O · 2CaO · 4SiO2 were obtained by means of the sol–gel technique. A solution of tetraethyl orthosilicate Si(OC2 H5 )4 (TEOS) and nitrates of the cations in a ratio of the desired compositions was prepared and combined with deionized water for the hydrolysis and methanol as solvent. 1 molar HNO3 was added to the solution to get pH = 1. The sol–gel films were derived by spin coating on thermally oxidized silicon wafers with an oxide thickness of 135 nm. The samples were subsequently densified at 773 K in air. Ellipsometry measurements yielded layer thicknesses of approximately 100 nm. This thickness was confirmed by compositional investigations of the layer by means of time-of-flight secondary ion mass spectrometry (TOF-SIMS). Atomic force microscopy revealed a surface roughness of about 1 nm. The samples were cut into pieces with lateral dimensions of 7 × 7 mm2 for the diffusion experiments. The non-crystallinity of each glass was checked by X-ray diffraction, no partial crystallization was found. The densities measured with a floating method are in good agreement with calculations given by Appen [11] and Huggins and Sun [12]. Glass transition temperatures Tg were obtained by means of a differential scanning calorimeter (Perkin-Elmer DSC 7) at a heating rate of 10 K min−1 . These measurements show no phase separation in the glass systems and a strong correlation between Tg and the radii of the cations [13–15]. Prior to the diffusion anneal a chloride solution with the radioactive isotopes 45 Ca, 85 Sr, and 133 Ba, respectively, was dropped onto the surface of the sample and dried. Diffusion anneals were performed in encapsulated silica ampoules filled with dried air or in an open quartz tube under a flow of forming gas at temperature up to 50 K below Tg . In both cases a pressure of about 1 atm was established. The forming gas consists of a mixture of hydrogen gas H 2 and nitrogen gas N 2 in the ratio of 1 : 9. During diffusion annealing the temperature was controlled within ±1 K with a Pt/PtRh thermocouple. The temperatures and times set for the diffusion experiments under dried air and forming gas are given in Tables 1 and 2, respectively. The diffusion process was stopped by cooling the sample in air to room temperature. Serial sectioning was accomplished via an ion-beam sputtering technique [16]. The sputtered material was collected on a foil, which was fed automatically after a specific time of 20 s to 10 min. The activity of each foil section was counted in a NaI(Tl) crystal scin-

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Table 1. Self-diffusion coefficients DCa of 45 Ca in Na2 O · 2CaO · 4SiO2 and K 2 O · 2CaO · 4SiO2 glass films obtained after diffusion annealing at temperatures T and times t in dried air. Q Ca and DCa,0 are the corresponding diffusion activation enthalpies and pre-exponential factors, respectively. 45

T (K)

Ca in Na2 O · 2CaO · 4SiO2 t (s) DCa (m2 s−1 )

638 650 663 684 711

778 500 345 600 180 000 68 400 2160

2.2 × 10−21 3.6 × 10−21 1.5 × 10−20 3.7 × 10−20 1.2 × 10−19

45

T (K)

Ca in K 2 O · 2CaO · 4SiO2 t (s) DCa (m2 s−1 )

726 751 772 821

606 180 241 320 100 200 18 000

DCa,0 (m2 s−1 ) 45

Ca in Na2 O · 2CaO · 4SiO2 Ca in K 2 O · 2CaO · 4SiO2

45

+163 −5.4 +89 −3.5

(5.6 (3.6

) × 10 ) × 10−7

−4

2.8 × 10−21 7.0 × 10−21 2.8 × 10−20 1.1 × 10−19 Q Ca (eV) 2.20 ± 0.20 2.03 ± 0.21

Table 2. Self-diffusion coefficients DCa of 45 Ca in A2 O · 2CaO · 4SiO2 bulk glasses with A = Li, Na, K obtained after diffusion annealing under a flow of forming gas at temperatures T and times t. Q Ca and DCa,0 are the diffusion activation enthalpies and pre-exponential factors, respectively. 45

T (K)

Ca in Li2 O · 2CaO · 4SiO2 t (s) DCa (m2 s−1 )

640 674 709 738 768

608 220 432 000 173 700 86 400 86 400 45

764 790 814

1.6 × 10−20 1.1 × 10−19 3.2 × 10−19 6.6 × 10−19 1.3 × 10−18

45

T (K)

Ca in Na2 O · 2CaO · 4SiO2 t (s) DCa (m2 s−1 )

669 717 741 767 795

852 000 272 640 166 620 86 400 85 440

1.0 × 10−20 1.2 × 10−19 3.7 × 10−19 8.1 × 10−19 4.2 × 10−18

837 865

86 400 86 400

1.6 × 10−19 5.0 × 10−19

Ca in K 2 O · 2CaO · 4SiO2 431 400 203 880 173 040

1.4 × 10−20 4.5 × 10−20 1.0 × 10−19

DCa,0 (m2 s−1 ) Ca in Li2 O · 2CaO · 4SiO2 Ca in Na2 O · 2CaO · 4SiO2 45 Ca in K 2 O · 2CaO · 4SiO2

45 45

+24 −3.0 +11 −2.2 +35 −6.8

(3.4 (2.7 (8.5

) × 10−9 ) × 10−6 ) × 10−8

Q Ca (eV) 1.42 ± 0.12 1.90 ± 0.10 1.93 ± 0.11

tillation counter. For the bulk material, the depths of the circular craters left from sputtering were determined by the density of the glass, the diameter of the crater, and the mass difference of the sample before and after sectioning. For the glass films, the crater depths were determined by means of an optical profilometer. Assuming constant sputter rates, which were checked by measurements of the ion flux through the sample holder during the sputter process,

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Fig. 1. Diffusion profiles of 45 Ca in sol–gel derived (a) Na2 O · 2CaO · 4SiO2 and (b) K 2 O · 2CaO · 4SiO2 glass films measured after diffusion annealing at various temperatures under dried air. Fits illustrated by the solid lines are obtained on the basis of a convolutional integral, which takes the instrumental broadening effects of the sputter system into account (see [16] for details). The dotted line indicates the interface between the mixed cation glass film and the SiO2 layer. The Ca diffusion coefficients and the corresponding activation enthalpies and pre-exponential factors are listed in Table 1.

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Fig. 2. Diffusion profiles of 45 Ca in A2 O · 2CaO · 4SiO2 bulk glasses prepared from the melt with A = Li (a), Na (b), K (c) measured after diffusion annealing under forming gas flow at various temperatures. Fits illustrated by the solid lines are obtained on the basis of a convolutional integral, which takes the instrumental broadening effects of the sputter system into account (see [16] for details). The Ca diffusion coefficients and the corresponding activation enthalpies and preexponential factors are listed in Table 2.

the activity vs. sputter-time profiles were transformed to activity vs. depth profiles. In the case when the sputter profiles of the glass films extend into the substrate, sputter rates separately determined for the glass films and the silicon substrate were used to calculate the penetration depth of the radiotracer. The penetration profiles were accurately described by a convolution of the thinfilm solution of Fick’s second law and a contribution, which takes into account the broadening of the profiles due to instrumental effects (see [16]). Figure 1 shows Ca profiles in sol–gel derived glass films obtained after diffusion annealing in dried air. Figure 2 illustrates Ca profiles in bulk glasses prepared from the melt after annealing in a flow of forming gas. The solid lines in Figs. 1 and 2 represent best fits to the experimental profiles. In order to determine the alkali ion mobility in glass films of the compositions listed in Table 3, conductivity measurements were performed. Sol–gel glasses were derived on silicon wafers with a sputtered platinum layer on top. Gold contacts were evaporated on top the glass film. Conductivities were measured at frequencies ranging from 101 to 106 Hz and at temperatures between 490 to 730 K using a HP Agilent 4192A and a Novocontrol α-S high resolution

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Fig. 2. Continued.

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Table 3. Activation enthalpies Q dc A determined from conductivity measurements of A 2 O · 2MO · 4SiO2 glass films with A = Li, Na, K and M = Mg, Ca, Sr. The results are compared to data obtained from bulk glasses of the same composition [20]. Thin film glasses This work Q dc A (eV)

Bulk glasses [20] Q dc A (eV)

Li2 O · 2MgO · 4SiO2 Li2 O · 2CaO · 4SiO2

1.06 ± 0.01 1.10 ± 0.01

0.95 1.11

Na2 O · 2MgO · 4SiO2 Na2 O · 2CaO · 4SiO2 Na2 O · 2SrO · 4SiO2

0.85 ± 0.01 1.10 ± 0.01 1.12 ± 0.01

0.90 1.11 1.19

K 2 O · 2CaO · 4SiO2 K 2 O · 2SrO · 4SiO2

1.11 ± 0.01 1.16 ± 0.01

1.08 1.17

Fig. 3. Conductivity spectra of a Li2 O · 2CaO · 4SiO2 glass film at different temperatures.

dielectric analyzer. Conductivity spectra of a Li2 O · 2CaO · 4SiO2 glass film are illustrated in Fig. 3 for different temperatures.

2.2 Diffusion of network formers For the investigation of the diffusion of the network formers O and Si, bulk glass samples were melted and the crystallinity, density, and Tg were measured

Q O (eV) 1.50 ± 0.10 1.99 ± 0.20 0.92 ± 0.16 1.66 ± 0.36 1.99 ± 0.20 2.81 ± 0.36 1.31 ± 0.22 1.99 ± 0.20 4.06 ± 0.25 1.31 ± 0.26 1.99 ± 0.20 1.94 ± 0.15

Glass composition

Li2 O · 2CaO · 4SiO2 Na2 O · 2CaO · 4SiO2 K 2 O · 2CaO · 4SiO2

Na2 O · 2MgO · 4SiO2 Na2 O · 2CaO · 4SiO2 Na2 O · 2SrO · 4SiO2

0.5Na2 O · 1CaO · 4SiO2 Na2 O · 2CaO · 4SiO2 1.25Na2 O · 2.5CaO · 4SiO2

3CaO · 4SiO2 Na2 O · 2CaO · 4SiO2 1.5Na2 O · 1.5CaO · 4SiO2

  −12 5.76+21.3 −4.53  × 10  −9 7.61+130 × 10 −7.19   −15 4.98+43.7 −4.47 × 10   −11 3.14+492 −3.12  × 10  +130 −9 7.61−7.19 × 10   −3 7.61+1522 −7.57 × 10   −13 5.25+135 −5.05  × 10  −9 7.61+130 × 10 −7.19   +5 6.55+292 −6.41 × 10   −14 2.90+84.2 −2.80  × 10  −9 7.61+130 × 10 −7.19   −8 2.38+20.7 −2.14 × 10

DO,0 (m2 s−1 )

− 5.2 ± 1.0 −

− 5.2 ± 1.0 5.3 ± 0.5

6.7 ± 0.8 5.2 ± 1.0 −

− 5.2 ± 1.0 8.5 ± 0.8

Q Si (eV)



−  969 734 9.2+8 × 109 −9.2 −

−  +8 969 734  9.2−9.2 × 109  +14 637  8.8−8.8 × 1011

−  9.2+8 969 734 × 109   −9.2 +5884 2.5−2.5 × 1025  +7116  2.3 × 1017  +8−2.3  969 734 9.2−9.2 × 109 − 

DSi,0 (m2 s−1 )

1039 873 813

893 873 869

880 873 834

812 873 895

Tg (K)

Table 4. Compositions of mixed cation silicate glasses and diffusion activation enthalpies Q and pre-exponential factors D0 determined from O and Si diffusion experiments. The Na2 O · 2CaO · 4SiO2 glass is shown four times for better comparison between glasses with varying alkali ions (first row), varying alkaline-earth ions (second row), varying number of non-bridging oxygens (third row), and varying alkali to alkaline-earth ratio (fourth row).

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by means of the methods mentioned in Sect. 2.1. Cylindrical glass samples were sliced and polished as described above. The bulk glasses used for the network diffusion experiments are listed in Table 4. These glasses cover a compositional range that allows us to investigate the dependence of O and Si diffusion on the type of the monovalent cations A and divalent cations M, the alkali to alkaline-earth ratio, and the concentration of non-bridging oxygens (NBOs). The latter comprises the oxygens bound to both alkali and alkalineearth ions. Accordingly, the number of NBOs can be calculated by the number of alkali ions and the number of alkaline-earth ions (x NBO = x A + 2x M ). For the O diffusion experiments the glass samples were annealed under isotopically enriched 18 O2 -atmosphere at temperatures between 673 and 973 K and an O2 pressure of approximately 1 atm. The 18 O-diffusion profiles were established due to surface exchange between network 16 O (naturally abundance 99.76%) and 18 O2 from the gas phase. The oxygen profiles were recorded by means of time-of-flight secondary ion mass spectrometry (TOF-SIMS) at TASCON GmbH Münster in Germany. The measurements were performed in the dual beam mode using 2 keV Cs+ ions for sputtering and 25 keV Bi+ ions for analysis. The profiles were normalized to the total amount of O (C 0O = C 16 O + C 18 O ) and fitted with the following error-function solution of Fick’s second law of diffusion   x (1) + C2 , C O (x) = (C 1 − C 2 ) · erfc √ 2 DO t where C 1 , C 2 , x, and D O are the surface and background concentrations of 18 O, the penetration depth, and the O diffusion coefficient, respectively. Although all O diffusion anneals were performed in 18 O-enriched ambient that represents an infinite 18 O source, some O profiles were better described with the thin film solution   −x 2 (2) + C2 C O (x) = (C 1 − C 2 ) exp 4D O t that is expected in the case of a finite 18 O source. On closer inspection of the TOF-SIMS data it became evident that the 18 O profiles, that are best fitted by Eq. (2), correspond to samples with a depletion of alkali and alkaline-earth ions near the surface. This change of glass composition in the surface region expands about 15 to 25 nm into the bulk and causes a retarded O diffusion in the depleted region compared to the bulk. This limits the O supply and is considered to lead to Gaussian shaped profiles. Oxygen diffusion profiles together with best fits based on Eqs. (1) and (2) are shown in Fig. 4a. Diffusion of silicon from the gas phase into glasses is not possible due to the low partial pressure of Si at the diffusion temperatures. Moreover, diffusion studies with radioactive Si isotopes are not feasible due to the short half-life of 2.6 h of the radioactive isotope 31 Si. In order to explore the diffusion of Si in

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Fig. 4. Normalized diffusion profiles measured with TOF-SIMS in mixed cation silicate glasses as indicated together with best fits. (a) 18 O-profiles were described by Eqs. (1) or (2) (see text for details). (b) 30 Si-profiles were fitted taking into account the pre-annealed profiles as initial and solving Fick’s law of diffusion numerically.

silicate glasses, we prepared thin layer structures with enriched stable Si isotopes by means of the sol–gel technique. For this purpose isotopically enriched TEOS was synthesized from highly enriched 28 Si. Details of the synthesis are described in Ref. [17]. Sol–gel layers of 28 SiO2 were prepared by spin-coating on polished bulk glasses of compositions listed in Table 4. Altogether nine

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different glass substrates were used for the deposition of an isotopically enriched 28 SiO2 film. However, the isotope layer on the Na2 O · 2SrO · 4SiO2 glass cracked during consolidation. Moreover, from the remaining eight glasses only four revealed a uniform distribution of alkali and alkaline-earth atoms in both the bulk material and the glass film after consolidation. These four glass samples were used for Si diffusion experiments under flowing O2 at temperatures between 773 and 973 K and an O2 pressure of 1 atm. After diffusion annealing the profiles of 30 Si were recorded by means of TOF-SIMS using 1 keV O2− ions for sputtering and 25 keV Bi+ ions for analysis. The profiles were normalized to the total amount of Si (C 0Si = C 28 Si + C 29 Si + C 30 Si ) and described by the solution of Fick’s second law of diffusion. The pre-annealed profiles measured after consolidation were considered as initial profiles for the solution that was determined numerically using a software package provided by Jüngling et al. [18]. Examples of 30 Si profiles and best fits are illustrated in Fig. 4b. The limited amount of isotopically enriched 28 Si restricted our study to the diffusion of Si in sol–gel derived glass films. No isotopically enriched bulk glasses could be prepared. The Si diffusion experiments with sol–gel derived glass films were conducted under open volume conditions in a gas flow of natural oxygen. In contrast, O diffusion experiments were performed with bulk glasses under closed volume conditions in 18 O2 . In order to investigate Si diffusion under closed volume conditions and O diffusion in sol-del derived glass films, we prepared Na2 O · 2CaO · 428 SiO 2 glass films on a Na2 O · 2CaO · 4SiO2 bulk glass as described above and annealed them in 18 O2 at a pressure of about 1 atm. By means of these experiments the impact of the ambient conditions on Si diffusion and differences between O diffusion in bulk and thin film glasses can be investigated. The diffusion anneals were performed at temperatures between 723 and 873 K and the resulting 18 O and 30 Si profiles were recorded with TOF-SIMS and analyzed as mentioned above.

3. Results and discussion 3.1 Dynamics of network modifiers 3.1.1 Mixed cation effect The impact of the type of cations and the alkali to alkaline-earth ratio on the diffusion of alkaline-earth ions in mixed alkali alkaline-earth silicate glasses with compositions A2 O · 2MO · 4SiO2 (A = {Li, Na, K, Cs}; M = {Ca, Sr, Ba}) and xA2 O · (3 − x)MO · 4SiO2 (A = {Na, K}; M = {Ca, Ba}) has already been published (see: [13,14,16,19]). These investigations revealed that the diffusion of the alkaline-earth ions is strongly affected by the alkali to alkaline-earth radii ratio resulting in a minimum of the alkaline-earth diffusion activation

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enthalpy in the case when both cations are of the same size. An unsymmetrical linear behavior in the diffusion activation enthalpy with increasing size difference is observed. These results support similar investigations with dynamic mechanical loss spectroscopy performed on the same glasses by Martiny et al. [20]. This peculiar mixed cation effect in the diffusion activation enthalpy of alkaline-earth ions in A2 O · 2MO · 4SiO2 glasses is accurately described by a model proposed by Kirchheim [21]. This model considers electrostatic forces between the cations and the non-bridging oxygens and elastic interactions between the diffusing cations and the glass network. Two cases are treated that are related to either smaller or bigger alkali ions compared to the ionic size of the alkaline-earth ions. In the first case the smaller alkali ions lead to a reduced mesh size of the glass network with respect to the bigger alkaline-earth ions. Accordingly, the bigger the alkaline-earth ions the higher is the network deformation associated with its jump. For radii ratios r M /r A > 1 the elastic interaction between the network and alkaline-earth ions mainly affects the mobility of the divalent cation and its diffusion activation enthalpy. For radii ratios r M /r A < 1, that is, the alkali ions are bigger in size than the alkaline-earth ions, the mesh size is expanded. In this case the smaller alkaline-earth ions can approach the center of negative charge of non-bridging oxygens closer than the alkali ions. This leads to a higher electrostatic interaction between the oxygen and the alkaline-earth ion. Therefore, for r M /r A < 1 the diffusion activation enthalpy of alkaline-earth ions also increases with increasing size difference. This impact of elastic and electrostatic contributions on the mobility of divalent cations is also supported by foreign atom diffusion experiments of calcium and barium in strontium containing glasses [19]. An other peculiar mixed cation effect in the diffusion of the alkalineearth ions in mixed cation silicate glasses is found in the composition dependence. The variation of x in the glass composition xA2 O · (3 − x)MO · 4SiO2 (A = {Na, K}; M = {Ca, Ba}) leads to no change in the diffusion activation enthalpy of the alkaline-earth ion in the case when both cations are of the same size [14]. This holds for a soda-lime and a potassium-barium silicate glass. This experimental result confirms molecular dynamic simulations of soda-lime glasses [22]. The simulations predict that the local structure near Ca is independent of the Na content. Neutron diffraction measurements and reverse Monte Carlo modeling on soda-lime silicate glasses by Karlsson et al. [23] also reveal similar local environments of Ca and Na, which are retained when both cations are mixed. This supports the model of Kirchheim [21,24] that predicts for cations of similar size a negligible change in the activation enthalpy of alkalineearth ions compared to a binary alkaline-earth silicate glass. According to Kirchheim, the decisive factor for a change in Q is an increasing size difference of the cations, which is almost zero for soda-lime and potassium-barium glasses. However, in glasses with mono- and divalent cations of dissimilar size, like in sodium-barium silicate glasses, an increasing Ba diffusion activation enthalpy is observed with increasing Na content. This reflects the reduction of the

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mesh size by the smaller Na ions resulting in a higher elastic contribution to the Ba diffusion activation enthalpy [21]. Glasses with dissimilar sized cations were also investigated by Karlsson et al. [23] by means of neutron diffraction measurement. In contrast to cations of similar size, the local environment of alkaline-earth ions in sodium-strontium containing glasses changes with varying ratio of the cations. Irrespective of the radii ratio, the alkaline-earth mobility increases with increasing alkali content in the glass [14,19]. This mixed cation effect is very different to the characteristics of the mixed cation effect in mixed alkali glasses, where the addition of a second alkali cation species constrains the mobility of the other alkali ions [25,26]. The impact of the alkali content on the mobility of the alkaline-earth ions is supported by NMR measurements [27,28]. These measurements reveal the formation of dissimilar cation pairs near non-bridging oxygens. A diffusion jump of a divalent ion induces a large electric dipole moment leading to a high backward jump probability to the nearby NBOs. This dipole moment associated with the diffusion jump of the alkaline-earth ion can be effectively reduced by an adjacent alkali ion jumping into the vacant site. This reduces the backward hopping probability of the alkaline-earth ion and leads to an increasing alkaline-earth diffusivity with increasing alkali content. 3.1.2 Impact of glass preparation The diffusion data obtained from the analysis of Ca diffusion profiles in sol–gel derived glass films (see Fig. 1) are listed in Table 1. The temperature dependence of the Ca diffusion coefficients are illustrated in Fig. 5 in comparison to the diffusion coefficients in the corresponding bulk glasses. The Arrhenius behavior shown by both glass systems is illustrated by the straight lines. It is evident that, within experimental accuracy, the diffusion coefficients and diffusion activation energies of the alkaline-earth ions in sol–gel derived glasses equal the diffusion behavior in the corresponding bulk material. The frequency and temperature dependent conductivity spectra of a Li2 O · 2CaO · 4SiO2 glass film illustrated in Fig. 3 exhibit a characteristic shape for ionic conducting glasses consisting of three different processes: the electrode polarization, the frequency independent dc plateau, and the power-law dispersion. At low temperatures the conductivity isotherms do not show the nearly constant dc plateaus, because the experimental frequency is not sufficiently low. Figure 6 shows an Arrhenius plot of the dc conductivities. The obtained activation enthalpies are consistent with the results for bulk glasses reported by Martiny et al. [20] (see Table 3). The good agreement between the mobility of the cations in bulk glasses prepared from the melt and glass films derived from the sol–gel technique indicates that the local structure of the cations in mixed alkali alkaline-earth glasses does not strongly depend on the way of glass preparation.

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Fig. 5. Temperature dependence of 45 Ca diffusion coefficients DCa in Na2 O · 2CaO · 4SiO2 and K 2 O · 2CaO · 4SiO2 glass films compared to the corresponding bulk glasses [19]. The Arrhenius behavior is illustrated by the solid lines. Diffusion annealing was performed under dried air.

Fig. 6. Arrhenius plot of the dc conductivities of A2 O · 2MO · 4SiO2 glass films with A = Li, Na, K and M = Mg, Ca, Sr.

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Fig. 7. Temperature dependence of 45 Ca diffusion coefficients DCa in A2 O · 2CaO · 4SiO2 with A = Li (a), Na (b), K (c) under forming gas compared to the diffusion measurements under dried air [19]. The striped square in (b) illustrates a diffusion anneal under air flow that is consistent with the results of the diffusion measurements in encapsulated silica ampoules under dried air. The Arrhenius behavior is illustrated by the solid lines.

3.1.3 Impact of ambient conditions The diffusion data determined from Ca diffusion under forming gas in bulk silicate glasses (see Fig. 2a–c) are summarized in Table 2. The temperature dependence of Ca diffusion is accurately described with an Arrhenius expression as illustrated in Fig. 7a–c. The corresponding Arrhenius parameters are listed in Table 2. Previous results [19] on self-diffusion of Ca ions in Li2 O · 2CaO · 4SiO2 , Na2 O · 2CaO · 4SiO2 , and K 2 O · 2CaO · 4SiO2 performed in encapsulated silica ampoules under dried air are shown in Fig. 7a–c, respectively, for comparison. It is demonstrated that reducing ambient conditions significantly influence the diffusion behavior of Ca. Whereas the mobility of Ca in Na-Ca and K-Ca glasses is retarded, its mobility is enhanced in Li-Ca glasses compared to diffusion annealing under air. For comparison we also performed a diffusion experiment of Ca in the soda-lime glass under air flow that confirms the data obtained for diffusion annealing under air (see Fig. 7b). This reveals that the change in the Ca diffusion behavior in soda-lime glasses is truly related to the reducing ambient and not to the gas flow. The activation energy Q Ca of

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Fig. 7. Continued.

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Ca diffusion for both the Na-Ca and the K-Ca glass is not significantly affected by annealing under reducing ambient, i.e., within experimental accuracy, the activation energies are equal for annealing under air and forming gas. However, a clear decrease in Q Ca is observed for the Li-Ca glass. The differences in the diffusion behavior of Ca under air and reducing ambient observed for various glasses is not related to compositional changes near the surface as verified by TOF-SIMS analyses. The change in the diffusivity of Ca in the various mixed cation glasses as function of the ambient conditions is likely a consequence of structural changes in the Ca environment associated with Si−O−Si bond breaking. The presence of hydrogen might form additional defects such as hydroxide groups. In the case of the Li-Ca glass, the formation of hydrogen related defects likely causes structural changes that favor an expansion of the mesh size. Thereby the elastic interaction between the glass matrix and the Ca ions is reduced. This increases the mobility of the Ca ions. On the other hand, a mesh size increase in Na-Ca and K-Ca glasses due to bond breaking likely leads to a higher electrostatic interaction between the Ca ion and the local network, because the calcium ions can approach the NBOs more closely. As a consequence, the binding energy between Ca ions and NBOs increases. For all glasses studied in this work, oxygen is less mobile than Ca (see Sect. 3.2). Accordingly, a larger Coulomb interaction between Ca ions and NBOs will retard the diffusion of Ca under hydrogen annealing. This should also affect the activation enthalpy of Ca diffusion. However, the experiments demonstrate only a small retardation of Ca diffusion under hydrogen annealing (see Fig. 7b and c). Accordingly, the change in the activation enthalpy, which can clearly be seen in lithium calcium glass, is expected to be small and hardly resolved within the experimental accuracy. Additionally, some diffusion sites for calcium might no longer be available due to the presence of hydrogen, which occupies these sites. In summary, the diffusion behavior of alkaline-earth ions in alkali alkalineearth silicate glasses under reducing ambient conditions can also be explained by the interplay of elastic and electrostatic interactions.

3.2 Dynamics of network formers 3.2.1 Impact of glass preparation and ambient conditions The impact of glass preparation on O diffusion in a Na2 O · 2CaO · 4SiO 2 glass is illustrated in Fig. 8a. Compared to the bulk glass of the same composition, the O diffusivity in the sol–gel derived glass is about one order of magnitude higher. The diffusion of O both in the bulk and the sol–gel derived glass is well described by an Arrhenius expression. The Arrhenius parameters for the bulk glasses are summarized in Table 4 and that of the sol–gel glasses are indicated in Fig. 8a. The higher diffusivity of O in the sol–gel glasses is associated with smaller activation enthalpies Q O compared to the bulk glass. Presumably, O

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Fig. 8. a) Temperature dependence of O diffusion in bulk Na2 O · 2CaO · 4SiO2 glass with and without a sol–gel derived Na2 O · 2CaO · 428 SiO2 layer. The Arrhenius behavior is illustrated by the straight lines. b) Temperature dependence of Si diffusion in sol–gel derived Na2 O · 2CaO · 4SiO2 /Na2 O · 2CaO · 428 SiO2 heterostructures annealed under flowing O2 and under 18 O2 in a closed volume, respectively. The straight line is the best fit to the flow-data. Annealing times as indicated.

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diffusion is facilitated by Si−OH hydroxyl groups in the sol–gel that soften the glass structure. Such hydroxyl groups are products of the hydrolysis of TEOS and likely remain in a sol–gel derived glass even after consolidation. During annealing in O-rich ambient the concentration of OH groups decreases due to the formation and outdiffusion of H2 O [29]. Accordingly, diffusion of O via hydroxyl groups should be a transient process. Actually, diffusion experiments at 773 K for 1 and 45 d indicate a smaller O diffusion for longer times (see Fig. 8a). The difference, however, is small, because in the closed volume a constant H2 O pressure is likely established that still leads to a higher diffusion coefficient of O compared to the bulk glass prepared from the melt. The impact of ambient conditions on Si diffusion in the sol–gel derived Na2 O · 2CaO · 4SiO 2 glass is illustrated in Fig. 8b. The filled triangles and open squares indicate the temperature dependence of Si diffusion under a gas flow of natural O 2 and in a closed volume under enriched 18 O2 , respectively. Both kinds of experiments were performed under an oxygen pressure of approximately 1 atm. An Arrhenius type temperature dependence with parameters listed in Table 4 describes the Si diffusion under gas flow conditions very well. No evidence of a transient Si diffusion behavior is found, because the outdiffusion of hydroxyl groups likely occurs too fast to affect the slow Si diffusion. In this respect it is also noted that the Si diffusion data for closed volume annealing equal the Si diffusivity under gas flow conditions for temperatures higher than 800 K. However, at lower temperatures the Si diffusion for closed volume annealing exceeds the data for gas flow conditions and shows a clear transient behavior. We assume that the enhanced diffusion of Si under closed volume compared to gas flow conditions for temperatures equal and below 773 K is related to the higher stability and slower outdiffusion of OH groups at these temperatures compared to higher temperatures. The OH groups are considered to enhance the Si diffusion by softening the glass structure. According to theoretical calculations of the formation and migration energy of Si-vacancies and -interstitials in silica, Si diffusion under O-rich conditions is likely mediated by vacancies rather than by interstitials [30–32]. Theory predicts formation energies of about 4 and 12.5 eV for vacancies and interstitials, respectively [31]. Taking into account theoretical results on the migration energy of Si-vacancies reported by Limoge et al. [32], the predicted activation energy of Si diffusion via vacancies is lower than via interstitials. In this picture, OH groups can favor the formation of Si vacancies by passivating the oxygen dangling bonds. As a consequence, Si diffusion in silicate glasses is enhanced by OH addition. Our results on Si diffusion under different ambient conditions demonstrate that the impact of OH residues on Si diffusion in the sol–gel derived glass layers can be neglected when the temperatures and annealing times are sufficiently high and long. Under such conditions an Arrhenius type temperature dependence is obtained that is considered to be also representative for Si diffusion in bulk glasses. In the following the Arrhenius parameters for Si diffusion

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Fig. 9. Temperature dependence of O diffusion in a) aNa2 O · bCaO · 4SiO2 glasses and in b) A2 O · 2MO · 4SiO2 glasses as indicated. The values for the Na2 O · 2CaO · 4SiO2 glass are shown in both figures for comparison. Solid lines represent the Arrhenius-type temperature dependence. c) Temperature dependence of Si diffusion in mixed cation silicate glasses. Annealing times as indicated.

are presented that were determined for different glasses after sufficient long annealing times. 3.2.2 Mechanisms of diffusion The diffusion of O and Si in mixed cation silicate glasses reveals an Arrhenius type behavior as illustrated in Fig. 9a and b for O and Fig. 9c for Si. The corresponding activation enthalpies Q and pre-exponential factors D 0 are summarized in Table 4 together with the respective Tg values. The glass transition temperature can be considered as a measure of the rigidity of the glass. With increasing Tg the glass network becomes stiffer. At first glance one might expect an increasing O and Si diffusion activation enthalpy with increasing Tg . This is actually confirmed for Si as demonstrated by Fig. 10a. However, the activation enthalpy of O diffusion as function of Tg behaves opposite as illustrated by Fig. 10b, that is, Q O decreases with increasing Tg . Obviously, the diffusion of O is not directly related to the rigidity of the glass. As the ionic radius of the alkaline-earth element M decreases from Sr to Mg, Fig. 10b also indicates that Q O decreases with decreasing ionic size of M in Na 2 O · 2MO · 4SiO 2 glasses with M = Mg, Ca, Sr. With decreasing size the alkaline-earth ions can approach negative charge centers of NBOs more closely, i.e., a binding energy

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Fig. 9. Continued.

E b between the alkaline-earth ion and the oxygen can be considered that increases with decreasing size of the divalent cation. This binding energy favors the formation of O−M pairs that then can facilitate O diffusion. The observed

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Fig. 10. Activation enthalpies of (a) Si diffusion and (b) O diffusion vs. the glass transition temperature Tg in glasses as indicated.

correlation between the activation enthalpy of O diffusion and the radii ratio r M /r Na is shown in Fig. 11 for Na 2 O · 2MO · 4SiO 2 glasses with M = Mg, Ca, Sr and is explained with an alkaline-earth assisted migration of O. The higher the binding energy, the higher is the probability to form mobile O−M pairs. This leads to a decreasing O diffusion activation enthalpy with increasing O−M binding energy. This is somewhat comparable with a vacancy-assisted

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Fig. 11. Activation enthalpy Q O and Q M of O and M diffusion, respectively, in Na2 O · 2MO · 4SiO2 glasses with M = Mg, Ca, Sr, Ba vs. the radii ratio rM /rNa of the divalent cation M and monovalent Na ion.

diffusion of dopants in semiconductors [33,34]. An attractive binding energy between the dopant (D) and the vacancy (V) due to electrostatic and/or elastic forces can favor the formation of D−V pairs. The mobile V acts as diffusion vehicle for the substitutional dopant. In the case of our glass system, the diffusion vehicle of oxygen is the alkaline-earth ion. Note, the formation of O−M pairs is not considered to facilitate the diffusion of M, since the pairs are likely less mobile than M2+ . As a consequence, O−M pairs mainly determine the diffusion of oxygen in alkali alkaline-earth silicate glasses but contribute to the overall diffusion of M only to a minor extend. Following the approach of Kirchheim [21], the binding energy E b = E C − E el of the O−M pairs comprises elastic (E el ) and electrostatic (E C ) contributions. A Coulomb interaction energy E C can be estimated via EC =

4e2 1 , 4π 0  r r M + r O + δ

(3)

where M2+ and O2− are considered as hard spheres in a distance of δ to each other.  0 and  r denote the vacuum permittivity and the permittivity of the glass, respectively, e is the elementary charge and r M and r O indicate the sphere radii of the particular cations. Assuming that δ is negligible compared to r M and r O and taking into account ionic radii r M for alkaline earth ions in sixfold coordination (see values listed in Ref. [21]), r O = 0.140 nm, and  r ≈ 8 [35], Eq. (3)

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yields for the O−M pairs (M ∈ {Mg, Ca, Sr}) E Mg C = 3.40 eV E Ca C = 3.00 eV Sr E C = 2.79 eV .

(4) (5) (6)

To estimate the elastic contribution to the binding energy of O−M pairs, we consider the elastic interaction E el of alkaline-earth ions in Na 2 O · 2MO · 4SiO 2 glasses with M = Mg, Ca, Sr. Within the approach of Kirchheim [21], the following values are obtained E Mg el = 0.019 eV Ca E el = 0.005 eV E Sr el = 0.070 eV .

(7) (8) (9)

Ca Ca Sr The values given by Eqs. (4) to (9) yield E Mg b − E b = 0.386 eV (E b − E b = 2+ 2+ 0.275 eV). This indicates that the binding energy between Mg (Ca ) and O2− is stronger than between Ca2+ (Sr2+ ) and O2− . A similar trend is observed in the difference of the activation enthalpies of O diffusion. Our results for O diffusion in Na2 O · 2MO · 4SiO 2 with M = Mg, Ca, Sr (see Table 4) yield Q Mg O − Ca Sr = −(0.33 ± 0.56) eV (Q − Q = −(0.82 ± 0.56) eV). The negative sign Q Ca O O O indicates that Mg−O (Ca−O) pairs are more strongly bond than Ca−O (Sr−O) pairs. A good agreement between the binding energy differences and activation energy differences is observed for glasses containing Mg and Ca. The energy differences of the Ca- and Sr-glasses reveal the same trend, but the activation energy difference exceeds the binding energy difference by about 0.5 eV. Nevertheless, both values are of the same order of magnitude and even agree within experimental accuracy. Figure 11 also illustrates the dependence of the alkaline-earth diffusion activation enthalpy on the radii ratio r M /r Na [19,20]. The correlation with a minimum in the activation enthalpy Q M at r M /r Na ≈ 1 (see Sect. 3.1.1) becomes even more pronounced when the diffusion activation enthalpies of the divalent ions in glasses with Li, K, and Cs are taken into account [19]. The interrelation between Q M and the radii ratio of the alkaline-earth to alkali ions is explained with the impact of electrostatic and elastic interactions on the diffusion of the alkaline-earth ions [21]. For r M /r Na < 1 electrostatic and for Q M at r M /r Na > 1 elastic interactions determine Q M . The formation of O−M pairs affects the overall diffusion of the alkaline-earth ions only to a minor extent, because the diffusion of the divalent cations is mainly assisted by the alkali ions (see Sect. 3.1). Therefore, the diffusion activation enthalpies Q M of the alkaline-earth ions shown in Fig. 11 are not directly comparable with the activation enthalpies Q O of the O−M assisted O diffusion. This is supported by the different trend in the diffusion activation enthalpy vs. the radii ratio r M /r Na . In contrast to Q M , Q O shows no minimum for r M /r Na ≈ 1 indicating that the

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Fig. 12. Temperature dependence of O (solid lines) and Ca (dashed lines) diffusion in xNa2 O · (3 − x)CaO · 4SiO2 glasses with x = 0, 1, 1.5.

O−M binding energy is mainly determined by the electrostatic interaction between O2− and M2+ . The formation of O−M pairs is also supported by our results on O diffusion in xNa2 O · (3 − x)CaO · 4SiO2 glasses as illustrated in Fig. 12. With increasing Ca content, i.e. decreasing x, the diffusion of both O and Ca decreases. However, the Ca diffusion decreases almost 4 orders of magnitude from x = 1 to x = 0, whereas the decrease of O diffusion in this x range is less than two orders of magnitude. As a result the mobility of Ca approaches the mobility of O with decreasing x. Finally, O diffuses only about one order of magnitude slower than Ca in the 3CaO · 4SiO2 glass. Ca diffusion in soda-lime glasses is assisted by the mobile Na+ ions that effectively reduce the electric dipole moment associated with the hop of the divalent ion to a next possible site (see Sect. 3.1). With decreasing x, i.e. decreasing Na content, a Na+ assisted migration of Ca2+ gets less significant and other contributions to Ca diffusion become more important. These contributions could comprise a diffusion via Ca2+ and O−Ca pairs. In particular, the significance of O−M pairs in the diffusion of Ca is expected to increase with decreasing x because the formation of neutral O−M pairs effectively reduces the electrostatic interaction of Ca2+ with the network. In summary, the convergence of the O and Ca diffusivity in xNa2 O · (3 − x)CaO · 4SiO2 with x → 0 also indicates a cooperative diffusion of O and Ca via O−Ca pairs, that are likely uncharged. The activation enthalpy of O diffusion in yNa2 O · 2yCaO · 4SiO 2 glasses with y = 0.5, 1.0, 1.25 decreases with increasing Tg as illustrated in Fig. 13.

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Fig. 13. Activation enthalpies of O diffusion vs. the glass transition temperature Tg in yNa2 O · 2yCaO · 4SiO2 glasses with y = 0.5, 1.0, 1.25. The dashed line serves to guide the eye.

Obviously, like in the case of Na2 O · 2MO · 4SiO2 glasses (see Fig. 10b), the O diffusion is not related to the rigidity of the glass. With increasing y the concentration of alkali and alkaline-earth ions and NBOs increases and the glass network gets more fragmented. Figure 13 suggests, in line with an M-assisted O diffusion, that the O−M binding energy decreases with increasing y in order to predict increasing Q O values. But why should the formation of O−M pairs be hampered when the number of NBOs increases? One possible explanation is that with increasing y stronger structural relaxations occur. Then the formation of O−M pairs costs more elastic energy and is hindered. This is supported by the increasing density of the glass with increasing y that leads to a smaller volume of the glass per mole oxygen and, in accordance with the model of Kirchheim [21], larger values for the elastic energy are expected.

4. Conclusions Combining diffusion studies with radioactive and stable isotopes and conductivity measurements, we investigated the dynamics of alkali (A) and alkalineearth (M) network modifiers and of the network formers silicon (Si) and oxygen (O) in various mixed alkali alkaline-earth silicate glasses that were prepared as bulk glasses from the melt and as glass films by means of the sol–gel technique. We investigated how the type of mono- and divalent cations, their composition ratio, the concentration of NBOs, the way of glass preparation, and the ambient conditions affect the diffusion of the glass components.

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The alkaline-earth diffusion is strongly affected by the alkali to alkalineearth radii ratio due to electrostatic and elastic interactions of the alkaline-earth ions with the network that hamper the alkaline-earth mobility with increasing size difference. The increasing alkaline-earth mobility with increasing alkali content is explained by the formation of dissimilar cation pairs, whose presence has been detected by NMR measurements [27,28]. The alkali and alkaline-earth diffusion is unaffected by the way of glass preparation. Diffusion experiments under reducing ambient show a retarded alkaline-earth mobility in soda-lime and potassium-calcium glasses, however the calcium mobility is enhanced in lithium-calcium glasses. This behavior can be understood by assuming additional defects that form in glasses under annealing in reducing ambient. The silicon diffusion is directly related to the glass transition temperature Tg and therewith to the rigidity of the glass. In contrast it was found that the activation enthalpy of oxygen diffusion decreases with Tg . This provides evidence of an alkaline-earth ion assisted diffusion of oxygen that assumes the formation of pairs between alkaline-earth (M) and oxygen (O) ions. Although these O−M pairs mainly mediate oxygen diffusion, their contribution to the overall diffusion of the alkaline-earth ions is minor, because the alkaline-earth diffusion is mainly alkali assisted and exceeds the oxygen diffusion by several orders of magnitude. The O−M pair formation hypothesis is supported by the difference of the oxygen and calcium diffusivity in soda-lime glasses with various sodium contents, that is, with decreasing sodium content the sodium-assisted diffusion of calcium becomes minor and the contribution of O−M pairs becomes noticeable. Considering soda-lime glasses with different concentrations of non-bridging oxygens (NBOs), the diffusion activation enthalpy of oxygen increases with increasing NBO concentrations. In the picture of an O−M pair assisted oxygen diffusion this is related to a decreasing O−M binding energy. This points to stronger structural relaxation in soda-lime glasses with an increasing number of NBOs that hinders the formation of O−M pairs.

Acknowledgement The authors acknowledge financial support by the Sonderforschungsbereich 458 of the Deutsche Forschungsgemeinschaft.

References 1. A. L. Laskar, Superionic Solids and Solid Electrolytes: Recent Trends, Academic Press, New York (1989). 2. W. E. Lee, M. I. Ojovan, M. C. Stennett, and N. C. Hyatt, Adv. Appl. Ceram. 105 (2006) 3. 3. N. P. Frolova, A. F. Perveyev, and B. P. Kryzhanovskiy, Sov. J. Opt. Technol. 40 (1973) 248.

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The Use of Radiotracer Diffusion to Investigate Ionic Transport in Polymer Electrolytes: Examples, Effects, and Their Evaluation By N. A. Stolwijk∗, M. Wiencierz, J. Fögeling, J. Bastek, and Sh. Obeidi Institut für Materialphysik, Universität Münster, Wilhelm-Klemm-Str. 10a, 48149 Münster, Germany (Received June 30, 2010; accepted in final form September 16, 2010)

Ion-Transport Model / Cation–Anion Pairs / Transference Number / Salt Precipitation / Oxide Nano-Particles / DC Conductivity The paper highlights some remarkable results obtained by applying the radiotracer diffusion (RTD) technique to the study of ionic transport in salt-in-polymer electrolytes. The technique is based on the determination of a radiotracer depth profile by serial sectioning following isothermal diffusion annealing. Unlike alternative methods, RTD is able to measure the self-cation and self-anion diffusivity even for systems dilute in salt. Another unique feature is the capability to investigate foreign-ion diffusion at extremely low concentration levels. Combined with DC conductivity data, RTD may provide a virtually complete picture of mass and charge transport in solid-like polymer electrolytes (SPEs). The paper describes the special SPE-related procedures used in the RTD experiments and their analysis. The advantages of the method will be demonstrated with selected examples of self- and foreign-ion diffusion in prototype SPE systems. We also present prominent examples of RTD dealing with the effects of salt precipitation and oxide nano-particles used as dispersed filler material.

1. Introduction Polymer electrolytes are obtained by dissolving an inorganic salt in a polymer matrix. Suitable combinations of a polymer substance with high molecular weight and a lithium salt may produce solid-like electrolyte materials with useful properties for battery applications. In ideal cases, such solid polymer electrolytes (SPEs) combine an appreciable ionic conductivity with a high mechanical flexibility but avoid the disadvantages of liquid-like ionic systems. In prototype solvent-free systems solely based on cation-coordinating polyethers, however, the conductivity is too low for practical uses [1–3]. * Corresponding author. E-mail: [email protected] Z. Phys. Chem. 224 (2010) 1707–1733 © by Oldenbourg Wissenschaftsverlag, München

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Over the past, many studies have been conducted in order to gain more insight into the mechanisms of ionic transport in SPEs. In most of these studies, impedance spectroscopy has been the primary experimental technique to determine the DC conductivity, which characterizes the overall charge transfer capacity due to the combined effect of cations and anions [2,4]. However, different and more detailed information is acquired by measuring the individual diffusivity of either ionic species. In particular, the combination of charge and mass transport data may shed light on the role of ion association and allow for the determination of key parameters controlling the formation of neutral ion pairs. Since the 1990s pulsed-field-gradient nuclear magnetic resonance (PFGNMR) has evolved to be the method of choice for determining diffusion coefficients in solid polymer (and liquid) electrolytes [5,6]. By contrast, the radiotracer diffusion (RTD) technique has been practically abandoned for this kind of studies after pioneering work done in the 1980s [7,8]. However, recent reports from our group have (re-)demonstrated the usefulness of RTD for accurate diffusion experiments in SPE materials [9–11]. It is the aim of this paper to review some salient results based on RTD that were obtained over the years 2006–2009 within the third project period of the Collective Research Centre (SFB 458) on “Ionic Motion in Materials with Disordered Structures” established at the University of Münster in January 2000. The outline of the paper is as follows: Sect. 2 describes the general characteristics of the RTD technique and the experimental procedures for its application to SPE materials. Section 3 deals with the mathematical analysis of measured radiotracer depth profiles. Section 4 presents an ion transport model which enables us to simultaneously evaluate RTD of individual ionic species and the overall DC conductivity. Section 5 demonstrates the suitability of RTD to determine the self-ion diffusivities in systems with an extremely low salt concentration while Sect. 6 exemplifies the unique possibility of RTD to measure foreign-ion diffusion. Section 7 shows that RTD is capable of determining ionic diffusivities in polymer electrolytes subject to salt precipitation. Finally, Sect. 8 gives examples how RTD can be used to explore the effects of nanoparticle additives to SPE systems.

2. The radiotracer technique for ionic diffusion studies in polymer electrolytes The use of RTD for SPEs requires suitable radioactive isotopes of relevant ionic species. A major criterion is the half-life time (τ 1/2 ) which should preferably be longer than several days. For the alkali metal cations, the situation is favourable owing to the principal availability of 22 Na (τ 1/2 = 2.6 a), 42 K (12 h), 86 Rb (19 d), and 137 Cs (30.2 a) [12]. In addition, the nobel metal 110m Ag (τ 1/2 = 250 d) and the divalent isotopes 65 Zn (244 d) and 109 Cd (453 d) may be useful for ionic systems while 64 Cu (13 h) forms a borderline case de-

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Fig. 1. Schematic of the RTD technique, which includes the following specific steps when employed for ionic diffusion in polymer electrolytes: (a) Deposition of the radiotracer by using a thin diffusion-source film of the same composition as the sample. (b) Annealing in an oil bath pre-heated at the diffusion temperature. (c) Sectioning of the sample in a serial manner with a rotary microtome at freezing temperatures. (d) Analysis of the sections’ radioactivity and evaluation of the radiotracer depth profile to determine the ionic diffusion coefficient D.

manding enhanced experimental skills. Unfortunately, there exists no suitable radiotracer for the technologically important cation lithium. For the anions, the choice is practically restricted to the isotope 125 I (τ 1/2 = 60 d). Other anions frequently employed in studies on polymer electrolytes, such as the triflate (CF3 SO3 − ) and TFSI (N(CF3 SO2 )2 − ) ion, are complex in nature. In principle, these anions can be labelled by replacing the natural sulphur constituent by radioactive 35 S (τ 1/2 = 88 d). However, isotope substitution is expensive and, if achievable at all, would involve radiochemistry with its inherent stringent safety requirements. In an early study, the 14 C-marked thiocyanate (SCN− ) anion was used [7,13]. A schematic representation of the RTD technique is given by Fig. 1. The experiments require macroscopic samples in cylindrical geometry that are obtained by hot pressing of the SPE material under investigation. In our case we used PFTE moulds with an inner diameter of 6 mm and an inner height of 8 mm. Since the method is destructive, every diffusion-profile measurement needs a separate sample, which is then lost for further experiments. Therefore, it is advantageous to synthesize the SPE mother substance in batches that are large enough to allow for measuring the full temperature dependence of both ionic species over the temperature range of interest. For this purpose, typically 10 to 20 samples are needed. In addition, the same production batch should provide specimens for conductivity measurements. SPE batches of sufficient size, i.e., containing typically 10 g of weight-in material (polymer plus salt), can be produced by dissolution of the components in a suitable organic liquid and subsequent vacuum evaporation [14]. The diffusion source is cut from a thin film (ca. 40 μm thick) with the same composition as the sample, but doped with a tiny amount of a solution containing the radiotracer. We frequently used 22 Na and 125 I in the form of aqueous 22 NaCl and Na125 I solutions, respectively. The source film was dried before use to remove residual water and organic solvent. The specific activity of the radiotracer (in units of Bq/g) is so high that the necessary total activity for a successful experiment does not significantly change the composition of the

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Fig. 2. Temperature-time profile of a diffusion anneal performed with a PEO30 NaI sample at 130 ◦ C in an oil-bath thermostat and stopped by quenching in water. The short heating-up and cooling-down periods show features of PEO melting and (partial) crystallization, respectively. Time tc and temperature Tc mark the onset of salt precipitation occurring in some polymer electrolytes such as, e.g., PEO30 RbI.

source film. In this way, the diffusion experiment is performed under virtual iso-concentration conditions. This characteristic feature of the RTD technique avoids the complicating effects of chemical diffusion which imply thermodynamic factors that deviate from unity [15,16]. Moreover, the closely similar compositions of sample and diffusion source reduce unwanted effects such as, e.g., built-up of a space-charge layer or tracer hold-up at the sample surface. Adding both cation and anion tracers to the same source film allows for the simultaneous measurement of two diffusion profiles, which reduces experimental error and saves time. After pressing one or more pieces of radioactive source film onto the flattened upper face of the sample and subsequent capping of the PFTE mould, the assembly is enclosed in an aluminium container and isothermally annealed in a pre-heated oil bath [14]. The effective diffusion time may be estimated from separate calibration runs with a dummy sample of similar SPE material to correct for heating-up delay and non-ideal quenching (in water) at the end of the diffusion treatment. Figure 2 shows a typical temperature-time profile that is representative of an oil-bath diffusion anneal at 130 ◦ C for nominally one hour [17]. Here, the temperature was recorded by a thermocouple inserted in a (non-radioactive) PEO 30 NaI sample encapsulated in the standard fashion. The heating ramp reveals an endothermic event near 65 ◦ C that reflects the melting of PEO. A corresponding exothermic feature is seen on the cooling ramp at a somewhat lower temperature. The nominal diffusion time tnom runs from the immersion of the sample in the pre-heated oil (130 ◦ C) at t = 0 till its quenching in water (ca. 20 ◦ C), indicated in Fig. 2 by the sharp T -decrease at the end

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Fig. 3. Radiotracer depth profiles of 22 Na and 125 I in PEO500 NaI resulting from simultaneous diffusion at 120 ◦ C for 70 min. The solid lines are least-squares fits based on Eq. (4), which represents a mixed Gaussian/erfc profile shape arising from exhaustion of the diffusion source in the course of annealing.

of the isothermal plateau. To obtain the effective diffusion time tdiff for the calculation of the diffusion coefficient tdiff = tnom − 2 min was taken as a suitable estimate. Radiotracer depth profiling is achieved by serial sectioning at sufficiently low temperatures using a rotary microtome and the subsequent detection of the activity of each section. In our experiments, a standard section thickness of 20 μm and cutting temperatures near −50 ◦ C proved to be effective. Thicker sections can be mimicked by putting together two or more standard sections. Depending on the radiotracer employed, activity counting can be done by γ or β-detectors of conventional type, taking into account corrections for background radiation. Two typical diffusion profiles resulting from simultaneous diffusion of 22 Na and 125 I in PEO 500 NaI are shown in Fig. 3. The profiles are composed of 70 to 100 sections, making up a total penetration depth of 2 to 3 mm. The wide activity range extending over 3 to 4 orders of magnitude and the absence of significant scatter lead to a distinct profile shape that is reproduced by the appropriate mathematical solutions of the diffusion equation (solid lines; see Sect. 3). Comparing RTD with PFG-NMR, the special features of either technique render them either more or less suitable for application to a particular SPE diffusion problem. Both methods rely on suitable probe atoms whose signals can be detected with a sufficiently high sensitivity. For PFG-NMR, the stable natural isotopes 7 Li and 19 F have been frequently used in the field of polymer ionics [18,19]. In the case of RTD, 22 Na and 125 I are useful radiotracers to investigate the diffusion behaviour of the corresponding ionic species. However, it proves to be difficult to find ionic systems that can be examined sufficiently

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well by both methods. For example, long-lived radioisotopes of lithium or fluorine do not exist, whereas suitable radiotracers such as 32 P are not commercially available as markers integrated in complex anions such as 32 PF6 − . On the other hand, 23 Na is detectable by NMR, but experiments suffer from very short spin relaxation times. This unfortunate constellation virtually precludes the possibility to directly crosscheck the results of RTD and PFG-NMR against each other. However, the benefits of both methods may be exploited in studies that use PFG-NMR and RTD on the same complex to investigate self-ion and foreign-ion diffusion, respectively [20] (see Sect. 6). RTD like PFG-NMR allows for the ion-specific measurement of diffusion coefficients, which may involve a concentration-weighted average over single ions, pairs, and higher-order aggregates in different electric charge states. By contrast, the DC conductivity comprises the combined effect of all positively and negatively charged species of any size. It will be shown in Sects. 4 to 8 how the simultaneous evaluation of mass and charge transport data may give rise to a fairly complete picture of long-range ion motion in polymer electrolytes.

3. Analysis of radiotracer depth profiles Deducing the diffusion coefficient D from a measured RTD profile involves its comparison with appropriate solutions of Fick’s second law [16,21]. Ideal profile shapes are described by the Gaussian function for instantaneous source conditions, i.e., c(x, t) = c0 exp(−x 2 /4Dt) or the complementary error function for a constant diffusion source, i.e., √ c(x, t) = c0 erfc(x/2 Dt)

(1)

(2)

where x denotes penetration depth, t is diffusion time, and c is concentration (in arbitrary units of specific activity). The pre-factor c0 is the volume concentration at the sample surface (x = 0), which is√constant in the erfc case but time dependent in the Gaussian case (c0 = M0 / πDt; M0 is the amount of radiotracer substance per unit area). signifiEquation (2) describes the situation, in which M0 is not becoming √ cantly exhausted during the diffusion process. Then M(t) = 2c0 Dt/π  M0 holds, where M(t) is the amount of diffused substance at the end of the annealing time t and given by the integration of Eq. (2) over x. However, for a high diffusivity, prolonged annealing, and/or a weak diffusion source, all the tracer atoms may become involved in the diffusion process, which then leads to a gradual changeover from an erfc profile to a Gaussian profile. In our experiments both types of profile were observed, as exemplified by the tracer distributions plotted in Fig. 4. However, in view of the indicated relationship

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Fig. 4. Radiotracer depth profiles of 86 Rb and 125 I in PEO30 RbI resulting from diffusion annealing at 120 ◦ C for 75 min and 80 ◦ C for 100 min, respectively. The solid lines are fits of the Gaussian function, Eq. (1), or the complementary error function, Eq. (2), as indicated.

between the Gaussian and the erfc function, it may not surprise that often profiles of intermediate shape were measured. Such profiles are mathematically described by [21]      h+x h−x 1 + erf √ (3) c(x, t) = c0 erf √ 2 2 Dt 2 Dt which includes the sum of two error functions and where h and c0 stand for the width and the tracer concentration of the source layer, respectively. Identifying zero penetration depth with x = h and shifting the depth scale accordingly leads to      x + 2h x 1 − erfc (4) c(x, t) = c0 erfc √ √ 2 2 Dt 2 Dt The x-translation corresponds to omitting the first one or two sections that contain the radiotracer source film of strength M0 = c0 h. We note that the profiles in Fig. 3 were fitted by Eq. (4), taking D, h, and √ c0 as free parameters. In fact, Eq. (3) converges to the Gaussian function for h/ Dt  1 and to erfc for √ h/ Dt  1. It was verified that in these extreme cases the D values resulting from fitting with Eq. (4) coincide with those from the Gaussian and erfc fits, respectively. Moreover, for all experimental profiles of intermediate shape, the D value obtained by fitting with Eq. (4) lies between the corresponding best estimates from fitting with Eqs. (1) and (2). In rare cases, the measured profile was better fitted by the ierfc √ function, which mathematically applies to the boundary condition c0 = k t [21]. This behaviour can be rationalized by assuming that the transition of radiotracer

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ions from the source layer to the sample may be retarded. Such retardation could arise from less perfect intermolecular contacts at the source-layer/sample interface. The observed profile shapes demonstrate that the diffusion coefficient is a constant at fixed temperatures and thus independent of depth (and tracer concentration). In turn, this provides evidence for the homogeneity of our samples and for the absence of chemical diffusion. The statistical error in D arising from least-squares fitting is usually less than a few percent, which adds to other sources of error, such as, e.g., the measurement of diffusion time and temperature, sample deformation due to thermal stress upon quenching, and the uncertainty in the location of zero penetration depth (x = 0).

4. A model to combine diffusion data with ionic conductivity measurements The self-diffusion and conductivity data can be simultaneously evaluated within a phenomenological transport model that offers a general framework for describing the interrelationship of mass and charge transfer in ionic systems [9,20,22]. The treatment will be restricted to self-ion transport in fully amorphous, solvent-free SPE systems, although the model has a wider applicability [23]. A basic assumption of the model is that long-range ion transport essentially takes place via charged single ions and neutral cation–anion pairs. As a consequence, ionic triplets and higher-order clusters are supposed to play a minor role, which relates to the relatively low salt concentration in the systems under consideration [24]. Below, the general features of the model will be outlined in detail, which involves the introduction of the pertinent model parameters. The practical use of this general framework for the fitting of experimental data requires additional simplifying assumptions to reduce the number of free parameters. Since these assumptions are not unequivocal, this eventually leads to a distinction between two different submodels, one of which (Model #1) has been employed in previous work by our group [9,22]. The other submodel (Model #2), footing on the results of recent molecular dynamic (MD) calculations [25], is preferred in our newer publications [20,26]. For convenience, a list of symbols designating the physical quantities and parameters used in the models is presented in Table 1. DC conductivity data (σ DC ) may be converted into charge diffusivities D σ by using the Nernst–Einstein equation Dσ =

σ DC k B T , C s e2

(5)

where k B denotes the Boltzmann constant, T is temperature, C s is salt concentration (i.e., number density), and e is elementary charge. Having C s in the denominator of Eq. (5) (instead of 2C s being the number density of all ionic species together), D σ stands for the net charge diffusivity per salt molecule,

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Table 1. Symbols used for physical quantities and model parameters. Symbol Description B M+ B A− Bp Cs ΔHp kB kp k p0 D∗M D∗A Dσ D M+ D A− Dp D0M+ D0A− D0p DNa p DIp rp rpNa rpI ΔSp σDC T0 Tc t+ tpM tpA

VTF parameter for cation M+ VTF parameter for anion A− VTF parameter for MA0 pair Salt concentration (number density) Enthalpy of MA0 self-ion pair formation Boltzmann constant Reaction constant for MA0 pair formation Pre-factor of MA0 pairing reaction constant Radiotracer diffusivity of cation species M Radiotracer diffusivity of anion species A Charge diffusivity deduced from DC conductivity Diffusivity of cation M+ Diffusivity of anion A− Diffusivity of MA0 self-ion pair Pre-factor of M+ diffusivity Pre-factor of A− diffusivity Pre-factor of MA0 self-ion pair diffusivity Diffusivity of pairs containing foreign cation Na Diffusivity of pairs containing foreign anion I MA0 self-ion pair fraction normalized to C s NaPF6 0 pair fraction normalized to total Na concentration LiI0 pair fraction normalized to total I concentration Entropy of MA0 self-ion pair formation DC conductivity VTF parameter (zero mobility temperature) Critical temperature of salt precipitation Cation transference number (related to M+ ) Pair-related transport number of cation M (= Li, Na) Pair related transport number of anion A (= PF6 , I)

that is, D σ comprises the joint effect of one cation and one anion. However, this does not imply that the salt is fully dissociated in the polymer matrix. Information about the degree of cation–anion pair formation will be obtained below from the comparison of D σ with the self-ion diffusion coefficients. In polymer-salt complexes, transport of the constituant ions may be described along the lines recently proposed by Stolwijk and Obeidi [9,22]. The basic features of their model are based on the assumption that long-range selfion transport can be essentially described by the occurrence of three distinct species, viz., charged single cations (M+ ), charged single anions (A− ), and neutral cation–anion pairs (MA0 ). The (fast) dynamic formation and decay of ion

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pairs may be expressed by the reaction M+ + A−  MA0 ,

(6)

in which the characteristic association/dissociation times (i.e., reciprocal rate constants) are conceived to be short with respect to the times relevant to the measurement of the pertaining transport data (diffusion time, inverse AC frequency). The (reduced) reaction constant k p adopts the conventional form   (7) k p = k p0 exp −ΔH p /k B T with the pair formation enthalpy ΔH p and the (reduced) pre-exponential factor k p0 = exp(ΔS p /k B ) containing the pair formation entropy ΔS p . According to the law of mass action, the pair fraction r p ≡ C p /C s results as 

(8) r p (T ) = 1 + 1 − 1 + 4k p /2k p . Hence r p is a T -dependent quantity which tends to coincide with k p for k p  1, whereas it comes close to unity for k p  1. In the latter case, the single-ion fraction 1 − r p converges to 1/ k p . It can be shown that the T -dependent tracer and charge diffusivities are described by the following set of coupled equations [9,22]: D∗M = (1 − r p )D M+ + r p D p D∗A = (1 − r p )D A− + r p D p D∗σ = (1 − r p )(D M+ + D A− )

(9) (10) (11)

which expresses our common-sense notion that cations and anions can be transferred both in their individual ionic state and as neutral pairs. The different contributions are weighted by the pair fraction and its complement, i.e., the single-ion fraction. In addition, charge transport is carried both by positively and negatively charged ions. Each ionic species X = M+ , A− , or p = MA0 is characterised by its own ‘true’ diffusivity D X , which generally obeys a uniform temperature dependence given by D X = D0X exp (−B X /(T − T0 )) .

(12)

In this Vogel–Tammann–Fulcher-like expression, the zero-mobility temperature T0 is a common parameter to all mobile species, characterizing the mechanical flexibility (viscosity) of the overall system. By contrast, the preexponential factors (D0X = D0M+ , D0A− , D0p ) and the pseudo activation energies (B X = B M+ , B A− , B p ) determine the magnitude of the diffusivity for each species individually. However, not all B X and/or D X parameters are expected to be independent, since ionic transport in SPEs is coupled with the segmental

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motion of the polymer chains [27–29]. This circumstance shall be used below to reduce the number of free parameters in data fitting. Recent MD simulations of ion transport in the PEO-LiI system have shown that the diffusivities of the cation and the corresponding cation–anion pair are closely similar, whereas the anion diffusivity appeared to be partly decoupled from the polymer-segment dynamics [25]. In the present formulation, this translates to D0p = D0M+  = D0A− and B p = B M+  = B A− . Altogether, the simultaneous fitting of the experimental data (D∗M , D∗A , D σ ) by Eqs. (9)–(11) comprises seven free parameters, i.e., ΔS p , ΔH p , D0M+ , D0A− , B M+ , B A− , and T0 . This specific variant of the general model will be preferentially used in the following sections for the evaluation of the experimental data. For historical reasons it will be referred to as Model #2 [20]. Earlier work by our group was based on the alternative constraints D0p  = 0 D M+  = D0A− and B p = B M+ = B A− , which also give rise to seven free parameters. However, this original variant of the general model, termed Model #1, is not supported by state-of-the-art MD studies. If appropriate, however, the results of Model #1 may be compared with those of the probably more realistic Model #2.

5. Self-ion diffusion in a salt-poor PEO-NaI electrolyte The high sensitivity of the radiotracer technique makes it feasible to monitor the ionic self-diffusivity in SPEs with low salt concentrations. This outstanding property of RTD was exploited in an extensive study of the prototype system PEO n NaI consisting of poly(ethylene oxide) solvating sodium iodide. In particular, we investigated the concentration dependence of ion transport by varying the monomer-to-salt ratio n from 20 to 1000. For each composition, the diffusivity of 22 Na and 125 I was compared with the charge diffusivity deduced from conductivity data obtained by impedance spectroscopy. The complete results of this study will be published in a separate report. Here we only present the data and their analysis for the composition PEO500 NaI. The salt concentration in this complex corresponds to C s ≈ 3 × 1019 cm−3 , which is too low to be analyzed by PFG-NMR, even when NaI is replaced by, e.g., lithium triflate (LiCF 3 SO 3 ) containing the suitable (natural) NMR isotopes 7 Li and 19 F. With the RTD technique, however, it is no problem to measure diffusion profiles in the low-concentration range, as demonstrated for PEO 500 NaI by Fig. 3. Figure 5 displays the tracer diffusivity of Na and I in PEO 500 NaI as a function of reciprocal temperature, together with the charge diffusivity D σ deduced from the DC conductivity. It is seen that at all temperatures D σ falls distinctly below D∗I , so that D σ  D∗Na + D∗I certainly holds. This finding may be interpreted in terms of NaI0 pair formation, since neutral ion pairs may contribute to mass transport but not to mass transport. Accordingly, the experimental data

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Fig. 5. Tracer diffusion coefficients (D∗Na , D∗I ) and charge diffusivity (Dσ ) in PEO500 NaI as a function of inverse temperature. The solid lines result from a simultaneous fit within an iontransport model (i.e., Model #2) that is based on the occurrence of neutral ion pairs in addition to charged single cations and anions.

Table 2. Parameters describing self-ion transport in various polymer electrolytes within Model #2 (see text). M+ denotes Na+ or Rb+ depending on the system considered. B M+ [K]

B I− [K]

T0 [K]

D0M+ [cm2 s−1 ]

D0I− [cm2 s−1 ]

ΔSp [k B ]

ΔHp [eV]

PEO500 NaI

1177 ±15%

1315 ±25%

186 ±10%

1.3 × 10−4 ±35%

4.4 × 10−2 ±125%

29.4 ±10%

0.613 ±25%

PEO60 NaI

973 ±5%

940 ±10%

209 ±5%

5.1 × 10−5 ±25%

4.8 × 10−3 ±50%

19.2 ±5%

0.381 ±15%

PEO30 RbI

935 ±10%

916 ±20%

218 ±5%

7.1 × 10−5 ±20%

4.7 × 10−4 ±65%

11.9 ±45%

0.301 ±50%

were analysed within the general ion-transport model outlined in Sect. 4. In fact, the solid lines in Fig. 5 represent the best fit resulting from simultaneous adjustment of D∗Na , D∗I , and D σ within the special model variant termed Model #2. Thus, this model is able to reproduce the data within experimental error by finding a set of best estimates for the seven free parameters, which are compiled in Table 2. From these parameter values, we can deduce several important quantities as a function of temperature characterizing the ionic transport behaviour in PEO 500 NaI. Figure 6 shows that the pair fraction r p given by Eqs. (7) and (8) is close to unity over the whole T -range investigated. This implies that most ions are bound in NaI0 pairs, which may be also recognized from the low fraction of free cations or anions (1 − r p ) additionally depicted in Fig. 6. It is remarkable

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Fig. 6. Ion-pair fraction rp and cation transference number t+ of PEO500 NaI as a function of inverse temperature (solid lines), as deduced within the ion transport model #2 from the experimental data given by Fig. 5. The single-ion fraction 1 − rp is also plotted (dashed line).

that 1 −r p decreases and hence r p increases with increasing temperature, which seems to be a general property of solvent-free SPE systems [30]. Analysing the present result in more detail, it may be concluded that the high r p values are due to the large positive value of ΔS p which overcompensates the large positive value of ΔH p (cf. Eq. (7)). Thus, although pair formation is less favourable for enthalpy reasons, it appears to be driven by entropy, in agreement with views expressed earlier [30,31]. Specifically, the polymer chains acquire more conformational freedom when the strong coordination of the ether oxygen atoms to the cations is released by ion association. We further note that the slope of equals ΔH p/2, since for r p ≈ 1 the fraction of free ions 1 −r p in Fig. 5 virtually converges to 1/ k p . Another relevant quantity plotted in Fig. 6 is the cation transference number t+ , which is defined as [32] t+ =

D Na+ . D Na+ + D I−

(13)

It is seen that t+ only weakly varies with temperature, adopting values near 0.007. Such a small transference number is far from favourable with respect to battery applications, as it reflects the low mobility of the cations. A direct comparison of D Na+ and D I− = D p is presented in Fig. 7. It should be mentioned that the diffusivity of the most mobile species, I− , is on the order of 10−4 cm2 s−1 at high T . Such values are typical for small molecules in liquidlike systems. The impact of ion pairing may be also expressed by the partial transport numbers tpNa and tpI , which are displayed in Fig. 8. According to its definition,

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Fig. 7. True diffusion coefficients of Na+ , I− , and NaI0 pairs in PEO500 NaI as a function of inverse temperature (solid lines), as deduced within the ion transport model #2 from the experimental data given by Fig. 5. The coincidence of DNa+ and Dp reflects the model-specific assumption Dp = DNa+ .

Fig. 8. Partial transport numbers tpNa and tpI in PEO500 NaI, which quantify the respective fractions of Na and I transport carried by NaI0 pairs (solid lines). The relative Na+ contribution to D∗Na , 1 − tpNa , is also given (dashed line).

i.e., tpNa =

r p Dp r p Dp = ∗ , r p D p + (1 − r p )D Na+ D Na

(14)

tpNa denotes the relative contribution of NaI0 pairs to the total diffusivity of cations given by D∗Na . Figure 8 shows that tpNa is close to unity, whereas 1 − tpNa varies from more than 0.0007 at high T to about 0.03 at low T . Thus, only a mi-

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nor fraction of sodium transport takes place via single Na+ ions. For iodine the situation is less extreme, since tpI lies between 0.9 at high T and 0.2 at low T . Hence, at lower temperature iodine transport is mainly carried by I− . Comparing the results among PEO n NaI complexes with different EO/Na ratios n, it was found that the tendency to ion pairing increases with decreasing salt concentration. Also this counterintuitive effect, like the unusual temperature behaviour, may be explained by the strong role of polymer entropy in the solvation of the salt, i.e., in the formation of coordinative bonds with the cations. This will be reported in detail in a future publication.

6. Foreign-ion diffusion in a polyether – LiPF6 electrolyte RTD is a classical method to study impurity diffusion in metals, semiconductors, and other materials. Generally, the use of radiotracers allows us to perform foreign-atom diffusion experiments at very low concentration levels, so that the properties of the host material remain virtually unchanged by the diffusion treatment. To our knowledge, the first report about foreign-ion diffusion in a polymer electrolyte was the study by our group on Rb diffusion in PEO 30 NaI [33]. It was found that Rb impurity diffusion proceeds faster than Na self-diffusion but distinctly slower than I self-diffusion. The selfand foreign-ion diffusivity data were simultaneously evaluated along with the charge diffusivity by using our original transport model (Model #1). Within this model variant, the results could be rationalized by an enhanced tendency of the Rb+ ions, compared to Na+ ions, to associate with I− . Since Model #1 generally predicts relatively low fractions of pairs with mobilities much higher than those of the corresponding single-cation species, the larger fraction of the RbI0 pairs with respect to that of NaI0 pairs gives rise to enhanced total transport of Rb. However, analysing the same data within Model #2 may lead to a different interpretation of Rb impurity diffusion in PEO 30 NaI. Recently we conducted an extensive study of foreign-ion and self-ion diffusion in the cross-linked polyether-based electrolyte PolyG 30 LiPF 6 [20]. The polyether host is essentially made of random poly(ethylene oxide/propylene oxide) copolymer of 3600 g/mol molecular weight with an EO/PO ratio of 4 : 1, which after cross-linking forms an elastomeric polyurethane. This SPE system constitutes a stable amorphous complex at ambient temperatures, which is a great advantage for potential battery applications compared to exclusively PEO-based systems exhibiting the melt transition at elevated temperatures (near 65 ◦ C). An earlier RTD study of a related system by our group dealt with self-ion transport in PolyG 20 NaI [10]. Self-diffusion in PolyG 30 LiPF 6 was measured by PFG-NMR using the 7 Li signal for the cation and the 19 F signal for the anion. In addition, foreign-ion diffusion was monitored with the radioisotopes 22 Na and 125 I. All data including the charge diffusivity are shown in Fig. 9.

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Fig. 9. Self-ion and foreign-ion diffusivities in PolyG30 LiPF 6 resulting from PFG-NMR analysis (D∗Li , D∗PF 6 ) and RTD experiments (D∗Na , D∗I ) compared with the charge diffusivity (Dσ ). Solid and dashed lines originate from a simultaneous fit within an extended ion-transport model.

It is seen that the cations are slower than the anions at the same temperature and that the foreign ions are slower than the self-ions of similar type. However, the latter difference tends to vanish for the anions at high T . The charge diffusivity takes an intermediate position between the diffusivities of the self-anion and the self-cation. Nevertheless, all data can be simultaneously fitted (solid and dashed lines) when allowance is made for pair formation between cations and anions of various kinds. To this aim, the ion-transport model in Sect. 4 was extended to include the diffusion of ionic impurities and the possible pairing of these impurities with the self-ions of opposite type [20]. Specifically, LiPF6 0 , NaPF6 0 , and LiI0 pairs were involved in the analysis, whereas NaI0 pairs do not occur at all (or only in negligible concentrations). Figure 9 shows the good agreement between the fits based on Model #2 (extended to include foreign ions) and the experimental data. The adjusted model parameters yield the pair fractions for the self-ions (r p ≈ 1) and the foreign anions (r pI ≈ 1), as displayed by Fig. 10. However, there is no information available on r pNa within Model #2 because NaPF6 0 pairs and Na+ ions have the same ‘true’ diffusivity or mobility according to the model-specific assumptions, so that their contributions to D∗Na cannot be distinguished from each other [20]. It is also seen in Fig. 10 that the corresponding single-ion fractions 1 − r p and 1 − r pI assume values close to 0.01. The weak “reverse” T -dependence revealed by these quantities relates to the low positive values of the pertaining pair formation enthalpies obtained by fitting. Figure 10 also exhibits the cation transference number t+ < 0.01, which relates to the dissimilar mobilities of the self-ions (Li+ , PF6 − ). This means that, in terms of Li-ion transfer, the performance of the PolyG 30 LiPF 6 electrolyte is predicted to be rather poor.

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Fig. 10. Pair fractions (rp , rpI ; upper solid line) related to LiPF 6 0 and LiI0 pairs, respectively, and pair-fraction complements (1 − rp , 1 − rpI ; dashed lines), as deduced within the combined self-ion/foreign-ion transport model from the experimental data on PolyG30 LiPF 6 in Fig. 9. The cation transference number t+ is also displayed.

Fig. 11. Partial transport numbers related to LiPF 6 0 self-ion pairs (tpLi ,tpPF6 ) and LiI0 foreign-ionrelated pairs (tpI ), as deduced within the combined self-ion/foreign-ion transport model from the experimental data on PolyG30 LiPF 6 in Fig. 9 (solid lines). The single-ion fraction of D∗Li , 1 − tpLi , is also displayed (dashed line).

Another important question is: how high are the relative contributions of neutral pairs to the self-ion and foreign-ion self-diffusion coefficients? Answers can be obtained by plotting the T -dependence of the various pair-related partial transport numbers, as done in Fig. 11. It may be recognized that with 1 − tpLi ≈ 0.01, the pair contribution to Li transport, tpLi , becomes indistinguishable from unity over the whole T -range. This implies that Li transport in

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Fig. 12. True diffusion coefficients of the self-ion species Li+ , PF6 − , and LiPF 6 0 (DLi+ , DPF 6 − , I Dp ) and the foreign-ion species Na+ , I− , NaPF6 0 , and LiI0 (DNa+ , DI− , DNa p , D p ), as deduced within the combined self-ion/foreign-ion transport model from the experimental data on PolyG30 LiPF 6 in Fig. 9 (solid lines). The coincidence of several diffusivities reflects the specific assumptions of the employed Model #2.

PolyG 30 LiPF 6 almost entirely takes place via LiPF6 0 pairs. For self-anion transport, the LiPF6 0 and PF6 − contributions appear to be of similar magnitude, as may be inferred from tpPF6 ≈ 0.3–0.4 for all temperatures displayed in Fig. 11. Concerning the foreign ions, Model #2 only provides the pair contribution tpI , given by tpI =

r pI DIp D∗I

=

r pI DIp r pI DIp + (1 − r pI )D I−

,

(15)

to the tracer diffusivity of iodine D∗I , whereas tpNa like r pNa cannot be resolved. It is remarkable that tpI decreases with increasing temperature, i.e., from 0.74 to 0.32 over the T -range investigated. Since 1 − r pI ≈ 0.01 only changes moderately (see Fig. 10), the observed behaviour must be due to a pronounced difference in the T -dependence of DIp and D I− . Indeed, close inspection of Fig. 12 reveals that D I− varies much stronger with T than DIp while the larger magnitude of D I− compensates for the low value of 1 − r pI . Also the other true diffusion coefficients are depicted in Fig. 12. The high mobility of PF6 − and I− may be due to the (partial) decoupling of anion transport from the motion of the polymer segments. By contrast, the low mobility of the Li- and Na-related species reflects the strong coordinative bonds between the cations and the ether oxygens in the polymer chain. In conclusion, the simultaneous evaluation of self-ion and foreign-ion diffusion data may yield valuable insight into the ion-transport properties of polymer electrolytes. To this aim, PFG-NMR with RTD offers a powerful

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combination of experimental methods, as demonstrated by our results on PolyG 30 LiPF 6 .

7. Salt-precipitation in complexes of PEO and alkali metal iodides Several SPE systems are subject to salt precipitation (SP) when the temperature exceeds a critical value Tc . This phenomenon can be understood from the role of polymer entropy in the solvation of the salt, similar to the increase of ion pair formation with increasing T, as discussed above. For a particular polymer matrix, the critical temperature Tc , marking the onset of precipitation, crucially depends on the pertinent cation and anion species of the salt to be dissolved. For unfavourable combinations of polymer and salt, no complexes are formed, which usually implies that Tc falls below room temperature. Recent work by our group showed that for PEO 30 MI electrolytes with a fixed concentration of alkali metal iodides MI (EO/MI = 30), Tc decreases with increasing size of the cation M (= Li, Na, K, Rb, Cs) [17]. More specifically, a linear decrease of Tc with decreasing lattice energy was found. Furthermore, for particular SPE systems, i.e., PPO-KSCN [34] and PEO-RbI [17], it was observed that Tc monotonically decreased with increasing salt concentration. Classical methods to detect and characterize precipitation phenomena in materials are thermal analysis, X-ray diffraction, and microscopic methods. In case of ionic systems, precipitation of the salt can be sensitively monitored by conductivity measurements, using, e.g., impedance spectroscopy. An example is presented by Fig. 13, which exhibits D σ data for the PEO 30 RbI complex (open circles). It is seen that upon heating the charge diffusivity starts to break down at Tc ≈ 120 ◦ C and continues to fall with increasing T . This decrease of D σ can be naturally explained with the removal of free ions from the electrolyte due to salt precipitation, as was independently indicated by an corresponding endothermic event in differential scanning calorimetry. Unambiguous evidence for the precipitation of the RbI salt was obtained from NMR magic angle spinning (MAS) analysis using the signals of 87 Rb and 127 I, since only for T > Tc the characteristic spectrum of pristine RbI appeared and became stronger with increasing T [17]. Additional information about ionic mobility in SPE systems that are subject to salt precipitation can be obtained by RTD experiments. Measuring diffusion of the radiotracers 86 Rb and 125 I in PEO 30 RbI for temperatures in the SP-free range T < Tc yields common profiles of Gaussian or erfc type, as expected [17]. The diffusion coefficients D resulting from fits with Eqs. (1) or (2) are plotted in Fig. 13 as solid triangles. However, RTD also provides diffusion data for T > Tc , where PFG-NMR would fail because of the low free-ion concentrations prevailing in this SP-active range. Examples for both radioisotopes are shown in Fig. 14. It is striking that the depth profiles reveal a distinct two-step shape that can be well fitted by the sum of two complementary error

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Fig. 13. Tracer diffusion coefficients (D∗Rb , D∗I ) and charge diffusivity (Dσ ) in PEO30 RbI showing salt precipitation for T > 120 ◦ C. Solid and open triangles originate from one-step and two-step diffusion profiles, respectively (see text). The solid lines result from a simultaneous fit within an ion-transport model (i.e., Model #2) that is based on the occurrence of neutral ion pairs in addition to charged single cations and anions.

Fig. 14. Two-step diffusion profiles of the radiotracers 86 Rb and 125 I, as measured in PEO30 RbI after annealing at 140 ◦ C for 60 min and 160 ◦ C for 25 min, respectively. The solid lines are fits of Eq. (16). The dashed lines illustrate the decomposition of the measured profiles in two separate erfc components.

functions, i.e.,



   x x c(x, t) = c1 erfc √ + c2 erfc √ , 2 D1 t 2 D2 t

(16)

represented by the solid lines. In this equation, the two diffusion coefficients (D 1 , D 2 ) and the two concentrations (c1 , c2 ) characterize the near-surface part (step 1) and the great-depth tail (step 2) of the profiles, respectively.

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In quantitative evaluation of the tracer diffusion coefficients by leastsquares fitting it was found that the D 2 data resulting from Eq. (16) for T > Tc were compatible with the unique D values from Eqs. (1) and (2) for T < Tc . This is apparent in Fig. 13 for both 86 Rb and 125 I from the fact that the open (D 2 ) and solid (D) triangles coincide with the smoothly curved solid lines, which represent the ion-transport model with the parameter values compiled in Table 2. By contrast, the D 1 values obtained from fits of Eq. (16) for T > Tc (data not shown) were found to be much lower than the corresponding D 2 values and did not exhibit a systematic T -dependence. The data constellation of Fig. 13, which also displays D σ values (open circles), strongly indicates that D 2 and D may be characteristic of the normal ionic diffusivity commonly observed for polymer electrolytes of similar type but without precipitation (cf. PEO 500 NaI). However, an explanation has to be found for the occurrence of the two-step profiles for T > Tc . Examining the evolution of the profiles with diffusion time, a clear difference between step 1 and step 2 was observed, both for 86 Rb and√125 I [17]. Whereas the deep-penetration step 2 appeared to obey the common D 2 t diffusion law, the near-surface step 1 did not reveal a significant time dependence. Instead, the transition from step 1 to step 2 remained located at essentially the same depth. The occurrence of profile step 1 can be understood from the precipitation phenomenon, as may be elucidated with the aid of the temperature-time profile in Fig. 2. Upon immersing the capsule in the pre-heated oil at time t = 0, the sample temperature starts to rise. Hardly any significant diffusion takes place until t = tm when the PEO melting temperature Tm at about 65 ◦ C is reached. However, after crossing this melt transition, which is indicated by a kink in the heating curve, all ions obtain a noticeable mobility (cf . Fig. 13). This situation persists until the sample temperature attains the RbI precipitation temperature (Tc ≈ 120 ◦ C in the example of Fig. 2) at t = tc . Upon further heating, precipitation sets in, which withdraws an equal amount of cations and anions from the diffusion process. When the diffusion temperature is reached shortly after tc , only a constant (small) fraction of either ionic species remains mobile until the end of diffusion annealing (t = 60 min in the example of Fig. 2), which is given by the onset of quenching in water. Under the conditions of an RTD experiment, the second term on the righthand side of Eq. (16) is likely to reflect the penetration of (the minority of) cations or anions that remain in solution during diffusion annealing. More specifically, D 2 characterizes the ionic diffusivity at the diffusion temperature and c2 /c1 + c2  1 represents the fraction of non-precipitated ions. The first term on the right-hand side of Eq. (16) essentially arises from the diffusion of all cations or anions during the initial annealing period, that is, from t = 0 till t = tc . Thus, c1 + c2 stands for the total tracer concentration at the boundary x = 0, which is indicative of C s , and c1 /c1 + c2 designates the precipitated ionic fraction at T > Tc . However, the penetration depth of profile step 1, character-

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√ ized by 2 D 1 t in Eq. (16), cannot be attributed to a single constant diffusivity D 1 operating during the entire annealing time t. Rather, this penetration must be only due to the initial annealing period 0 − tc , over which √ temperature and diffusivity are monotonically increasing with time. In fact, 2 D 1 t should be replaced by 2 D 1 tc in Eq. (16), where D 1 denotes the mean diffusivity over the time interval 0 − tc . This interpretation is supported by quantitative estimates based on the measured diffusivities for T < Tc (cf. solid triangles in Fig. 13) [17]. The reason for the appearance of two-step diffusion profiles relates to the fact that the salt precipitation in SPEs is an endothermic process and thus takes place at T > Tc . This leads to a physical picture that may be effectively characterized by D 1 < D 2 and c1  c2 . Usually, precipitation processes in material systems are exothermic and occur at low temperatures. Consequently, during the heating-up period of diffusion annealing, the number of mobile tracer atoms tends to be lower than at the diffusion temperature T > Tc . In the present formalism, this would mean that D 1 < D 2 combines with c1  c2 . In this common scenario, however, profile step 1 will be overwhelmed by step 2 and thus will not be visible in the measurement. By contrast, the unique combination of D 1 < D 2 and c1  c2 occurring, e.g., in PEO 30 RbI, makes it feasible to extract additional information from the diffusion experiment. In particular, the fraction of non-precipitated (mobile) ions c2 /c1 + c2 may be inferred from the radiotracer profiles. Moreover, RTD enables us to determine the ionic diffusivity in the T -range subject to precipitation, where PFG-NMR suffers from the low concentration of mobile ions.

8. Effects of oxide nanoparticles in PEO60 NaI Since more than two decades, it has been recognized that the ionic conductivity in SPEs may be enhanced by the addition of nano-sized oxide particles, such as SiO 2 , Al2 O 3 , and TiO 2 [35,36]. Particularly strong effects were observed near room temperature when large weight fractions (10 wt. % and more) of so-called oxide fillers were added to solvent-free PEO-based systems with a relatively high salt concentration (indicated by mole ratios EO/salt typically greater than 10). However, at T < 65 ◦ C PEO-salt complexes exhibit a semi-crystalline microstructure, which comprises salt-poor PEO crystallites, precipitate particles of a highly salt-concentrated crystalline phase (e.g., PEO 3 NaI), and amorphous regions in between. The detailed microstructure in general and the degree of crystallinity in particular depend on the type of salt, its concentration C s , and the thermal history of the complex. On the other hand, the ionic conductivity in this T -range below the melting point of pure PEO is relatively low. A major factor in the observed conductivity enhancement turned out to be the lower degree of crystallinity caused by the dispersed oxide nano-particles. This can be rationalized by the notion that ion migration proceeds much faster

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Fig. 15. DC conductivity σDC of pristine PEO60 NaI (without filler) compared to σDC in the same complex with TiO2 (5 wt. %, 4 nm) or Al2 O3 (10 wt. %, 5 nm) nano-particle dispersions, as indicated. The three sets of data exhibit a closely similar temperature dependence but are different in magnitude, as evidenced by the parallel solid lines serving to guide the eye.

in amorphous than in crystalline regions. It has been also speculated that the special interactions between the cations and the surface of the oxide particles may play a crucial role, so that for large weight fractions of filler material a percolative network of highly conductive pathways may be established. Indeed, such interactions between Li ions and the oxygen atoms of nano-particle surfaces have been detected by sophisticated NMR techniques [37,38]. Nevertheless, the conductivity enhancements in the fully amorphous phase field of the salt-in-polymer complexes were found to be rather small. This motivated us to carry out RTD experiments on PEO-NaI electrolytes with nano-particle dispersions to explore the filler effect on the diffusivity of the individual ion species. Figure 15 shows σ DC as a function of 1/T for PEO 60 NaI with different amounts of TiO 2 (5 wt. %, 4 nm) and Al2 O 3 (10 wt. %, 10 nm) compared to the same complex without filler [39]. It is seen that the changes in σ DC are small, also in view of our general experience that the sample-to-sample differences or – even more relevantly – the batch-to-batch differences of home-synthesized PEO-salt complexes may amount to 20%. It should be further realized that the additives may affect the overall ion density and induce local blocking of ion motion in the bulk of the electrolyte. This may explain that the oxide fillers even give rise to a slight decrease of the conductivity, which amounts to about 14% and 27% for TiO 2 and Al2 O 3 , respectively. Altogether, the present results seem to confirm previous data by several groups that also showed a small influence of nano-sized particles on σ DC in completely amorphous SPEs [37,40]. However, σ DC comprises the joint contributions of cations and anions. Therefore, a weak net effect on σ DC may, in principle, arise from two opposite

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Fig. 16. Tracer diffusion coefficients (D∗Na , D∗I ) and charge diffusivity (Dσ ) in PEO60 NaI without (open symbols and solid lines) and with (solid symbols) TiO2 nano-particles (5 wt. %, 4 nm) as a function of inverse temperature. The solid lines result from a simultaneous fit within an iontransport model (i.e., Model #2) that is based on the occurrence of neutral ion pairs in addition to charged single cations and anions.

relatively strong changes in the ion-specific mobilities, for instance, an increase of the Na+ mobility and a decrease of the I− -one. This issue can be well explored with the RTD technique. Figure 16 compiles the tracer diffusion coefficients (D∗Na , D∗I ) and charge diffusivity (D σ ) of PEO 60 NaI with 5 wt. % of TiO 2 nano-particles with an average diameter of 4 nm (solid symbols). For comparison, the corresponding data for pristine PEO 60 NaI are displayed as well in Fig. 16 (open symbols and solid lines). Obviously, the changes in D∗Na and D∗I are even smaller than those in D σ , with an maximum diffusivity increase of 15% for the cation and a similar maximum decrease for the anion. In converting σ DC to D σ based on the Nernst–Einstein equation (5), we used the mean salt concentration C s (number density) deduced from the measured mass density of the complex and the weight-in fractions of polymer, salt, and oxide particles. Therefore, comparisons of D σ may be more meaningful than those on the σ DC scale, as the former include a correction for the altered ion density. It may be concluded from Fig. 16 that the charge diffusivity has decreased by up to 20% due to the admixture of TiO 2 . It should be further noted that the abovementioned geometrical blocking effects will influence D∗Na , D∗I , and D σ in a closely similar way. Thus particle-induced changes of different sign and magnitude in these experimentally determined diffusion coefficients are meaningful in principle. Attempts to consistently interpret the present results within the iontransport model were less successful. Although it turned out to be easy to obtain good reproductions of the experimental data by simultaneous fitting

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of D∗Na , D∗I , and D σ for either system separately, as demonstrated for pristine PEO 60 NaI by the solid lines in Fig. 16 (see Table 2 for parameter values), the comparison between the systems did not show a consistent picture. This probably relates to the smallness of the nano-particle effects and the strong impact of NaI0 pair formation on the transport properties. Preliminary analysis indicated a reduction of the pair formation enthalpy ΔH p upon the addition of nano-particles. However, more experiments with larger weight fractions of TiO 2 are necessary to arrive at a reliable interpretation. In conclusion, it can be safely stated that the influence of oxide-particle dispersions on long-range ion motion in fully amorphous SPE systems tends to be weak. This is not only true for charge transfer but also holds for the diffusion of cations and anions individually, as revealed by RTD.

9. Conclusions We have shown that radiotracer diffusion experiments can be successfully employed in studies of ionic transport in solid-like polymer electrolytes. The RTD technique profits from its high reliability, sensitivity, and accuracy, which is not only well proven in connection with metals, semiconductors, or oxide glasses, but has been demonstrated now to hold for SPE systems as well. Several examples and effects dealing with long-range ionic motion in prototype salt-in-polymer complexes have revealed the outstanding features of RTD, which include the unique capability of measuring diffusion coefficients at extreme low concentrations of self- and foreign ions. For monitoring ion diffusivities, RTD supplements PFG-NMR rather than being an alternative, since both methods utilize their own suitable probe isotopes, which exhibit very little or no overlap. Combining RTD with electrical conductivity measurements may provide a virtually complete picture of ionic mass and charge transfer. To simultaneously evaluate such data, we have presented a comprehensive theoretical framework involving cation–anion pairs as the dominant form of ion association in SPE systems. In particular, data fitting within this model yields quantitative information about the following physical properties: The ion-pair fraction as a function of temperature, The enthalpy and entropy of pair formation, The contribution of pairs and single ions to the total diffusivity of each ionic species, The true diffusivity (or mobility) of single ions and pairs, The cation transference number as a function of temperature. Data of this kind provide a deeper insight into ion-transport processes in SPEs, which, in turn, may contribute to the development of improved electrolytes for batteries, chemical sensors, or dye-sensitized solar cells in the near future.

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Acknowledgement The authors thank Th. J.-K. Köster, M. Kunze, M. Schönhoff, and L. van Wüllen for their active cooperation in parts of this work as well as all other members of the Collective Research Center (SFB 458) established at the University of Münster during the years 2000–2009 for helpful discussions. We appreciate the continuous support by S. Divinskyi in all matters dealing with the use of radiotracers. One of the authors (N. A. S.) is indebted to K. Funke and H. Mehrer for proposing the radiotracer technique to study ionic transport in polymer electrolytes. Financial support by the Deutsche Forschungsgemeinschaft (Project B12 within SFB 458) is gratefully acknowledged.

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Local Li Cation Coordination and Dynamics in Novel Solid Electrolytes By Leo van Wüllen∗, Thomas Echelmeyer, Nadine Voigt, Thomas K.-J. Köster, and Gerrit Schiffmann Institut für Physikalische Chemie, Westfälische Wilhelms-Universität Münster, Corrensstr. 28–30, 48149 Münster, Germany (Received July 19, 2010; accepted in final form August 19, 2010)

Ionic Transport Research on solid ionic conductors for use as electrolytes in all solid state batteries still constitutes a rather vivid branch of today’s materials science. Despite enormous efforts, neither the development of a solid electrolyte fulfilling the key requirements such as mechanical stability and high ionic conductivity at ambient temperature has been successful nor has an extended understanding of the local Li coordination motifs in the often amorphous systems been obtained. In this contribution, recent progress both in the development of novel solid state electrolytes with high ionic conductivity and mechanical stability and in the characterization of the local Li coordination motifs in these electrolytes from our laboratory is presented. The work was performed as a project within the framework of the Collaborative Research Centre SFB 458 “Ionic Motion in Materials with Disordered Structures – From Elementary Steps to Macroscopic Transport”. Results will be given for polymer electrolytes based on polyethylene oxide (PEO), polyphosphazene (PPZ) and polyacrylonitrile (PAN) with various Li salts, nano-composites of these polymer electrolytes and Al 2 O3 as a ceramic filler, novel inorganic/organic hybrid electrolytes, in which a mixture of an ionic liquid and Li salt is confined within the pore system of a SiO2 glass, and a crystalline electrolyte, Li5 La 3 Nb 2 O12 . Employing a range of advanced solid state NMR methodologies including dipolar based NMR techniques and pulsed field gradient (PFG) NMR and impedance spectroscopy we were able to obtain a detailed knowledge about the local Li cation coordination motifs and the mechanism of Li transport in these electrolytes. Especially the hybrid electrolytes and the salt rich PAN based polymer electrolytes were identified as rather promising materials which combine a high ionic conductivity and mechanical stability.

1. Introduction Electrolytes constitute a salient ingredient to Li ion batteries [1–7]. Among the important properties a successful candidate has to fulfil are a high ionic con* Corresponding author. E-mail: [email protected] Z. Phys. Chem. 224 (2010) 1735–1769 © by Oldenbourg Wissenschaftsverlag, München

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ductivity accompanied by a vanishing electronic contribution, a large electrochemical window, chemical inertness, environmental compatibility, mechanical flexibility and stability. Due to the favourable ionic conductivities, liquid electrolytes constitute today’s standard in battery technology, although the associated disadvantages such as environmental problems due to leaking, fire hazards, limited electrochemical window etc. comprise serious challenges. Solid electrolytes often offer increased mechanical stability and chemical robustness, however, the ionic conductivites generally do not meet the requirements for application as electrolytes in Li ion batteries. Although recent years have witnessed considerable progress in the development of polymer and crystalline Li ion conducting electrolytes – triggered by the ever increasing demand for battery systems for mobile electronics and the automotive industry (electric vehicles) – the ideal solid electrolyte is still waiting to be developed. A large variety of solid state materials has been studied and evaluated during the last two decades, ranging from all-crystalline solid electrolytes, nano-scaled composite systems, polymer based electrolytes or glasses. In the following, we will give a very brief introduction to the various classes of materials.

1.1 Polymer based electrolytes The vast majority of studies in the field of solid electrolytes for use in Li ion batteries has been devoted to polymer electrolytes. Especially their low weight and usually high mechanical flexibility render polymer electrolytes a distinctively promising class of materials. Polymer electrolytes are usually classified according to their polymer/salt ratio. Systems in which the polymer constitutes the major component and the salt concentration is low have been coined saltin-polymer electrolytes, whereas systems in which the Li salt contributes the major component are termed polymer-in-salt-electrolytes. 1.1.1 Salt-in-polymer-electrolytes PEO based electrolytes represent the archetype of this material class ever since the pioneering work of Wright et al. [8,9] and Armand et al. [10,11] who observed that Li salts can be dissolved in polyethylene oxide and exhibit ionic conductivity. A rather important characteristic of the PEO based polymer electrolytes is their strong tendency towards crystallisation and the concomitant formation of a mixture of different phases. The coordination of the Li-cations in crystalline salt-in-polymer PEO based polymer electrolytes is mainly accomplished by the ether oxygen atoms, as has been shown by Bruce et al. [12–15] employing X-Ray diffraction techniques. Only in the case of a higher salt concentration ((PEO)n : LiX; n < 6) a co-coordination of Li by the ether-oxygen atoms and the anions is observed. Employing NMR techniques, Berthier et al. [16] could show that the ionic conductivity predominantly occurs in the amorphous parts (above the glass transition temperature Tg ) of the often multiphased PEO based poly-

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mer electrolytes. Consequently, considerable effort has been directed towards a suppression of the crystallisation within these materials, e.g. by addition of plasticizers or the addition of cross-linkers. Cross-linking indeed surpresses the crystallisation (and offers the additional benefit of an increased mechanical stability), but simultaneously deteriorates the segmental mobility and with this the ionic conductivity. Another route towards a reduction of the crystallinity involves the use of inorganic polymers such as polyphosphazene [17–20] and polysiloxane based materials [21–23], in which polyether or related Li coordinating residues are attached to an inorganic backbone. The rather high flexibility of the inorganic backbone entails both, low Tg values and a reduced tendency towards crystallisation and thus offers the promise of high ionic conductivity. Successful examples include MEEP : LiTf (MEEP = methoxyethoxyethoxypolyphosphazene; Tf = CF3 SO3 − ) and BMEAP : LiTf (BMEAP = bis(2-methoxy-ethyl)amino) polyphosphazene) polymer electrolytes [24,25]. First accounts of the local Li coordination motifs in these polymer electrolyte systems have been given by Allcock and Luther. Whereas Luther et al. [26] – based on the results of solution NMR data – suggests a direct interaction of the Li cations with these sites, Allcock et al. [27] find no interaction between Li cations and the basic backbone nitrogen atoms in PPZ based electrolytes. A considerable disadvantage of these electrolytes is found in the reduced ability to coordinate Li cations. This consequently entails an increased number of ion pairs and higher aggregates, as confirmed by Frech and coworkers [28] for PPZ : LiTf electrolytes. 1.1.2 Nanocomposites Another possible route to enhanced ionic conductivities in polymer based electrolytes involves the addition of often nanoscaled ceramic fillers such as Al2 O3 , TiO2 or SiO2 to form composite materials. Initially, the incorporation of the ceramic particles was stimulated by their ability to improve the mechanical stability of the polymer electrolytes; however, the observation of an increase in the ionic conductivities of up to three orders of magnitude [29–32] initiated intense research in this field. A reduction of the amount of crystalline phases as a first explanation [30,33,34] was later challenged by the observation that the presence of such an effect critically depends on the nature of the Li salt [35], which then was ascribed to a considerable interaction between the Li salt (anions and/or cations) and the surface of the ceramic particle. Several models to explain such an interaction, all of which relying on electrostatic interactions between the surface of the particles and the various anionic or ionic species present in the polymer electrolyte have been put forward. In the Maier model [36–41], these adsorptive or desorptive interactions entail an increase of one ion species in the space charge layer in the interfacial regions of electrolyte and particle [41,42]. The electrostatic interactions between the ionic species and the particle surface may as well be viewed as

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Lewis acid/base interactions [43–48]. Predominantly Lewis acidic ceramic surfaces then interact with the present Lewis bases, either inducing a stiffening of the network and concomitant increase in the glass transition temperature for dominating surface/polymer interactions or entailing an increase in the Li transference number in the case of dominating surface/anion interactions. Such an interaction has been inferred from a shift in the Raman band for the ClO4 and CF3 SO3 anions in the systems LiClO4 , LiTf/PEO/TiO2 [49] and LiClO4 , LiTf/PMEO/TiO2 . Donoso and coworkers [50] used 19 F decoupled 7 Li-NMR to find a decrease in the anion–cation interaction in PEO/LiBF4 /Al2 O3 composites which has been interpreted in terms of an interaction between X− and the ceramic surface. Apart from the mere presence of ceramic particles the particle size seems to be an important parameter. This has been discussed by Heitjans et al. [51] In an investigation of the conductivities of nano-structured composites of Li2 O and B2 O3 [52] these authors observed a monotonic decrease of the Li conductivity with increasing x B2 O3 when using microcrystalline mixtures. Contrasted to this, a maximum of the conductivity at x B2 O3 = 0.5 was observed in the case of nano-scaled Li2 O/B2 O3 mixtures. The observed effects could be consistently explained employing a percolation model, assuming an increased Li conductivity in the interfacial regions between Li2 O and B2 O3 . 1.1.3 Polymer-in-salt electrolytes The increase of the Li salt content usually leads to a deterioration of the ionic conductivity due to an increase in the glass transition temperature. The work of Angell [53] who observed that some extremely salt rich systems with PEO or PPO (poly propylene oxide) as the minor component indeed exhibit a high ionic conductivity however triggered a paradigm shift. In these systems, the ionic conductivity is not restricted to temperatures above the glass transition temperature, indicating a distinct decoupling of cationic transport and segmental motion of the polymer chains. Judged from this high decoupling of cationic and polymer dynamics these systems more resemble those of inorganic glasses. Intensively studied systems comprise PAN or variants thereof as the polymer and LiTf as the Li salt. Based on a detailed study of the above system, Forsyth and MacFarlane [54] suggested a morphology-related conduction mechanism for the Li transport in these salt rich polymer electrolytes. The limited coordinative power of the nitrile group entails the formation of amorphous salt aggregates. Within these aggregates, the Li cations are not coordinated by the polymer and are mobile. At high salt concentrations, these aggregates connect, thus forming a percolation pathway for the macroscopic Li transport. 1.1.4 Crystalline electrolytes Despite immense effort, it still seems to be a formidable task to find polymer based electrolytes in which the optimization of the ionic conductivity does

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not at the same time entail a deterioration of other key properties such as mechanical or electrochemical robustness. Especially with respect to the latter two properties crystalline electrolytes might evolve as promising alternatives. As an additional benefit, the Li cations usually represent the only mobile species within an immobile anionic framework. Apart from the long known “classic” crystalline Li ion conductors such as Li-β-alumina, LiX·H2 O (X = Cl, Br, I) [55,56], Li3y La(2/3)−y TiO3 (0.04 < y < 0.14), [57–59] Li3 N, [60,61] LiSICON (Li14 ZnGe4 O16 ) [62], Li1.3 Al0.3 Ti1.7 (PO4 )3 [63] or Li2.88 PO3.73 N0.14 (LiPON) [64] in recent years especially members of the argyrodite family (e.g. Li6 PS5 Br) [65–67], garnets (e.g. Li5 La3 Nb2 O12 ) [68–76], and sulfidic systems (e.g. Li7 P3 S11 ) [77–79] have attracted considerable attention. Fine tuning of the ionic conductivity in these crystalline systems is often possible via aliovalent doping. As an example, for the best reported member of the garnet family to date, Li7 La3 Zr 2 O12 , (replacement of Nb+V by Zr+IV ) an ionic conductivity of 3 × 10−4 S cm−1 has been observed [80]. The field of Li ion conducting glasses and glass ceramics will not be covered here [81].

1.2 Scope of the work In this contribution, recent progress in the synthesis and characterisation of novel solid electrolytes from our laboratory is highlighted. While a good portion of the data presented here is new and has not been published before, reproduction of data recently published from our laboratory seems unavoidable to obtain the full picture of the project. The goal of the project the results of which are presented here is to find novel solid electrolytes with promising property profiles. Special attention is directed towards an elucidation of the mechanism of Li ion transport in these materials. As a rather important ingredient to an in-depth understanding of the Li ion transport – this denotes the first step on the path towards materials with optimized key parameters – we have to obtain knowledge about the local Li coordination within these electrolytes. Results will be presented for polymer electrolytes based on polyethylene oxide (PEO), polyphosphazene (PPZ) and polyacrylonitrile (PAN) with various Li salts, nano-composites of these polymer electrolytes and Al2 O3 as a ceramic filler, novel inorganic/organic hybrid electrolytes, in which a mixture of an ionic liquid and Li salt is confined within the pore system of a SiO2 glass, and a crystalline electrolyte, Li5 La3 Nb2 O12 from the garnet family.

1.3 Methodology Since many of the studied systems are at least partially amorphous, X-raydiffraction techniques – as employed by P. G. Bruce for the elucidation of coordination motifs in crystalline PEO based polymer electrolytes [15] – prove

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to be only of limited applicability. Among the various alternative characterisation tools such as TEM, XANES, IR, Raman, XPS or solid state NMR, the latter technique has evolved as the most informative tool to obtain structural information on short and intermediate length scales. Information about the immediate environment (length scale 1–2 Å) of a given nucleus, this is referred to as the short-range order, can be obtained from simple NMR or MAS (magic angle spinning) NMR spectra, which supply chemical shift and/or quadrupolar information. The elucidation of structural motifs on intermediate length scales (−8 Å) can be achieved employing homo- or heteronuclear dipolar double resonance techniques such as REDOR [82–84] (rotational echo double resonance) and its variants [85–87]. With REDOR, the structural motifs are evaluated via the determination of internuclear distances between two nuclei I and S, measuring the heteronuclear dipolar coupling between these two nuclei, which scales with the inverse cubic power of the distance. For further details of these advanced NMR techniques the reader is referred to recent excellent reviews of the field [84,88,89].

2. Results The organization of this section is as follows: The first part is devoted to polymer based electrolytes, specifically, PEO based electrolytes and composites [90–92], PPZ based electrolytes and composites [93] and salt rich PAN based systems [94,95]. Subsequently, results on novel hybrid electrolytes, in which a mixture of an ionic liquid and Li salt is confined within the pore system of a SiO2 glass will be presented [96]. The section will conclude with a report on the analysis of the mechanism of Li transport in the crystalline electrolyte Li5 La3 Nb2 O12 [97,98]. Details of the syntheses and experimental procedures will not be given here, the reader is referred to the literature.

2.1 Polyethylene oxide based electrolytes Polymer electrolytes in the system PEO : LiTFSI (TFSI = (CF3 SO2 )2 N) are characterized by an especially rich complex superposition of various crystalline and amorphous phases. Starting with pristine LiTFSI, the first mixed phase which is being formed with increasing PEO content is (PEO) 2 : LiTFSI, then (PEO)3 : LiTFSI and (PEO) 6 : LiTFSI [90,99]. All these phases are crystalline, intermediate compositions contain a superposition of the two adjacent phases. However, only the crystal structure of (PEO)3 : LiTFSI has been solved (cf . Fig. 1) [100]. In an attempt to obtain information about the local Li coordination motifs in this polymer electrolyte system we performed an extended solid state NMR study employing 19 F{7 Li}and 13 C{7 Li}-REDOR NMR spectroscopy [90,92]. (PEO)3 : LiTFSI served as a model compound to validate the applicability of the approach. The coordina-

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Fig. 1. Local Lithium coordination in (PEO)3 : LiTFSI: red: Li; black: C, green: F; blue: N; grey: S; white: O [100].

tion of the Li cations by the polyether chain may be directly evaluated employing 13 C{7 Li}-REDOR NMR spectroscopy. The data for (PEO)3 : LiTFSI and (PEO) 6 : LiTFSI (data not shown; cf . [92]) produces Li−C distances of 3.15 ± 0.2 Å, in nice agreement with the crystal structure of (PEO)3 : LiTFSI (average distance 3.1 Å [100]) or the values (3.2 Å) found for (PEO)6 : LiPF6 by Reichert et al. [101] using REDOR-NMR. A co-coordination of the Li cations by the oxygen atoms from the SO 2 groups of the TFSI anions can be checked employing 19 F{7 Li}-REDOR NMR spectroscopy. The corresponding data for (PEO)3 : LiTFSI is shown in Fig. 2. In the 19 F MAS NMR spectrum, two different 19 F signals can be identified and assigned to the two crystallographically distinct CF3 groups of the TFSI anion. In (PEO)3 : LiTFSI, only one of the CF3 SO2 -residues per anion coordinates to the lithium ions, whereas the second residue points away from the PEO helices to the interchain space. The mean F···Li distance for the CF3 group closer to the PEO helix (denoted (1) in Fig. 1) is 4.5 Å, whereas it is 5.9 Å for the second CF3 group (denoted (2) in Fig. 1). In the dipolar evolution curves obtained from the 19 F{7 Li}REDOR-NMR experiment (cf . Fig. 2 bottom), qualitatively, the initial slope of the dipolar evolution curves is inversely proportional to the internuclear distance, hence the fluorine nuclei represented by the signal at − 77.4 ppm belong to the CF3 groups in closer proximity to the Li cations (i.e. (1) in Fig. 1) than the second CF3 group, represented by the signal at − 80.0 ppm (cf . Fig. 2). A quantitative analysis of the REDOR curves is possible by a full simulation of the dipolar evolution curves employing the SIMPSON [102] software. With the MAS frequency and the employed rf field amplitudes for 7 Li and 19 F as input parameters for the simulations and assuming 19 F–7 Li spin pairs, 3 the value for the dipolar coupling constant D (in Hz) = (γLi γF hμ 0 )/(r Li−F 8π 2 ) is varied until the calculated curve fit the experimental data. From an analy-

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Fig. 2. Top: 19 F-MAS NMR spectrum for (PEO)3 : LiTFSI: two signals from the two crystallographically distinct sites are observed. Bottom: 19 F{7 Li}-REDOR NMR data: dots denote experimental data points for the signal at − 77.4 ppm (black; upper points) and − 80.0 ppm (red; lower points); solid curves are the results of numerical simulations assuming F−Li distances of 4.5 Å (upper curve) and 5.9 Å (lower curve), respectively.

sis of the REDOR data we obtain 19 F–7 Li distances of 4.5 and 5.9 Å for the signals at − 77.4 and − 80.0 ppm, respectively, in excellent agreement with the values determined from the crystal structure. The signals at − 77.4 ppm consequently corresponds to 19 F nuclei in the CF3 SO2 residue coordinating the lithium ions and the line at − 80.0 ppm to the residue that points into the interchain space.

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Fig. 3. 19 F-MAS NMR spectra and results from (top) and (PEO) 6 : LiTFSI (bottom).

19

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F{7 Li}-REDOR NMR for (PEO) 2 : LiTFSI

The REDOR approach was then used to analyse the ion coordination in (PEO) 2 : LiTFSI and (PEO) 6 : LiTFSI, whose crystal structures are unknown. For (PEO) 2 : LiTFSI (cf . Fig. 3) the two 19 F NMR lines exhibit different dephasing behavior in the REDOR experiment, with resulting 19 F–7 Li nuclear distances of 4.7 and 5.2 Å for the signals at − 78.3 ppm and − 80.3 ppm, respectively. Obviously, the local Li coordination motifs in (PEO)2 : LiTFSI very much resemble that of Li in PEO3 LiTFSI with the TFSI anions directly coordinating to the lithium ions. With increasing O : Li ratio, however, this coordination motif is more and more abandonded in favour of an exclusive coordination of the Li cations by the PEO oxygen. This is borne out by the results for (PEO) 6 : LiTFSI the REDOR data of which rules out any direct cation– anion coordination (cf . Fig. 3), reminiscent of the situation in (PEO)6 : LiPF6 , in which the PF 6 − anions occupy positions between the PEO chains and do not contribute to the coordination of the lithium cations [15]. Thus, the exclusive coordination of Li cations by the PEO ether oxygens seems to be the prefereable coordination motif, as long as the PEO offers sufficient oxygen coordination sites. Only in the salt-rich compositions, in which the polymer alone

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Fig. 4. Top: fraction of mobile Li ions and TFSI anions in (PEO)n : LiTFSI at ambient temperature as a function of polymer electrolyte composition; bottom: fraction of mobile Li cations and TFSI anions in (PEO) 2 : LiTFSI as a function of temperature. Red dots: TFSI; black dots: Li+ .

cannot saturate the coordination environment of the cations, the anions help in the coordination. Such a coordination in the salt-rich compositions seems to inhibit any extended Li mobility. Considerable Li mobility is exclusively observed for samples with n ≥ 6. In addition, cations and anions share the same onset temperature for the mobility (cf . Fig. 4). These compositions are characterized by a mixture of pristine PEO, crystalline (PEO) 6 : LiTFSI and an amorphous phase of approximate composition (PEO) 11 : LiTFSI, as borne out by the DSC and X-ray diffraction patterns [90,92], contrary to the phase diagram as proposed by Lascauld et al. [99], which predicts the presence of a single amorphous phase. Upon cooling, the amorphous part of the mixture in the sample of composition (PEO) 10 : LiTFSI further segregates into crystalline PEO 6 : LiTFSI and pristine PEO. This can be inferred from the appearance of two new signals at

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Fig. 5. 13 C-CPMAS NMR spectra for (PEO) 10 : LiTFSI in the temperature range 193 K ≤ T ≤ 293 K.

71.5 ppm and 72.3 ppm in the 13 C-CPMAS NMR spectra for (PEO)10 : LiTFSI, as shown in Fig. 5. The 13 C nuclei associated with these two signals do not exhibit any dipolar coupling to Li nuclei as exemplified in a 13 C{7 Li}-REDOR NMR experiment, performed at T = 208 K, well below the glass transition temperature of the material. From the isotropic chemical shift values and the absence of any Li···C dipolar coupling these signals could be assigned to carbon in pristine PEO. This segregation was found to be completely reversible. 2.1.1 Effect of inorganic ceramic fillers As outlined in the introduction, the addition of – often nanoscaled – inorganic filler materials may entail a significant increase in the ionic conductivities. Dipolar based solid state NMR techniques offer an elegant approach to study the suggested interactions between the salt ions and the ceramic surface [91]. The binary system LiX/Al2 O3 (X = Tf, TFSI; using γ -Al2 O3 with three different surface modifications acidic, neutral and basic) was chosen as a model system. In the 7 Li MAS NMR spectra, apart from the signal of the pristine LiTFSI salt at − 1.4 ppm, a new signal at approx. 0 ppm, the intensity of which in-

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Fig. 6. 7 Li MAS NMR (left) and 19 F MAS NMR spectra (right) for binary LiTFSI/Al2 O3 samples: i pristine LiTFSI; ii LiTFSI/ Al2 O3 acidic ; iii LiTFSI/Al2 O3 neutral ; iv LiTFSI/Al2 O3 basic .

creases with the basicity of the Al2 O3 surface (cf . Fig. 6) can be identified. Employing 1 H{7 Li}-CPMAS-{27 Al}-REAPDOR NMR spectroscopy, this new signal could be safely assigned to Li cations in direct contact with the alumina surface. In this experiment, specifically designed to detect Li cations in close proximity to the Al2 O3 surface, Li cations in proximity to protons (i.e. at the alumina surface) are selected via an initial Li CP step (cf . Fig. 7). Then, the Li magnetization prepared in this way is fed through a 7 Li{27 Al}-REAPDOR sequence which detects dipolar couplings between these nuclei. REAPDOR (Rotational Echo Adiabatic Passage Double Resonance) constitutes a variation of the REDOR sequence, specifically optimized for quadrupolar nuclei on the dephasing channel. From a simulation of the obtained data employing the SIMPSON software a second moment of M2 = 7 × 107 rad2 s−2 can be deduced. The van Vleck second moment M2 relates to zeroth order to an effective dipolar coupling D eff according to M2 =

4 4π 2 I(I + 1)D2eff , 15

with I denoting the nuclear spin of the dephasing nucleus. Since a single Li···Al dipolar coupling, assuming an internuclear distance of 3.1 Å [91], contributes 1.5 × 107 rad2 s−2 to the second moment, the calculated M2 value thus translates to the presence of 4–5 Al nuclei within a 3.1 Å

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Fig. 7. Identification of Li cations associated with the Al2 O3 surface employing a specifically designed dipolar NMR experiment 7 Li{1 H}-CPMAS-{27 Al}-REAPDOR NMR spectroscopy. a) Pulse sequence; b) 7 Li-MAS NMR spectrum of LiTFSI/Al2 O3 neutral . In the first step of the sequence, the cross polarisation step of the sequence selects those 7 Li nuclei exhibiting a close proximity to protons; c) In the second step, the dipolar coupling of these selected Li nuclei to 27 Al nuclei is monitored employing 7 Li{27 Al}-REAPDOR NMR. The resulting REAPDOR evolution curves are shown in d) black dots: LiTFSI/Al2 O3 neutral ; red circles: Al2 O3 basic ; solid curve: result of a full simulation of the experiment using the SIMPSON [102] software assuming an effective second moment of 7 × 107 rad2 s−2 .

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sphere around a central Li cation. From this a strong interaction between the Li cations and the Al2 O3 surface was deduced. At the same time, an anion interaction with the alumina surface can be observed. In the 19 F MAS NMR spectra for the various samples (cf . Fig. 6) two different signals can be identified and assigned to TFSI anions within an LiTFSI type of environment and TFSI anion coordinated to the alumina surface, based on the results of 19 F{7 Li}-REDOR NMR and 19 F{27 Al}TRAPDOR (Transfer of population double resonance) NMR spectroscopy (data not shown) [91]. Having established the interaction between Li cations and the alumina surface in the binary Li-salt/Al2 O3 systems, we investigated the effect of addition of alumina to the PEO : LiTFSI system. In the salt poor concentration regime (n = 10) we indeed did observe a roughly 50% increase in the amount of mobile Li ions, based on the results of static 7 Li NMR experiments. We did however not find any indication for a direct interaction between Li cations and the alumina surface. Instead, the Al2 O3 addition seems to entail a considerable increase in the fraction of the amorphous phase. This can be inferred from the 13 C-CPMAS NMR spectra of the Al2 O3 containing samples, in which the intensity of the broad signal (which can be assigned to the amorphous phase [90]) is considerably increased (data not shown, cf . [90]). Thus, the predominant effect of alumina in PEO based polymer electrolytes seems to be a suppression of the formation of the crystalline (PEO)6 : LiTFSI phase, entailing an enhancement in the ionic conductivity of the samples.

2.2 Polyphosphazene based electrolytes The most striking difference between PEO based and PPZ based polymer electrolytes is the extreme reluctance of the latter towards crystallisation. In addition, compared to the situation in PEO based polymer electrolytes, the coordinative power of the BMEA ligand in BMEA-PPZ based polymer electrolytes is considerably reduced. Therefore, the coordination of Li by the BMEA side chains is supposed to be less dominant, even in the salt poor samples. This was studied on a range of BMEA PPZ based polymer electrolytes with varying amounts of LiTf as Li salt [93]. Some of the samples contained small amounts (< 10 wt. %) of nano-scaled Al2 O3 . In Fig. 8, the 7 Li MAS and 7 Li{1 H} CPMAS NMR spectra of a BMEA PPZ polymer electrolyte (BMEA PPZ; 10 wt. % LiTf, 8 wt. % nano-scaled Al2 O3 ) are shown. Judged from the DSC results, the PPZ based samples are completely amorphous. In the 7 Li spectra, four different Li signals can be identified: (1) The signal at − 0.55 ppm can be assigned to Li cations coordinated by the BMEA side chains of the PPZ. The assignment is based on the results of 7 Li{1 H}-REDOR and CPMAS NMR experiments from which a strong Li−H interaction for the Li nuclei contributing to this signal and protons from the side chains was deduced [93]. (2) The rather broad signal at 0.16 ppm, accompanied by a set of spinning sidebands

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Fig. 8. 7 Li-MAS NMR (top) and 7 Li-CPMAS NMR spectrum for a BMEAP polymer electrolyte at room temperature containing 10 wt. % LiTf and 8 wt. % nano-scaled Al2 O3 . Four different signals can be discerned, the assignment of which is given in the text.

(not shown) does not change its appearance even at temperatures T > 365 K, indicating that the corresponding Li cations remain completely immobile even at these elevated temperatures. The signal is completely absent in a 7 Li{1 H}CPMAS NMR experiment, excluding any spatial proximity to the BMEA residues. The relative fraction of this signal is increasing with the relative fraction of LiTf in the sample, increasing from 35% (5 wt. % LiTf) to 65% (15% LiTf). Thus, the Li cations contributing to this signal may be assigned to Li cations located in larger LiTf agglomerates. (3) The rather narrow signal, contributing 10–15% to the total signal intensity, could be assigned to rather mobile Li cations. Since the mobility of these Li cations was found to be closely linked to the glass transition temperature of the PPZ, [93] we tentatively assigned this species as Li cations loosely connected to the polyphosphazene backbone, as e.g. suggested by Luther [26]. Such an interaction should principally be detectable via 31 P{7 Li}- or 15 N{7 Li} REDOR experiments at temperatures below the glass transition temperature, at which the polymer mobility is frozen. Unfortunately, since the signal for these Li cations considerably broadens and severerly overlaps with the other Li signals at temperatures below the glass transition temperature, such an experiment is not feasible.

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Fig. 9. 7 Li{1 H}-CPMAS-{27 Al}-REAPDOR data for a BMEAP sample containing 8 wt. % nano-scaled Al 2 O3 and 10 wt. % LiTf. Filled circles: normalized difference intensities for the 7 Li signal at − 0.27 ppm; filled diamonds: normalized difference intensities for the 7 Li signal at − 0.55 ppm.

2.2.1 Effect of inorganic ceramic fillers (4) In the samples containing nano-scaled Al2 O3 , a fourth signal could be detected in the 7 Li{1 H}-CP-MAS NMR experiment (cf . Fig. 8). Here, the CP experiment acts as a dipolar filter, only those Li cations with spatial proximity to protons will contribute to the signal. Apart from the signal at − 0.55 ppm, which could be assigned to Li cations in close proximity to the BMEA residues, a second signal at − 0.27 ppm can be identified. This signal is only visible in the Al2 O3 containing samples; its intensity scales inversely with the Al2 O3 particle size. These findings suggest that the Li cations contributing to this signal are associated with the Al2 O3 surface, obtaining the magnetization from the 1 H nuclei at the alumina surface (from Al-(OH)-Al or terminal Al-OH units). Direct evidence for this assumption comes from 7 Li{1 H}-CPMAS-{27 Al}REAPDOR NMR spectroscopy (vide supra). As obvious from inspection of Fig. 9, only the signal at − 0.27 ppm exhibits a sizeable REAPDOR effect indicating spatial proximity between the corresponding Li cations and the alumina surface, whereas the Li cations associated with the BMEA residues (signal at − 0.55 ppm) do not exhibit any 7 Li{27 Al} dipolar effect. In Fig. 10, the four different local Li coordination motifs are sketched. Thus, comparing (salt-poor) PEO and PPZ based polymer electrolytes, conceptional differences in the local cation coordination and the influence of ceramic fillers are observed. In PEO based polymer electrolytes the Li cations are efficiently coordinated by the polyether oxygen atoms; additionally offered coordination sites from the alumina surface can not compete with this ener-

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Fig. 10. The four different identified local Li coordination motifs in BMEAP based polymer electrolytes with alumina particles.

getically favorable situation. Thus the influence of alumina as ceramic filler is restricted to inhibiting the crystallisation of (PEO) 6 : LiTFSI, thereby enhancing the ionic conductivity. Coordination to the alumina surface is only feasible if the coordination sites offered by the polymer (e.g. by the BMEA residiues in the examined PPZ materials) do not provide sufficient stable coordination sites.

2.3 Polyacrylonitrile based electrolytes Compared to PEO and PPZ based polymer electrolytes the coordinative power of the nitrile group of PAN based electrolytes is even further reduced. As a consequence, the Li salts will not be dissolved as efficiently as in e.g. PEO. On the other hand, the reduced coordinative power entails a decoupling of the ionic transport from any matrix motional process, i.e. the ion transport is not bound to a support by the segmental motion of the polymer chains. Often, PC, EC or related molecules are added as plasticizers. In an attempt to obtain more information about the local Li coordination motifs and dynamics in PAN based electrolytes we investigated the systems PAN/LiBOB (BOB = bisoxalatoborate) and PAN/LiBF 4 . A detailed account of the results of these studies will be given elsewhere, [94,95] here we want to focus on the salient features which contrasts the PAN based electrolytes from the PEO and PPZ based electrolytes discussed above. In a typical synthesis procedure, polyacrylonitrile and the Li salt were dissolved in DMSO, stirred for 36 h and the solvent subsequently re-

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Fig. 11. Static 7 Li NMR linewidth (given in full width at half height) for a PAN/LiBOB sample with 80 wt. % LiBOB (full diamonds). The corresponding data for pristine LiBOB (full circles) is given for comparison.

Fig. 12. 7 Li-MAS NMR spectrum of a LiBOB/PAN sample prepared employing deuterated d6 DMSO. The assignment of the three signals to the indicated local Li coordinations is based on the results of 7 Li{1 H}- and 7 Li{2 H}-REDOR NMR experiments as outlined in the text. Sv denotes a solvent molecule DMSO.

moved in vacuo at 90 ◦ C for 72 h. For the PAN/LiBOB system, compositions in the extreme salt rich regime (salt content ≥ 80 wt. %) were investigated. Compared to the situation in pristine LiBOB, the samples exhibit a considerably increased Li mobility as evidenced from an analysis of the evolution of the static 7 Li NMR line width with temperature. The 7 Li NMR line width –

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which scales with the inverse of the mobility – begins to narrow at temperatures as low as 230 K (cf . Fig. 11). In the 7 Li-MAS NMR spectra of the PAN/LiBOB samples, mainly three different signals at + 0.3, − 0.7 and − 1.1 ppm can be identified (cf . Fig. 12): The signal at + 0.3 ppm can be safely assigned to Li cations within crystalline LiBOB domains based on the isotropic chemical shift, a clear assignment of the remaining two signals however is not possible without the assistance of dipolar based NMR experiments. The observations that a) the intensity of the signal at − 1.1 ppm increases with the PAN content of the samples, b) does not exhibit any spinning sideband intensities and c) constitutes the dominant signal in the LiBOB/PAN sample with 80 wt. % LiBOB suggest that this signal originates in mobile Li cations within the framework offered by the PAN polymer. The signal at − 0.7 ppm on the other hand is accompanied by a set of spinning sidebands, from which a quadrupolar coupling constant of 68 kHz can be calculated. Thus, this signal may be assigned to an immobile Li species. In a 7 Li{1 H}-CPMAS NMR experiment at room temperature, only the signal at − 0.7 ppm can be detected; performing the experiment at T = 200 K entails a detection of the signals at − 0.7 and − 1.1 ppm. Thus both species exhibit a spatial proximity to protons. In addition, the observations support the assignment of the − 1.1 ppm signal to a mobile species (no CP possible) at room temperature. (The signal for the Li cations within crystalline LiBOB naturally does not exhibit any 7 Li{1 H}CPMAS intensity.) To further investigate the nature of these two signals it has to be recognized that there are two potential sources of protons in the sample, from which magnetization can be transferred to 7 Li nuclei in the CP experiment: a) the methylene and methine protons of the PAN backbone and b) the methyl protons from some residual DMSO solvent molecules. To separate the effect of these two proton sources on the CP and REDOR spectra we prepared a LiBOB/PAN sample using perdeuterated d 6 -DMSO. In the 7 Li{1 H}-REDOR NMR experiment, performed at 200 K to eliminate the influence of any cation motion, the signal at − 1.1 ppm clearly exhibits a much more pronounced dipolar coupling to 1 H nuclei as does the signal at − 0.7 ppm (see Fig. 13 top). In the 7 Li{2 H}-REDOR NMR experiment performed at T = 200 K, on the other hand, the 7 Li{2 H} dipolar coupling is more pronounced for the signal at − 0.7 ppm (cf . Fig. 13 bottom). From this, it is clear that the 7 Li signal at − 0.7 ppm originates in Li cations in close coordination to DMSO molecules. Based on the observation that this signal also occurs in a sample which was prepared by dissolving LiBOB in DMSO and then removing the solvent as described above and also occurs in a salt rich PEO/LiBOB polymer electrolyte with unchanged isotropic chemical shift value and quadrupolar parameters, we can assign this signal to a Li(DMSO) n BOB complex isolated from the polymer host. For the signal at − 1.1 ppm the REDOR results support the assignment to Li cations within the PAN matrix which are rather mobile at room temperature and only frozen at temperatures below 230 K. Based on the

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Fig. 13. Top: 7 Li{1 H}-REDOR NMR data and bottom: 7 Li{2 H}-REDOR NMR data for the d6 DMSO LiBOB/PAN sample. Filled blue circles: data for the signal at − 1.1 ppm; filled red circles: data for the signal at − 0.7 ppm. The experiments were performed at T = 200 K to exclude any bias due to dynamics.

7

Li{2 H}-REDOR NMR results we have to assume that these Li cations as well are coordinated by (a smaller number of) DMSO molecules. Due to the fact that in these samples at room temperature we always observe a superposition of immobile and mobile DMSO molecules we were however not able to get more quantitative information, i.e. to determine the number of DMSO molecules per Li(DMSO) n BOB complex and per Li cation within the PAN matrix. In PAN based electrolytes with LiBF 4 as Li salt the situation proved to be quite different. Here no immobile Li(DMSO) complex is being formed and we were able to calculate the number of DMSO molecules associated with the Li cations [95]. Since all DMSO molecules present in these samples are found to be mobile at room temperature (based on 2 H-NMR data on deuterated samples, data not shown) and PAN proves to be completely immobile, we could obtain the number of DMSO molecules per PAN unit via a quantitative analysis of the 1 H MAS NMR spectra. Although we have to note that such an analysis may be biased to some extent [103] we can take the results (1–2 DMSO molecules per Li cation) as a rough estimate. In Fig. 12 the three different local Li environment found in LiBOB/PAN samples are sketched. The 7 Li MAS NMR

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Fig. 14. 7 Li-MAS NMR spectrum a) and ionic conductivity as a function of inverse temperature b) for sample PAN/LiBOB 1 : 1.

spectrum may thus be taken to directly evaluate the quality of the prepared electrolyte. Only if the signal at − 1.1 ppm contributes the dominant fraction to the total signal intensity will the sample exhibit high Li mobility. This is corroborated by the room temperature conductivity for sample PAN/LiBOB 1 : 1, which almost exclusively exhibits the − 1.1 ppm signal and the ionic conductivity was found to exceed 3 × 10−4 S cm−1 (cf . Fig. 14).

2.4 Hybrid inorganic–organic polymer electrolytes containing ionic liquids In the course PEO-PPZ-PAN the decoupling index as introduced by Angell [104] (in essence a degree of cooperative motion of the Li cations and the hosting matrix) finds itself successively reduced. Perpetuating this concept of decreasing the Li coordination power of the hosting matrix and increasing the decoupling index leads to the employment of polymers such as polyethylene or to inorganic systems such as glassy materials. In such a hybrid system, the host provides the mechanical stability, whereas a second phase – e.g. a Li salt dissolved in an ionic liquid – contributes the good ionic conductivity. This situation has to be contrasted to conventional inorganic glassy ionic conductors, in which the Li cations – e.g. in a Li2 O-P2 O5 glass – constitute a salient part of the glass network, determining e.g. the amount of connectivity among the phosphate units. In our approach, we employed an in situ synthesis of the hosting SiO2 glass matrix using a non-aqueous sol–gel process with the ionic liquid

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Fig. 15. Photograph of ionogel glass 0.5 LiTf : 1 SiO2 : 2 [BMIM]BF4 .

[BMIM]BF 4 (1-butyl-3-methylimidazolium tetrafluoroborate) and the Li salt (LiTf or LiTFSI) being present in the starting mixture [96,105]. Confining ionic liquids into a SiO2 matrix via a sol–gel process has been shown to produce stable solid membranes [106–111]; the resulting materials have been considered as materials for dye-sensitized solar cells [112,113], electrolytes for fuel cells etc.. In a typical synthesis procedure, tetraethoxysilane (TEOS) was added to a mixture of LiTf and formic acid (HCOOH). Upon addition of [BMIM]BF 4 to the clear solution, gelation occurred within less than 1 min. After drying at 50 ◦ C overnight the samples were stored at 110 ◦ C for several days, resulting in transparent glass monoliths (cf . Fig. 15). The host matrix was characterized employing 29 Si-MAS NMR spectroscopy and nitrogen sorption (cf . Fig. 16). The 29 Si-MAS NMR spectra of the ionogel glasses exhibit a clear dependence on the amount of ionic liquid added to the sol gel mixture. Whereas the spectra of IL free and Il poor samples are characterized by the presence of three different signals at −91, −101 and −110 ppm, the spectra for the IL containing samples only exhibit a single signal at −110 ppm. Employing the well established Q (n) -notation, in which Q (n) denotes a SiO 4 group connected to (n) further SiO 4 tetrahedra, the signals can be assigned to Q (2) units (−91 ppm), Q (3) units (−101 ppm) and Q(4) units (−110 ppm). Thus in the ionic liquid containing samples we find a fully condensed SiO2 network, whereas the gel to glass conversion in the IL free samples (containing a large amount of terminal Si−OH groups) is far from complete. The N 2 sorption isotherms for the ionogels indicate the presence of a mesoporous silica network with pore sizes around 2–2.5 nm and a surface area of ca. 600 m2 /g. For a good solid ion conductor, the interaction between the hosting matrix and the guest – here the ionic liquid/Li salt mixture – plays a pivotal role. The presence of a strong interaction would entail a significant reduction of the ionic conductivity of the guest phase, whereas in the case of but a marginal interaction problems due to leaking of the guest phase might arise. One way of probing such an interaction between hosting matrix and guest phase involves measuring the magnetic dipolar couplings between nuclei from either of both phases, e.g. employing 29 Si{19 F}-CPMAS or REDOR spectroscopy.

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Fig. 16. Characterization of the host matrix for the hybrid electrolytes. a) 29 Si-MAS NMR spectra for different ionogel samples: i 0.5 LiTf : 1 SiO2 : 0.1 [BMIM]BF 4 ; ii 0.5 LiTf : 1 SiO2 : 2 [BMIM]BF4 ; iii 0.5 LiTFSI : 1 SiO2 and iv 0.5 LiTFSI : 1 SiO2 : 2 [BMIM]BF4 . b) N2 sorption isotherms for ionogel 0.5 LiTf : 1 SiO2 : 2 [BMIM]BF4 (filled dots), pristine SiO2 gel as reference (black squares) and the ionogel after washing with H2 O/EtOH (open circles).

This has been investigated using a binary sample consisting of the pristine ionic liquid confined within the pores of the SiO2 host. The possibility to transfer magnetization from the 19 F nuclei (of the ionic liquid anion BF 4 − ) to 29 Si (as

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Fig. 17. NMR spectroscopic evidence for the host-guest interaction in the ionogel 0.5 LiTf : 1 SiO2 : 2 [BMIM]BF4 : top: 19 F-MAS NMR spectrum: The presence of (tiny) spinning sideband intensity indicates a deviation from isotropic mobility of the ionic liquid anions as a consequence of the confinement. Further, the existence of a 19 F{29 Si}-CPMAS NMR spectrum (middle) and a 29 Si{19 F}-CP-REDOR NMR effect (bottom) requires the presence of (non-averaged) magnetic dipole interaction between these nuclei.

an atom from the host matrix), i.e. the mere presence of a 29 Si{19 F}-CPMAS NMR spectrum (cf . Fig. 17) conclusively indicates that the confinement of the [BMIM]BF 4 within the pore system of the host puts some constraints on the mobility of the ionic liquid entailing a not completely averaged dipolar coup-

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Fig. 18. Static 7 Li NMR spectra (left) and 19 F-MAS NMR spectra for pristine LiTFSI (top), 0.5 LiTFSI : 1 SiO2 (middle) and the ionogel 0.5 LiTFSI : 1 SiO2 : 2 [BMIM]BF4 (bottom).

ling between 19 F and 29 Si. This is corroborated by the results of a 29 Si{19 F}CP-REDOR NMR experiment (cf . Fig. 17, bottom). The mobility constraints are also revealed by the presence of (minor) spinning sideband intensity in the 1 H (from the [BMIM] cation) and 19 F (from the BF 4 anion) MAS NMR spectra. Having established the stability of the hosting network and a sizeable interaction between the host matrix and the ionic liquid/Li salt guest phase ensuring the stability of the complete system the next step involves an analysis of the ionic mobility within the amorphous hosting network. The dynamics of the ionic liquid and the Li salt within the pore system of the confining rigid silica network was analysed employing solid state 19 F and 7 Li NMR spectroscopy. The 19 F-MAS and static 7 Li spin-echo NMR spectra for pristine LiTFSI bear the typical characteristics for immobile TFSI anions and Li cations with the observed numerous spinning sidebands in the 19 F-MAS NMR spectrum indicating non-averaged chemical shift anisotropy (19 F-MAS) and the broad m = 1/2 → m = −1/2 central transition in the static 7 Li NMR spectrum (FWHM = 3.5 kHz) indicating non-averaged homo- and heteronuclear dipolar interactions (cf . Fig. 18). The situation changes considerable in a binary LiTFSI/SiO2 glass system, in which the SiO2 matrix has been prepared in situ. The spinning sideband intensity in the 19 F-MAS NMR spectrum is distinctively reduced, the line width in the static 7 Li NMR spectrum reduced to approx. 500 Hz. Both observations indicate extended anion and cation mobility in this sample. At this moment, we can only speculate about the origin of

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Fig. 19. Ionic conductivity as a function of inverse temperature for ionogel glasses 0.5 LiTf : 1 SiO2 : 2 [BMIM]BF4 (open circles); 0.5 LiTf : 1 SiO2 : 1 [BMIM]BF4 (filled diamonds) and 0.5 LiTFSI : 1 SiO2 : 2 [BMIM]BF4 (filled circles).

ion mobility in this binary system. We note, however, that this system may be viewed as a mixture of nano-dispersed SiO2 and LiTFSI particles, thus forming a solid electrolyte in which the interfacial regions between these particles could be involved in the ionic transport. The incorporation of the ionic liquid and the LiTFSI into the SiO2 matrix is accompanied by a further reduction of the 7 Li NMR line width and 19 F-MAS NMR spinning sideband intensity. This reduction in spinning sideband intensity and line width of the center signal indicates a corresponding increase in the respective ion mobility. This increase in the ion mobility was found to scale with the ionic liquid/Li salt ratio. The solid state NMR results thus indicate an almost liquid like dynamic of the present cations (Li+ , [BMIM]+ (from 1 H and 13 C NMR, data not shown)) and anions (Tf or TFSI, BF 4 − ) within the pore system of the fully condensed SiO2 network. This very high ionic mobility is continued down to subambient temperatures, extending to temperatures as low as 240 K (data not shown) [96]. The results of an impedance analysis for two different ionogels with a high IL/Li salt ratio indicate a high ionic conductivity in these samples. The material with the best performance – 0.5 LiTFSI : 1 SiO2 : 2 [BMIM]BF 4 – exhibits an ionic conductivity (T = 298 K) of 5 × 10−3 S cm−1 (Fig. 19). It is important to note, however, that this overall ionic conductivity is related to the contribution of four different ions: [BMIM]+ , BF 4 − , TFSI (or Tf) and Li+ . A convenient way to disentangle the individual contributions involves the determination of the individual self diffusion coefficients for the various ionic species employing pulsed field gradient (PFG) NMR experiments. Thus, 1 H-, 7 Li- and 19 F-PFG NMR experiments were performed to obtain the diffusion coefficients for the [BMIM]+ (1 H), BF 4 − (19 F), Li+ (7 Li), and

[BMIM]BF4 1 [BMIM] BF4 : 1 LiTf 0.5 LiTf : 1SiO2 : 2 [BMIM]BF4 0.5 LiTFSI : 1SiO2 : 2 [BMIM]BF4

Sample

11 4.5 4.5 6.3

− 1.5 1.3 2.0

− 1.9 2.8 6.3 11 2.5 3.6 4.3

Diffusion coefficients D/(10−12 m2 s−1 ) [BMIM]+ Li+ Tf− / BF4 − TFSI− − − 15 21

− − 1.1 1.7

− − 2.3 5.4

− − 12 14

− − 30 42

Ionic conductivity σ/(10−4 S cm−1 ) Calculated from D for individual ions From impedance [BMIM]+ Li+ Tf− /TFSI− BF4 − spectroscopy

Table 1. Room temperature diffusion coefficients from PFG NMR and ionic conductivities (from impedance spectroscopy for two selected ionogel glasses). The values for the pristine ionic liquid and the ionic liquid/Li salt mixture are included for comparison.

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TFSI (19 F) ions. The resulting diffusion coefficients for the two best performing samples are collected in Table 1. With respect to the pristine ionic liquid, the diffusion coefficients of the ions in the ionic liquid/LiTf mixture decrease by a factor of 2 for the [BMIM] cation and 4 for the BF 4 anion. We note that the Li cation – although being the smallest among the ions – exhibits the lowest diffusion coefficient, indicating substantial anion cation interaction in the mixtures, in accordance with published data for various binary ionic liquid/lithium salt systems [114,115]. Confinement of this mixture within the pore system of the SiO2 glass matrix does not entail a further reduction in the ionic mobilities; the diffusion coefficients are found to be more or less identical to those of the mixture. From the diffusion coefficients we may obtain a rough estimate of the individual contributions of the ions to the overall ionic conductivity using   σ= σ ion ∝ x ion Dion ion

ion

and ignoring any correlation effects such as ion pairing or the formation of larger anion–cation agglomerates (cf . Table 1). Although the ions from the ionic liquid provide the dominant contribution to the total ionic conductivity, the ionic conductivity obtained for the Li cations (1.7 × 10−4 S cm−1 at ambient temperature for the ionogel glass 0.5 LiTFSI : 1 SiO2 : 2 [BMIM]BF 4 ) presents an promising high value for a solid Li ion conductor.

2.5 Crystalline electrolytes As mentioned in the introduction, crystalline electrolytes from the garnet family offer a combination of high mechanical and chemical stability and reasonable ionic conductivity. In Li5 La3 Nb2 O12 , the La3+ cations occupy the cubic A site of the garnet structure; Nb5+ cations are located on the B sites, octahedrally coordinated by six oxygen atoms. A rather vivid discussion emerged about the location of the Li cations within the garnet structure. Since the C sites of the garnet structure can only accommodate 3 Li cations per formula unit, the excess Li has to be accomodated by the six coordinate octahedral (or trigonal prismatic) sites. These are unoccupied in the original garnet structure. Despite a number of X-ray and neutron diffraction studies on powders and single crystals, the distribution of the Li cations among the various possible sites has been controversely discussed. Whereas according to Mazza [116] the C site is fully occupied with the two excess Li cations being accommodated by the octahedral voids, the structure following Hyooma’s results [117] has all Li cations on octahedral positions with the tetrahedral C sites left completely empty. Based on the results of their neutron diffraction study, Cussen et al. [74] rule out both suggestions and put forward a model in which 1/3 of the tetrahedral voids and 2/3 of the octahedral voids are filled with Li cations, translating into 20% of

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Fig. 20. 7 Li NMR linewidth of the central (m = 1/2 → m = −1/2) transition as a function of temperature for Li5 La3 Nb2 O12 . From the onset temperature of the line narrowing, an activation energy E A = 42 kJ mol−1 can be deduced.

Fig. 21. 6 Li-MAS NMR spectrum for Li5 La3 Nb2 O12 : Two different signals can be identified and assigned to Li cations in an octahedral ( + 0.7 ppm) and Li cations in an tetrahedral coordination by oxygen ( − 0.2 ppm).

the Li cations adopting a tetrahedral and 80% an octahedral coordination. Following these authors, predominantely the octahedrally coordinated Li cations contribute to the ionic conductivity. In an attempt to aid in a settlement of this open question and to obtain information about the details of ionic transport within the garnet structure we performed a range of advanced solid state NMR experiment on a Li5 La3 Nb2 O12 sample annealed at 900 ◦ C. For the details of the synthesis, the reader is referred to the literature [98]. The dependence of the line width of the central transition of the static 7 Li NMR spectra, obtained in the temperature range

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Fig. 22. 6 Li-MAS NMR (left) and 6 Li{7 Li}-CPMAS NMR spectra (right) for Li5 La3 Nb2 O12 for the indicated temperatures. The bars are guides to the eye only.

150 K < T < 350 K is shown in Fig. 20. The decrease in the line width indicates the averaging of the homo- and heteronuclear dipolar couplings to 7 Li due to a dynamic process. Following the empirical Waugh–Fedin relation the onset temperature of Li motion (T = 270 K) translates into an activation energy of approx. 42 kJmol−1 . According to these results, most if not all Li ions in Li5 La3 Nb2 O12 are mobile at room temperature. Employing 6 Li-MAS NMR we were able to identify individual Li positions. The 6 Li MAS NMR data is shown in Fig. 21. The two signals at + 0.7 ppm (covering 85% of the total intensity) and − 0.2 ppm (contributing 15% to the total intensity) have to be assigned to Li in octahedral and tetrahedral positions, respectively, since the tetrahedral positions can accommodate a maximum of 60% of the Li cations. Being aware of the fact that this assignment is in conflict with the usual view that an increase in the Li coordination number entails an upfield shift of the corresponding signal we performed 6 Li{7 Li}-REDOR NMR experiments to aid in the assignment of the two signals. The results (data not shown, cf . [97,98]) clearly support the above conclusions. Having identified the distribution of Li cations among the tetrahedral and octahedral sites we then addressed the question whether or not both Li species are mobile. For this, we employed 6 Li{7 Li}-CPMAS NMR spectroscopy as a dynamic filter. Since the magnetization transfer necessary in the cross polarization step relies on the magnetic dipolar coupling between the two nuclei involved, the intensity of a CP signal in principle scales with the inverse of the mobility. Comparing the results of temperature dependent 6 Li-MAS NMR and

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6

Li{7 Li}-CPMAS NMR spectra then allows to separate signals from mobile and immobile Li cations. In Fig. 22, the 6 Li-MAS NMR spectra, exemplarily shown for temperatures of 260 and 350 K, are dominated by the signal of the octahedrally coordinated Li cations irrespective of temperature (as expected, since this experiment is more or less independent of any Li mobility). On the other hand, in the 6 Li{7 Li}-CPMAS NMR spectra, the intensity of this signal is considerably decreased at the higher temperature. These results clearly indicate that the octahedrally coordinated Li cations constitute the (exclusive) dynamic species. Li cations on tetrahedral positions seem to be trapped, not contributing to the Li mobility. Thus, the results are in excellent agreement with the findings of Cussen [74,75]. Employing two-dimensional 6 Li{7 Li}-CPMAS-Exchange NMR spectroscopy we could show that a jump from one octahedral position to another constitutes the basic step of Li diffusion within the garnet structure, with the tetrahedral sites being bypassed [98].

3. Conclusions Recent progress in the synthesis and characterization of novel solid state electrolytes from our laboratory was highlighted. Employing advanced solid state NMR strategies we were able to elucidate the details of the local Li cation coordination in various polymer electrolytes based on PEO, PPZ and PAN, inorganic/organic hybrid electrolytes and a crystalline electrolyte. In (PEO) n : LiTFSI polymer electrolytes, the local Li cation coordination motif was analysed as a function of the salt content. Whereas in samples with n < 6 a co-coordination of the Li cations by polyether oxygen atoms and oxygen atoms from the anions could be identified, an exclusive coordination by the polyether oxygen atoms is observed for n ≥ 6. The mode of operation of ceramic fillers to polymer electrolytes seems to be dependent on the nature of the polymer. A direct interaction between Li cations and the surface of γ -Al2 O3 added as ceramic filler could only be identified for BMEA-PPZ composite electrolytes. For PEO based composite electrolytes we did not observe such an interaction. In these polymer electrolytes, the ceramic additive entails a partial suppression of crystallization of the (PEO) 6 LiTFSI complex. Thus it seems that in the competition for Li cations the Al2 O3 surface can only succeed in composite systems in which the polymer does not supply sufficient and stable coordination sites, as e.g. in BMEA PPZ composite electrolytes. With the salt rich, PAN based polymer electrolytes containing at least 80 wt. % Li salt and the hybrid electrolyte, in which a mixture of ionic liquid and Li salt is confined within the pore system of a SiO2 glass matrix we were able to introduce two novel electrolyte systems with rather promising ionic conductivities and mechanical stability. For the crystalline electrolyte,

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Li5 La3 Nb2 O12 , the distribution of the Li cations among the various possible sites and the migration pathways of the cations could be identified.

Acknowledgement Financial support by the Deutsche Forschungsgemeinschaft (SFB 458) is gratefully acknowledged. T. K. thanks the NRW International Graduate School of Chemistry and the Fonds der Chemischen Industrie for a doctorate stipend.

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Transport Mechanisms of Ions in Graft-Copolymer Based Salt-in-Polymer Electrolytes By Miriam Kunze1 , Alexander Schulz1 , #, Hans-Dieter Wiemhöfer2 , Hellmut Eckert1 , and Monika Schönhoff1 , ∗ 1 2

Institut für Physikalische Chemie, Westfälische Wilhelms-Universität, Corrensstr. 28/30, 48149 Münster, Germany Institut für Anorganische und Analytische Chemie, Westfälische Wilhelms-Universität, Corrensstr. 28/30, 48149 Münster, Germany

(Received August 5, 2010; accepted in revised form September 30, 2010)

Polymer Electrolyte / Nuclear Magnetic Resonance / Spin Relaxation / Pulsed Field Gradient NMR / Conductivity Salt-in-polymer electrolytes based on graft copolymers with oligoether side chains and added LiCF 3 SO3 (LiTf) are investigated concerning the transport and dynamics of the ionic species with respect to applications as Li ion conductors. Polymer architectures are based on polysiloxane or polyphosphazene backbones with one or two side chains per monomer, respectively. NMR methods provide information about molecular dynamics on different length scales: The mechanisms governing local dynamics and long range mass transport are studied on the basis of temperature dependent spin-lattice relaxation rates and pulsed field gradient diffusion measurements for 7 Li, 19 F and 1 H, respectively. The correlation times characterizing local ion dynamics reflect the complexation of the cations by the oligoether side chains of the polymer. 7 Li and 19 F diffusion coefficients and their activation energies are rather similar, suggesting the formation of ion pairs and clusters with similar activation barriers for cation and/or anion long-range transport. Activation energies of local reorientations are generally significantly smaller than activation energies of long range diffusion. Long range transport is affected by (1) the coupling of conformational side chain reorientations to the cation movement, and (2) the correlated diffusion of cations and anions within ion pairs. Ion pairs and their dissociation play a major role in controlling the resulting conductivity of the material. Guidelines for material optimization in terms of a maximized conductivity can thus be derived by identifying a compromise between high ionic mobility and good Li complexation by the coordinating side chains.

* Corresponding author. E-mail: [email protected]

# Current address: Lehrstuhl für Makromolekulare Materialien und Oberflächen, DWI an der RWTH Aachen, Pauwelsstr. 8, 52074 Aachen, Germany

Z. Phys. Chem. 224 (2010) 1771–1793 © by Oldenbourg Wissenschaftsverlag, München

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1. Introduction Polymer electrolytes, consisting of a salt dissolved in a coordinating polymer, are a promising class of solid electrolytes for electrochemical applications. The archetype of polymer electrolytes is polyethylene oxide (PEO) due to the coordinative bonds it provides for small cations. The first systematic study of PEO as an ion conducting salt-in-polymer system was published by Wright and co-workers [1]. Subsequently, Armand et al. explored their electrochemical potential and pointed out the promising application in future lithium ion batteries [2,3]. A “solid” polymer electrolyte offers the advantage to avoid short circuits due to lithium dendrite growth. It thus prevents the leaking and enhances the safety of corresponding batteries [4]. For lithium-ion batteries, conductivities of the electrolyte equal to or higher than 10−3 S/cm at room temperature are required in order to deliver high current densities and efficiencies. However, in polymer electrolytes based on pure salt-in-PEO electrolytes, the conductivities below 60 ◦ C are rather small ranging between 10−6 S/cm and 10−7 S/cm [5]. Employing pure PEO as the polymer, the only parameter apart from the salt/polymer ratio that can be tuned for optimization is the molecular weight: At low molecular weight the electrolyte becomes fluid, whereas at high molecular weight the conductivity is low. To avoid these limitations, many groups have worked on an optimization of the polymer architecture and studied a range of different polymer electrolytes [6,7]. State of the art polymer electrolytes now reach conductivities of nearly 10−4 S cm−1 at room temperature [8]. Some rare examples were reported with even higher conductivities ranging between 10−4 S/cm and 10−3 S/cm [9,10], although such materials most often lack the dimensional stability required for electrolyte membranes. The polymeric component in a salt-in-polymer electrolyte itself needs to fulfil at least three important criteria to act as polymer host: (i) a glass transition temperature Tg below room temperature, (ii) coordination sites at a suitable distance to stabilize the cations, and (iii) low barriers of bond rotation for high internal flexibility [7,11]. Access to such properties is provided by various types of inorganic polymers. Among the novel polymer materials investigated, polymers with an inorganic backbone, in particular polysiloxanes [12–15] and polyphosphazenes [16,17], are considered as very promising. The primary advantages of these inorganic polymers are their highly flexible backbone with enhanced segmental mobility together with a high chemical as well as thermal stability. Another advantage is the ease of modifying their properties by a broad choice of different side groups attached to the silicon or the phosphorus atoms, respectively. In particular the grafting of short flexible oligoether (EO) chains, which act as coordination sites for Li ions is an attractive approach. A widely used polymer family are the polyphosphazenes with poly(bis(methoxyethoxyethoxy))phosphazene (MEEP) as one of the most prominent

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ones. This material was also the first polyphosphazene showing ionic conduction if a lithium salt is dissolved [16]. In the meantime polyphosphazenes have covered more application areas, for example as constituents in ion conducting self-assembled polymeric films [18,19]. The potential of polysiloxanes with dissolved lithium salts to serve as polymer electrolytes was discovered in the 1980-ies by various groups [20–22]. Comb like polysiloxanes with well defined solvating properties have been exploited using hydrosilylation in the presence of platinum catalyst as a means to introduce oligoether side chains [13]. In the last two decades, further investigations on poly- and cyclosiloxane based polymer electrolytes have shown that this is a promising class of polymer electrolytes with good electrochemical stability as well as good ionic conductivities [9,15,23–27]. A further advantage is the ease of chemical modification and the possibility of integrating polysiloxanes in copolymers and hybrid materials [9,15,23–28]. Such graft-copolymers can reach conductivities up to 10−4 S cm−1 at room temperature, which represented a big step from PEO based materials at that time. Nevertheless, it has been only in the last ten years that the number of investigations on these inorganic polymers has increased. The systems investigated here even show a conductivity at 30 ◦ C of up to 1.3 × 10−4 S cm−1 (or 6.9 × 10−5 S cm−1 after cross-linking) [24]. While many authors only use the achieved dc conductivity as a criterion for a successful electrolyte material, more detailed studies have evolved in the meantime, which aim at an understanding of the mechanisms of ion coordination and transport. Detailed characterizations were carried out by methods like infrared spectroscopy [29–31], nuclear magnetic resonance (NMR) [32–40] and viscosimetry [41–44] in order to clarify the interaction of the cations with a polymer network and their associated transport dynamics. This contribution briefly reviews the progress made in our research program, where we combined systematic NMR studies for the characterization of short range and long range mobilities of the cations, anions, and the polymer moieties. We focus on the ion dynamics in polysiloxane and polyphosphazene based model polymer electrolytes with varying oligoether graft chains. Key questions in salt-in-polymer materials concern the relative contributions of cation and anion transport to the effective total conductivity, the extent of correlation between cation and anion transport as well as between ion transport and segmental motion of the polymer and their corresponding temperature dependences [7]. NMR techniques represent an excellent approach to study these questions, since each of the system’s constituents, namely cations, anions and polymer, can be studied in an element-selective way utilizing the 7 Li, 19 F and 1 H isotopes. In the present contribution, we describe the local mobilities of all the species on the basis of temperature dependent spinlattice relaxation times T1 . The dynamic parameters extracted from these measurements are compared with the respective diffusion coefficients, determined by pulsed field gradient (PFG) NMR as a measure of the long range

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Scheme 1. Structures of polysiloxane and polyphosphazene graft copolymers described in this review, left: PSO5 employed with m = 5 and n ≈ 35, or PSOm with m = 2, 3 or 4 and n ≈ 22, respectively; centre: PSO5-X with n/x = 9 and n ≈ 35; right: MEEP with n ≈ 500.

(μm) transport. This approach enables to correlate short range dynamics and long range transport and provides an improved understanding of transport mechanisms.

2. Materials and methods 2.1 Materials Two different types of graft copolymers were employed as the basis of the polymer electrolytes, i.e. oligoether-grafted polysiloxanes and oligoethergrafted polyphosphazenes, respectively. One type of polysiloxanes (CH3 RSiO)n and the cross-linkable ([R(CH3 )SiO]n−x [R (CH3 )SiO]n ) (see Scheme 1) was synthesized from poly(methylhydrosiloxane) (PMHS, Mn = 1700–3200, Aldrich) with R = −CH2 CH2 CH2 O(CH2 CH2 O)4 CH3 and R = −CH2 CH2 CH2 Si(OCH3 )3 via Pt catalyzed hydrosilylation according to a synthetic procedure described earlier [24]. This yielded the structures PSO5 and PSO5-X in Scheme 1. The polymerisation degree n was between 28 and 53 with an average around 35. For the cross-linked salt-in-polymer samples, n/x was chosen as 9 as a good compromise between good conductivity and mechanical stability of the resulting polymer electrolyte membranes. To investigate the effect of the side chain length on ion transport poly(((2methoxyethoxy)propyl) methyl siloxane) (PSO2), poly(((2-(2-methoxyethoxy)ethoxy)propyl) methyl siloxane) (PSO3) and poly(methyl((2-(2-(2-methoxyethoxy)ethoxy)ethoxy)propyl)-siloxane) (PSO4) were synthesized according to the same procedure as PSO5, see Scheme 1, left structure with m  = 5 [24,45]. The polyphosphazene poly(bis(2-(2-methoxyethoxyethoxy))phosphazene) (MEEP) (see Scheme 1) was synthesized by living cationic polymerization.

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The synthesis starts from lithium bis(trimethylsilyl)amide (97%, Aldrich), phosphorus trichloride (PCl3 , 99%, Merck) and sulfuryl chloride (SO 2 Cl2 , 98%, Merck), yielding the monomer trichloro(trimethylsilyl)phosphoranimine. With phosphorus pentachloride (PCl 5 ) the living cationic polymerization was initiated and with 2-(2-methoxyethoxy)-ethanol (MEE, 99%, Aldrich) poly(bis(2-(2-methoxy-ethoxy)ethoxy)phosphazene) was obtained. The reactions were done according to a previously described procedure [46,47]. The chain length n was controlled by the concentration of PCl5 (chain growth initiator). Average values were typically chosen between 400 and 600. The polydispersity index (PDI = weight average molecular weight divided by number average molecular weight) was between 1.4 and 1.6, as determined by gel permeation chromatography. Lithium triflate, LiSO 3 CF 3 (LiTf), was obtained from Sigma Aldrich (battery grade) and tetrahydrofurane (THF) from Sigma Aldrich, purified and dried via distillation.

2.2 Polymer electrolyte samples Salt-in-polymer electrolyte samples were prepared by addition of LiTf to the respective polymer by mixing in solution: Both components were first separately dissolved in THF, appropriate volumes of the solutions mixed and stirred for one hour. The solvent was completely evaporated by placing the sample in a vacuum at 40 ◦ C for three days. The salt concentrations are described by the ratio of oxygens to Li in the sample, abbreviated as O : Li ratio, in order to facilitate the comparison of different materials. To mixtures with cross-linkable polysiloxane a small amount of concentrated hydrochloric acid was added as a cross-linking catalyst. The solvent was removed by drying the corresponding sample in a vacuum for a couple of hours and afterwards in a drying oven for two days at 40 ◦ C at atmospheric pressure for cross-linking to take place, and finally in a vacuum ( p = 10−3 bar) for at least two days. In the following, the non-crosslinked and cross-linked salt-inpolysiloxane systems are abbreviated as PSO and PSO-X, respectively.

2.3 NMR experiments All NMR measurements were carried out on a Bruker 400 MHz Avance NMR spectrometer with a liquid state probe head (Bruker, Diff 30), which provided a maximum gradient strength of 11.8 T m−1 for diffusion measurements. Selective RF inserts optimized for the 1 H, 19 F and 7 Li Larmor frequencies, respectively, were employed. The temperature was calibrated by a GMH 3710 controller with a Pt100 thermocouple (Greisinger electronics, Germany) and is controlled with a precision of 0.25 K. Samples for NMR measurements were placed in 5 mm glass NMR tubes (Wilmad), which were flushed with nitrogen and finally flame-sealed.

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Spin-lattice relaxation times T1 of the cation (7 Li), the anion (19 F), and the polymer (1 H, in the oligo-ether side chain) were measured with the inversion recovery sequence as a function of temperature as described earlier [48]. Pulsed-field-gradient NMR (PFG-NMR) was used to measure the selfdiffusion coefficients of 7 Li, 19 F, and 1 H as described previously [48]. The decay of the signal in dependence on the gradient strength g resulted in the diffusion coefficient by fitting the exponential decay function    δ (1) I(g) = I 0 exp −γ 2 g2 δ2 D Δ − 3 where γ is the gyromagnetic ratio, δ is the gradient pulse length and Δ is the observation time. Diffusion echo decays were well described by a single exponential fit, resulting in the diffusion coefficient D. Accordingly, using Eq. (1) for the signals of the respective nucleus, we determined the diffusion coefficients of lithium D Li (this contains diffusion information about free lithium ions as well as Li in ion pairs or other clusters), of the protons in the oligoether side chains of the polymer, D H , and of the fluorine-containing anions, D F , (containing diffusion of single anions and any pairs or clusters).

2.4 Information about local motions from spin-lattice relaxation rates The local molecular motions are generally described by the spectral density function J(ω), which is the Fourier transform of the correlation function. The spin lattice relaxation rate R 1 is given by R 1 = A (J(ω0 ) + 4J(2ω0 )) ,

(2)

where J(ω0 ) and J(2ω0 ) are the spectral densities at the Larmor frequency ω 0 and at 2ω0 . Several models and methods can be employed to derive information from experimentally determined relaxation rates. Because the local dynamics are generally thermally activated with correlation times τ c typically following Arrhenius behaviour, τ c = τ c 0 exp (E 1 /RT ) ,

(3)

a plot of R 1 in dependence on temperature results in a maximum, where it is ω0 τ c ≈ 1. In a model-free approach, the activation energy E 1 can be directly determined from the slope of the high-temperature branch of relaxation rates in dependence on inverse temperature [49] employing a fitting function   E1 . (4) R 1 = R 1,0 exp RT

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In cases where the temperature range available for experiments indeed yielded a linear dependence of the logarithm of the spin-lattice relaxation rate on inverse temperature, activation energies E 1 of the local motions were extracted from linear fits according to Eq. (4). On the other hand, with the assumption of motional models, the spectral density function can be analytically expressed, and model parameters determined from a comparison of Eq. (2) with the experimental data. The simplest such model is the assumption of isotropic rotational diffusion, yielding J(ω) =

τc , 1 + ω2 τ c2

(5)

which is termed BPP model according to Bloembergen, Purcell and Pound [50]. The constant A in Eq. (2) depends on the type of relaxation mechanism, for example electric field gradient fluctuations in quadrupolar interaction in the case of 7 Li [51,52], or fluctuating magnetic dipole–dipole interactions, in the case of 1 H and 19 F, where A is an additive function of the internuclear dipole– dipole distances [53]. Equations (2), (3) and (5) predict a symmetric R 1 (1/T ) curve with a maximum at the temperature where ω 0 τ c,m = 0.616. Using standard least-squares fitting methods the prefactors A and τ c 0 and the local activation energy E 1 can be extracted from such data. Systematic deviations of the experimental data from such predicted R 1 (1/T ) curves, in particular the observation of asymmetric maxima in the temperature dependence are frequently observed and attributed either to nonexponential autocorrelation functions or dynamic heterogeneities (i.e. distributions of correlation times). To account for such deviations the spectral density suggested by Cole and Davidson (CD) [54–56] has been frequently used in place of Eq. (5)    2 sin β arctan ω0 τ c,CD

 , (6) JCD(ω) = β/2  ω0 1 + ω2 τ 2 0 c,CD

where the parameter β (0 ≤ β ≤ 1) characterizes the width of the correlation time distribution. In the systems studied here, we showed that the BPP model can be applied to PSO and PSO-X for 19 F and 1 H, whereas for 7 Li it is not valid and data had to be evaluated with the Cole-Davidson model [48]. In the case of polyphospazenes, activation energies were extracted from the data using the model-free approach of Eq. (4) [57]. The errors of the diffusion measurements can be estimated to 2% for fluorine diffusion and 3% for lithium and proton diffusion. Errors of the relaxation rates are 3%, and for the activation energy E 1 and for τ c 0 an error of −5% is estimated.

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Fig. 1. Conductivities in dependence of salt concentration, described by the O : Li ratio, for PSO5 with LiTf at 303 K. Data are extracted from [59].

3. Results, data analysis and interpretation 3.1 Conductivity For the application in batteries the most prominent property is the ionic conductivity. For PSO5, it is given in Fig. 1 in dependence on the concentration of added LiTf salt [58]. The behaviour shown here is typical for the salt concentration dependence and can be explained bearing in mind that the conductivity is the sum over all mobile charge carriers summing up the product of their mobilities μ i , charges q i and number densities Nv,i : Nv,i μ i q i . (7) σ dc = i

At low salt concentrations, the increase of the number density of dissociated salt ions, which predicts a linear increase, leads to an increasing conductivity. This increase is, however, not linear at higher salt concentrations, the conductivity even goes through a maximum and decreases above a O : Li ratio of about 8. Two effects can contribute to this: A decrease of the mobilities with increasing salt content, which overcompensates the increase of the ionic number densities, and possibly a decrease of the degree of dissociation with increasing salt content, such that the increase of the charge carrier density is much weaker than that of the total salt content. Viscosity experiments (not shown here) show that the overall fluidity of the polymer electrolyte monotonically decreases with salt content [58]. The salt may have a temporary crosslinking effect here, since the cations mediate an attractive coordination interaction between the chains [48] (see Sect. 4.3).

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Fig. 2. Self-diffusion coefficients of 7 Li (squares), 19 F (circles) and 1 H (triangles) in a) the non-crosslinked system and b) the cross-linked system with a salt content of 5 wt. % LiTf, corresponding to an O : Li ratio of 34 : 1 (filled symbols), and 20 wt. % LiTf, corresponding to an O : Li ratio of 9 : 1 (open symbols) [48].

3.2 Ionic self-diffusion Detailed information about long range transport of the different molecular constituents can be obtained from self-diffusion coefficients. Diffusion coefficients were determined for a variety of conditions involving the polymer materials shown in Scheme 1 with varying concentrations of LiTf and at varying temperature. Figure 2 shows an example of the results, in this case PSO (Fig. 2a) and PSO-X (Fig. 2b) is presented at two different salt concentrations, i.e. O : Li = 34 : 1 (filled symbols) and O : Li = 9 : 1 (open symbols), respectively [48]. The temperature dependence of diffusion coefficients is generally of Arrhenius type. We note that PFG experiments are restricted to a limited temperature range, and there might be deviations from an Arrhenius law occurring over a larger range. Nevertheless, in the investigated temperature range the transport over μm length scales is characterized by a defined activation energy, which will be discussed further below. There are significant differences between the diffusion coefficients for the different nuclei: While the values measured for lithium and fluorine are always very similar, the chain diffusion coefficients, as monitored by the oligoether proton signal are always about one order of magnitude lower. The similarity of the lithium and fluorine diffusion is due to the fact that a large fraction of LiTf exists in form of ion pairs, which undergo a correlated motion. This is further discussed in Sect. 4.1. Another general feature found in all systems is the fact that the anion diffusion is somewhat faster than the Li diffusion, another conse-

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quence of the Li being coordinated to the chain, while the anions have no strong interactions to polymer moieties. While a chain diffusion slower than the ionic diffusion is expected, it is surprising that this difference amounts to only about one order of magnitude. This can be seen as a first indication of a strong coordinative interaction, which correlates the ionic motion to the chain motion and makes the former much slower than expected for the diffusion of small spherical objects in a non-interacting viscous matrix. The addition of salt can have different effects on the diffusion in crosslinked or non-crosslinked systems, as further discussed in Sect. 4.3. The crosslinking generally decreases all diffusion coefficients by a factor of up to 10. In the case of protons it is surprising that this decrease is not more pronounced, since a much larger difference would be expected between the diffusion of single chains and that of chains in a fully cross-linked network. The cross-linking might be not complete, and single non-crosslinked chains might lead to the observation of an unexpectedly large chain diffusion coefficient even in the network.

3.3 Local ion dynamics The long range motions, as monitored by dc conductivity data or by PFG NMR, are describing the molecular transport over the length scale of micrometers. A more detailed understanding of the interactions governing the molecular mechanisms of such long range transport can be gained by analysing the local dynamics of each species, again in a multinuclear NMR approach. Here, relaxation rates of the three nuclei 19 F, 1 H and 7 Li yield information about the local dynamics at the position of the nucleus acting as the label. Figure 3 shows relaxation rates of the three nuclei in dependence of inverse temperature for different systems [48]. In the PSO polymer electrolyte systems, in all cases there is a maximum which can be well determined. From this position, the prefactor A in Eq. (2) can be quantified and detailed temperature dependent correlation times can be extracted from these data on the basis of dynamic models: While for 19 F the BPP model can be applied, 7 Li data had to be evaluated by the Cole–Davidson model, both described in Eqs. (5) and (6), respectively. The exponent β CD in the latter model was generally about 0.8. We note that in other materials such as MEEP or PSOm with n = 22, where no R 1 maximum could be detected, the high temperature branch of T1 was directly evaluated by an Arrhenius activation behaviour (data not shown) [57,60]. Figure 4 shows a clear distinction of the correlation times extracted from these data for the different nuclei. τ c values of 7 Li are always larger than those of the protons, and generally up to one order of magnitude larger compared to 19 F. The fast dynamics of 19 F can be attributed to the free rotation of the CF 3 group. In comparison to similar polymer electrolytes, in the graft-copolymer system investigated here, the correlation times are about 1–2 orders of magni-

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Fig. 3. Relaxation rates R 1 of 7 Li (left), 19 F (centre) and 1 H (right) for LiTf in polymer electrolytes based on PSO (squares) and PSO-X (circles). Filled symbols: O : Li = 34 : 1; open symbols: O : Li = 9 : 1. The solid and dashed curves are fits based on BPP theory in the case of 19 F and 1 H, and fits based on the Cole–Davidson model in the case of 7 Li.

Fig. 4. Arrhenius plot of the correlation times τc of a) the PSO5 system, and b) the PSO5-X system, extracted from T1 of 7 Li (squares), 19 F (circles), and 1 H (triangles). Filled symbols: O : Li = 34 : 1; open symbols: O : Li = 9 : 1. The lines represent linear fits.

tude smaller than in a blockcopolymer PEO-b-PPO with LiN(SO2 CF3 )2 [33]. The increased mobility in the present system is probably a direct consequence of the short oligoether chains, and shows the success of graft chains in providing a larger ion mobility. In all cases, also for the salt concentrations not presented in Fig. 4, the temperature dependence of the correlation time is

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Fig. 5. Activation energies for the dynamics of different nuclei of PSO5 (filled symbols) and PSO5-X (open symbols) in dependence on the oxygen to Li ratio. Squares: E 1 of local motions (T1 ), triangles: E A of long range motion (diffusion). Lines are guides to the eye.

very well described by a single activated process. The activation energies do not show a major dependence on the salt concentration and are presented in Fig. 5. The fact that the proton local dynamics is fast compared to that of Li can be rationalized as follows: if Li is coordinated to 5 oxygens in the side chains, which Maitra and Heuer [61] show by computer simulations, then the local Li dynamics depend on the statistics of binding and unbinding to all these oxygens. In contrast, for each single ethylene oxide group and the local dynamics of its protons, the binding and unbinding statistics of just one single O−Li coordinative bond is relevant. As a matter of fact, the activation energies (see Fig. 5) of the proton local reorientations do not even depend on salt content, i.e. on the extent to which the sidechains are involved in lithium ion binding. This result suggests that – at least in the range of O : Li ratios of 341 : 1 to 9 : 1 studied here – the lithium ion binding makes a negligible contribution to the activation energy governing the local motions of the protons in the sidechains.

3.4 Activation energies of local and long range motion In order to compare long range and short range mobility with each other and to test whether motions on the short range directly induce long range transport, it is interesting to analyse activation energies. For the data presented as examples in Sects. 3.2 and 3.3 they can be directly derived from the slope in Arrhenius plots of the local motion correlation time τ c (Fig. 4) and of the self-

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diffusion coefficients D (Fig. 2). Figure 5 summarizes the activation energies of the PSO5-LiTf system. Activation energies show a clear separation: While the activation of the local motion of all nuclei requires less than 20 kJ/mol, all long range activation energies are clearly above that. For the local motions, activation energies in our graft copolymer architectures can be compared to simple linear chain architectures: In a PEO-PPO electrolyte system with LiTFSI (LiN(CF3 SO2 )2 ) [53] and in an electrolyte system consisting of PEG with LiTf [62], the activation energies of local motions for 7 Li were higher than in both systems presented here, i.e. PSO5 and PSO5X. The activation of the lithium motion in our graft-copolymer electrolytes is thus facilitated. The fluorine activation energies for PEO-PPO and our graft chain electrolytes, however, are in the same range [53], indicating that the large anion is not influenced by the structure of the polymer. This finding is not unexpected considering that the anion is coordinated to the cation and not directly to the chain.

4. Discussion of factors influencing ion transport 4.1 Pair formation The degree of dissociation of the Li salt involved is a parameter of high relevance for the achievable conductivity, since the fraction of dissociated salt determines the density of charge carriers and thus the conductivity. A simple qualitative estimate of dissociation vs. pair formation is obtained from a comparison of the diffusion coefficients of the salt species Li and Tf to the diffusion coefficients D σ of the charged species as calculated from the Nernst–Einstein equation Dσ =

σ DC k B T . e2 Nv

(8)

Here e is the elementary charge, k B the Boltzmann constant and Nv the number density of the charge carriers. Knowing the number of dissociated ions one can calculate the diffusion coefficients of the charged species D σ from the conductivity. In our case the degree of dissociation, α, is not known, thus, for the calculation of D σ , α is assumed to be 1 and Nv is the total ion concentration. Thus, if the salt ions are not fully dissociated, the effective ion concentration Nv∗ is Nv∗ < Nv , and D σ calculated from Eq. (8) is reduced. The comparison is shown in Fig. 6, where the diffusion coefficients D Li and D F obtained from PFG-NMR are compared to D σ as calculated from corresponding conductivity data [24,59]. Information about pair formation is given by the fact that both D F and D Li represent a weighted average of the respective ion in a salt molecule and in the dissociated state. With p being the fraction

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Fig. 6. Diffusion coefficients in PSO5 with the salt concentrations a) O : Li = 34 : 1 and b) O : Li = 9 : 1. Squares: lithium diffusion; circles: fluorine diffusion; down triangles: polymer diffusion all taken from [48]; up triangles: diffusion coefficients Dσ calculated from conductivities [58] using Eq. (8). The lines for lithium, fluorine, and the protons are linear fits, while the lines for Dσ are guides to the eye.

of pairs, D p the diffusion coefficient of pairs and D Li+ and D Tf − the diffusion coefficients of the ions, it is D Li = (1 − p)D Li+ + pD p , D F = (1 − p)D Tf − + pD p .

(9)

These equations present averages of quantities determined by NMR in the limit of fast exchange between two sites, which is valid here. In addition, it is 1 D σ = (1 − p) (D Li+ + D Tf − ) 2

(10)

as an average diffusion coefficient over both charge carriers, reduced by a factor (1 − p) due to pair formation as described above. The extent of pair formation can be estimated from the deviation of D σ from D Li and D F , respectively. In the semi-logarithmic plot of Fig. 6, the difference between D Li and D F and D σ is rather large, and an estimate shows it corresponds to a pair fraction of at least 90%, which is increasing with decreasing temperature. In the system with the higher salt content, Fig. 6b, the pair fraction seems to be somewhat larger. This trend is reasonable, since with increasing salt content the number of available oligoether sites per Li ion for coordination is decreasing. The temperature dependence reflects the fact that the entropic contribution to the dissociation equilibrium causes the pair form to be favored, because it is less strongly bound to the oligoether chains. With the above set of equations (8 to 10) one could attempt to quantify the pair fraction and the unknown diffusion coefficients of the species Li+ , Tf− and pairs. However, this renders four unknowns, while having only three equations. Stolwijk et al. have treated this problem by making several assump-

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tions about the activation enthalpies of dissociation and diffusion of different species, and thus exploiting the temperature dependence for a quantitative analysis of pair fractions [63,64]. This approach worked for radioactive tracer diffusion data of salt-in-polymer electrolytes of linear chains such as PEO and random copolymers. Even for a system with additional foreign ions, combining PFG-NMR and tracer diffusion experiments, such models could be employed and yielded estimates of pair fractions [65]. For the present graft copolymer systems, however, the temperature range over which diffusion coefficients could be determined was too small such that in view of the errors, such assumptions do not yield reliable data. One reason for this difference is probably the fact that the pair fractions in graft copolymer electrolytes are indeed very large, since the short oligoether graft chains are not flexible enough to provide an optimum coordination, such that the interaction of the cation with the anion remains relevant and leads to correlated motion as a pair. Nevertheless, conductivities of the graft copolymer electrolytes are high, which leads to the conclusion that ionic mobilities are even higher than in classical polyethyleneoxide homopolymer systems – here the reduced coordination of the cation has a positive effect. On the local scale, the correlation of the ion dynamics is much less pronounced. The similar magnitude of all local activation energies derived from Fig. 4, and displayed in Fig. 5, which differ only by a factor of two, shows that these local dynamics involve similar interaction energies of binding and unbinding. Coordination of the cation to the oligoether on the one hand and ion pair formation on the other hand appear to have similar interaction energies. The interplay of these interactions is of fundamental importance for the degree of dissociation of the salt in salt-in-polymer electrolytes, since this controls the conductivities that can be achieved. In summary, it can be stated that ion pairs contribute with a correlated motion to the mass transport. On the other hand, the correlation of their local dynamics is less pronounced, and a strong correlation to the chain dynamics is always present. Ion pairs can thus be envisioned as contact ion pairs, see Scheme 2, where Li is coordinated to the chain but still forms a part of the ion pair, as the anion is loosely bound to the cation with lower activation energy of its local motion, but still strongly correlated to the position of the cation even over long time scales.

4.2 Effect of cross-linking Since for application in Li ion batteries materials with a large mechanical stability are required, chemical cross-linking of the chains offers an easily feasible method to enhance stability in the present graft chain materials as well. This has the drawback, however, that the system becomes more rigid, conductivities are decreased [24] and this is also reflected in the molecular properties: As an effect of cross-linking, all the diffusion coefficients in the polymer electrolyte

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Scheme 2. Sketch of the coordination of Li+ (red circles) to oligoether side chains in a salt-ingraft-copolymer electrolyte. Anions are represented by light blue circles.

system are strongly reduced as compared to the non-crosslinked system. The differences amount to almost an order of magnitude, see Fig. 2a and b [48]. Interestingly, not only the chains, but even the ions are reduced in their diffusivitiy, which is an effect of their coordination, as discussed in the previous section. Activation energies of diffusion of Li and Tf are drastically increased and partly even exceed those of the chains, see Fig. 5. Though cross-links covalently connect the main chains of the polymers, and thus should be mainly effective on a long range scale, there are even effects on the local motions. All correlation times are slightly increased, compare Fig. 4a and b. Activation energies of local motions increase due to crosslinking, see Fig. 5. A complete decoupling of local and long range dynamics – which would be desirable to provide materials with high mechanical stability and high ion mobility – is not feasible this way. The increase of activation energies of local motions is not drastic, but there is a clearly detectable effect due to the restrictions occurring on a longer length scale.

4.3 Effect of salt concentration The amount of salt in the salt-in-polymer electrolyte has an interesting influence on the dynamics. With increasing salt content there is a small increase of the diffusion coefficients measured for all the nuclear species in PSO5. This can be attributed to a plasticizing effect: The dissolved salt leads to an increased separation of the chains from each other and therefore to a larger free volume and an enhanced mobility. The diffusion coefficients are displayed in dependence on the O : Li ratio in Fig. 7, where the increase with salt concentration can be directly observed. A similar increase, at least for 7 Li was also found in MEEP, see Fig. 7b [57].

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Fig. 7. Effect of salt concentration (O : Li ratio) on the diffusion coefficients, a) PSO5, b) MEEP. Squares: 7 Li, circles: 19 F.

In the cross-linked system PSO-X, Fig. 2b, however, the chain diffusion is hardly affected by the salt content and the 7 Li and 19 F diffusion coefficients are reduced instead of increased by more salt. Plasticization does not seem to play a role here, due to the polymeric network structure formed by cross-linking. Possibly the ions or ion pairs are hindering each other and are immobilised between covalently crosslinked chains. As discussed in Sect. 3.1 already, an increasing amount of salt in noncrosslinked electrolytes enhances the viscosity and this can be attributed to ions acting as temporary crosslinks between chains. The effect of this cross-linking on the local dynamics is, however, very different from the effect induced by chemical cross-linking: Due to salt addition the motional correlation times all decrease, whereas they show an increase due to chemical cross-linking, see Fig. 4a. This is explained by the ions only forming temporary cross-links. A reduced local motion correlation time suggests a smaller coordination energy per ion, as a direct effect of the competition of a larger number of ions for coordination sites.

4.4 Effect of oligoether chain length The oligoether chain is important for coordinating the Li ions and stabilizing the ionized form Li+ relative to Li bound in an ion pair. It can be speculated which length of the oligoether chain is optimal. On the one hand too short chains might not lead to sufficiently stable coordinative bonds, on the other hand too long chains would reduce the advantages of graft copolymers over PEO homopolymers. There are hints from other materials on the relevance of the chain length, since it was found that side chains consisting of only 3 ether groups are too short for efficient Li complexation [66,67].

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Fig. 8. Activation energies of local Li dynamics, obtained from direct Arrhenius fits of relaxation rates [60]. Squares: PSO2, circles: PSO3, triangles: PSO4. Lines are guides to the eye.

Here, we compare results from PSOm with the number of ether groups in the side chain m varying from 2 to 4. In these materials, the conductivities are increasing with m when compared at the same oligoether oxygen to Lithium ratio O : Li [60]. Salt concentrations are given here in form of this ratio to provide comparability in terms of the number of available coordination sites, even at varying chain length. The increase of conductivity with m is accompanied by a decrease of the local mobility, as Fig. 8 shows: Here, the activation energies of the local Li dynamics are increasing with m [60]. This implies that in the range from two to four oligoether groups the local coordination becomes stronger, as expected. Since the conductivity is still increasing up to m = 4, this enhanced local coordination apparently does not have a negative effect on conductivity. The explanation for this is that while very short side chains can themselves move only within a limited volume, longer chains have a longer accessible range, and thus chain motion can enhance the Li transport over very small length scales. Another aspect might contribute to a larger conductivity for longer side chains: In this range of rather short oligoether chains enhanced chain lengths still improve the Li coordination and a reduction of the pair fraction can be expected. This trend still continues with an activation energy of about 15 kJ/mol when m is increased to m = 5, as comparison to Fig. 5 shows. On the other hand, comparison of long range transport parameters is not appropriate, since the main chain length and thus the viscosity are different for PSO5 as compared to the PSOm material with m = 2–4 that was studied here.

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Fig. 9. Activation energies of diffusion for 7 Li (black symbols) and 19 F (red symbols) in PSO5 (closed symbols) and MEEP (open symbols).

Another interesting result supporting the above arguments is the strong dependence on salt concentration in Fig. 8. It shows that even at O : Li ratios far above the number of EO groups coordinating a Li ion of 5–6 [61]. there is already a competition of Li ions for coordination sites and the activation energies become strongly dependent on salt concentration. For even longer side chains (m = 5, see Fig. 5), the salt dependence is less pronounced. It seems that especially for short side chains a sufficient number of coordinating oxygens is sterically not available for coordination.

4.5 Comparison of PSO and MEEP architectures Though salt-in-polymer electrolytes based on polysiloxane structures PSO, or on polyphosphazene structures MEEP, respectively, show very similar properties in many respects, there are some differences that arise due to their deviating molecular structures. The main difference is, of course, the structure of the main chain being a siloxane or a phosphazene, and in addition PSO carries a methyl group and thus only one oligoether chain per monomer, see Scheme 1. MEEP carries two oligoether chains per monomer. However, since these only carry two EO groups each, plus an additional oxygen next to the phosphorus in the main chain, the total number of coordinating ethylene oxide groups per monomer can be considered similar in PSO5 as in MEEP. It is thus only the distribution, but not the number of coordination sites that is varied, and we can interpret the differences in this respect. First of all, the discrepancy of D σ vs. D F and D Li is much larger in MEEP, suggesting a larger pair fraction. Furthermore, the diffusion coefficients are smaller in MEEP and their activation energies larger; the latter are compared for MEEP and PSO5 in Fig. 9.

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It thus seems that the distribution of coordinating sites onto more, but shorter graft chains in MEEP hinders coordination, which results in a larger fraction of pairs. Nevertheless the diffusion coefficients are smaller though the local motions are more rapid at a given temperature. It can be speculated that the transport might be hindered by a very dense arrangement of graft chains. An additional aspect is the fact that MEEP carries a N group in the main chain, the free electron pairs of which might equally act as coordination sites for Li. Apparently the distances of the coordination sites and their flexibility are not suitable for an optimized coordination in MEEP. PSO5 thus is the optimum candidate from the choice of materials that we studied in this project.

5. Conclusion In summary, our investigations of several types of oligoether-grafted inorganic polymers show that on the on hand they have indeed interesting properties as salt-in-polymer electrolytes. On the other hand, the mechanistic studies of ion transport and dynamics reveal rather complex interdependencies of the dynamics of ions and chains in systems based on graft-copolymers. An important issue is the salt dissociation. It is remarkable that graft-copolymer systems reach very high conductivities, though they suffer from high pair fractions. In comparison to classical PEO homopolymers, their conductivity is enhanced due to much larger ionic mobility, but this occurs at the cost of lower fractions of dissociated ions. Further optimization of salt-in-polymer materials requires a further tuning of the Li coordination sites, which promote dissociation, but on the other hand too strong coordination causes undesired Li immobilisation. Comparison of local dynamics and long range transport in this work sheds more light on the complex mechanisms, and noteworthily, they are not directly correlated to each other: The example of the chain length dependence, Sect. 4.4, shows nicely that the enhancement of local dynamics does not always lead to an enhancement of the long range transport. Long range transport is rather controlled by the delicate balance of the local interactions, when in particular Li coordination is competing with ion pair formation. Therefore, studies that shed light on the local mechanisms of transport and interactions are indeed very important in the optimization of novel polymer electrolyte materials.

Acknowledgement This work is the result of the research program of project B15 within the collaborative research centre “SFB 458”, funded by the Deutsche Forschungsgemeinschaft. We thank N. Kashkedikar for providing the MEEP polymer, Y. Karatas for providing the PSO and PSO-X polymers and A. Hoffmann-zu Höne, S. Kloss for synthesis of PSO-m with varying side chain length, all within project A2, SFB 548.

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Nanoanalysis and Ion Conductivity of Thin Film Battery Materials By G. Schmitz∗, R. Abouzari, F. Berkemeier, T. Gallasch, G. Greiwe, T. Stockhoff, and F. Wunde Institute of Material Physics, WWU Münster, Germany, Wilhelm-Klemm-Str. 10, 48149 Münster, Germany (Received September 20, 2010; accepted in revised form September 30, 2010)

Thin Films / Battery Electrodes / Microscopical Analysis / Glass Membranes / Ionic Conductivity Thin film ion-conductive materials may represent essential components of future all-solidstate batteries. Besides, they are also of interest from a fundamental point of view. In this article, the deposition of thin films of complex oxides and amorphous glasses is discussed. Methods of local chemical analysis by analytical electron microscopy and atom probe tomography are described and studies of atomic transport in sputtered network glasses are presented. As experimental examples, thin films of Li borate and silicate glasses, LiCoO2 , V2 O5 , and Li 4 Ti 5 O12 are addressed.

1. Introduction Electrochemical devices such as batteries, fuel cells, chemical sensors or electro-chromatic switches, require significant ionic conductivity to achieve high power efficiency or fast response. High atomic mobility and at the same time sufficient mechanic stability are conflicting properties that are hardly combined in a solid material. Therefore, the development of all solid state devices is a delicate matter, especially when they should be used at room temperature. In order to realize them, one may search for distinguished materials of utmost specific conductivity, as they were found in the few cases of so-called superionic conductors. As an alternative strategy however, one may also try minimizing the required transport distances. Modern Li ion batteries follow this concept when they make use of powder type electrodes. Active materials and conductivity additives are combined in a heterogeneous short-scaled mixture thereby reducing the length of slow transport paths to the necessary minimum. * Corresponding author. E-mail: [email protected] Z. Phys. Chem. 224 (2010) 1795–1829 © by Oldenbourg Wissenschaftsverlag, München

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In order to separate the two half cells of a secondary battery, an ionconductive membrane is required. In commercial batteries usually some liquid or at least a polymer with liquid-like behavior is used to achieve sufficient mobility. However, one may apply here also the concept of reducing length scales. Reducing solid membranes down to a few tens of nanometers in thickness, the required absolute conductivity may be achieved even with materials of low specific conductivity. To realize the concept of reduced length scales and also to study respective chemical and physical properties from a fundamental point of view, nanometric ionic devices must be produced in a controlled manner. This is most easily achieved in multilayer architectures. Clearly defined planar interfaces allow the characterization of reactions during charging and de-charging in greater detail than it is possible in conventional particle geometries. Therefore, the growth of thin films has attracted considerable attention recently, see e.g. [1–5]. Obtaining correct lattice structures in deposited thin films and maintaining the desired stoichiometry is usually a delicate problem. This article describes deposition of relevant materials by ion-beam sputter deposition. Suitable sputter conditions to grow complex oxides are identified. With electron energy loss spectroscopy and atom probe tomography, we present two distinguished means of microscopic chemical analysis which will in future allow investigating electrochemical functions even with subnanometer resolution. Finally, we will discuss the conductivity of thin glassy membranes. In the case of crystalline materials, un-isotropic growth usually leads to considerable roughness of interfaces so that thin membranes fragment easily. Furthermore, heterogeneous ionic transport along grain boundaries may destroy planar layer structures after short time of operation only. In order to avoid these difficulties, amorphous materials represent suggested alternatives. In this work, we focus on borate and silicate network glasses which obtain significant conductivity for alkali ions by suitable doping. Although this article partly reviews work that was published elsewhere, it does not represent a review in its classical sense. Selection and presentation of topics follow strictly the scientific interest of the authors. Certainly, the present article cannot provide a complete overview on all relevant work that has been published in literature by our distinguished colleagues.

2. Thin film deposition Controlled deposition of thin films can be achieved by various techniques. While chemical vapor deposition requires a dedicated chemistry matched to a specific material, the advantage of physical deposition is its general applicability to a broad range of different materials. Laser ablation and sputtering are perhaps the most popular techniques. Almost every material can be ablated or sputter-eroded by laser beams or energetic particles. However, it is by

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Fig. 1. Schematic view of an ion beam sputtering chamber.

no means natural to achieve the right stoichiometry and atomic structure in deposited films. Also preparation of targets may become a delicate matter in the case of materials with high melting point or very brittle materials. In this work therefore, we explore deposition of thin films by a particular variant of sputtering, the so-called ion beam sputtering technique. This method offers certain advantages, if experimental thin films should be produced in a versatile manner. For the reported thin film growth, a custom made deposition chamber was designed and assembled. The experimental setup is illustrated by the sketch in Fig. 1. A beam of energetic particles, argon or a mixture of both, argon and oxygen, is produced in an external gun (Rau&Roth, RF source, 4 cm ∅). A plasma at the exit of the gun decharges the accelerated ions so that targets are hit by neutral particles. Thus, correctly the method should be termed “neutral particle beam sputter deposition”. Several targets can be mounted on a water cooled revolver so that switching between different layer materials is possible just by rotating the target head without breaking the vacuum. Substrates are mounted on a rotatable water-cooled sample stage. Controlled deposition temperatures up to 700 ◦ C are achieved by an UHV suited ceramic heating plate. The geometry of the sample stage allows turning the substrates in front of the gun to become directly exposed to the beam. In this way, cleaning surfaces just before deposition is easily possible. Growth rates are measured in-situ by a quartz balance attached close to the substrate. In addition to the oxygen fraction in the energetic beam, the vacuum chamber may be flood by other process gases. However, the background pressure during the process must be always limited to a few times 10−4 mbar to warrant a reliable function of the ion gun. In comparison to conventional magnetron sputtering, the important advantage of generating a beam in a separate gun chamber lies in the fact that targets and substrates can be held in field-free space and are hit by neutral particles. In this way, any potential charging of target and substrates is reliably prevented,

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Fig. 2. Sputter targets: (a) transparent glass disk of (Li2 O)0.2 -(B2 O3 )0.8 glued to a Cu support, (b) sintered powder LiCoO2 target in stainless steel pan. (diameter of disks amounts to 8 cm). Detail (top) shows required pressing molt.

which may become quite important, if thin structures of materials with low electronic conductivity are deposited on complex substrate geometries. For versatile experimental studies, a flexible supply of targets is a prerequisite. The ion beam sputtering method needs only rather small and thin targets so that also experimental materials can be processed, of which only a small amount is available. Circular disks of 8 cm in diameter are required for the setup shown in Fig. 1. In the case of metals, these disks were just cut from suitable foils. Network glasses were produced by melting appropriate mixtures of alkalicarbonates and silicates or borates. The melt was poured in an appropriate crucible to obtain transparent disks of about 4 mm in thickness, see Fig. 2a. This technique fails, if brittle and complex oxides, such as LiCoO2 or Li4 Ti5 O12 should be processed. For the latter, a pan-shaped container made of stainless steel was developed in which powders of the required average composition are filled and compacted under a pressure of 150 MPa, followed by a sinter heat treatment (see Fig. 2b). It should be noted that the target does not need to present the material in its final structure desired in the growing layers. In extreme cases, even heterogeneous powder mixtures of raw materials can be used which only offer the appropriate total concentration of components. Lattice structure and final composition are determined during growth by choosing correct growth rate, temperature and gas atmosphere. With the described experimental setup, complex multilayers can be produced. Even experiments supposed to be rather simple, such as conductivity

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Fig. 3. Cross section micrograph of a layer stack prepared by ion beam sputter deposition to measure ionic conductivity of a thin (15 nm) Li borate glass layer.

measurement of a single glass layer, may require a complicated layer sequence for practical reasons. For example, a layer structure is shown in Fig. 3, which allows measurement of ionic conductivity of thin borate glass membranes down to 7 nm thickness in current perpendicular to plane (CPP) geometry [6,7]. The membrane is sandwiched between two metallic electrodes, however in a complex manner: In order to avoid interface roughness which induce electrical short circuits, polished Si substrates are first coated with a very thin electrode (10 nm) made of an Al(Li) alloy. Thicker electrodes of crystalline metals had induced too much roughness. On the other hand, a reliable contact to spring loaded pins must be warranted at the surface side. The electrode must provide chemical and mechanical stability so that the thin glassy membranes are not damaged by the spring load. Therefore, a thicker Al(Li) alloy (100 nm) is deposited, followed by a Ta diffusion barrier (20 nm) and a Au top layer (50 nm) preventing the device from surface oxidation. With the described deposition technique, thin films of the cathode materials LiCoO2 and V2 O5 as well as the anode material Li4 Ti5 O12 were produced. Obtaining correct structure and stoichiometry is a delicate matter. Therefore, any new material requires a series of preliminary calibration experiments in which

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Fig. 4. XRD of LiCoO2 layers sputter-deposited under a 3 : 2 mixture of argon : oxygen at different temperatures. The bottom spectrum belongs to the powder material of the target. Inset demonstrates peak splitting into the (006) and (012) reflections which appears for temperatures of at least 500 ◦ C, if oxygen is present in the sputter atmosphere.

optimum sputter conditions are determined so that either bulk properties of the materials are matched as good as possible or desired non-equilibrium structures are formed. Already the correct chemical composition is a difficult problem, since atomic species may differ in their sputter rates or are potentially lost as gaseous component. If required, the precursory composition of the target must differ from the desired final composition to compensate for potential losses. In deposition of oxides, controlling the correct oxidation state is most important. To this aim, the layers produced in this work were first characterized by standard methods, such as X-ray diffractometry (XRD), cyclo-voltammetry (at constant voltage rate) and chrono-potentiometry (at constant current) to warrant correct structures and check electrochemical properties. An exemplary study of LiCoO2 films by XRD is shown in Fig. 4 [11]. Diffractograms of layers achieved in an Ar : O 2 ratio of 3 : 2 at different substrate temperatures are compared to the calibration spectrum of commercial powder material (bottom). Besides the ideal rhombohedrical equilibrium structure, also a metastable low temperature modification is known which however reveals a significantly lower charge capacity [8].

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Fig. 5. Cyclic voltammetry of a V2 O5 film, 140 nm in thickness: (a) current vs. voltage against Li+ /Li in the first cycles. Scan rate was 0.1 mV/s. (b) Development of capacities during 25 cycles.

The desired equilibrium phase is distinguished by a pronounced (003) peak and a reflex splitting between e.g. the (006) and (012) peaks which appears only for the rhombohedrical phase. Comparing the spectra in Fig. 4, it becomes obvious that a deposition temperature of at least 500 ◦ C is required to form this optimum structure of LiCoO2 [11].

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Fig. 6. Cyclo-voltammogram (a) and chrono-potentiometry (b) of sputter-deposited Li4 Ti5 O12 layers (deposition conditions see Table 1).

Electro-chemical function tests are illustrated by the cyclo-voltagramm of a V2 O5 layer shown in Fig. 5. For this measurement, thin films were sputterdeposited under the identified optimum conditions onto ITO covered glass slides that provide the required contact. Li wires served as counter and reference electrodes. The liquid electrolyte consisted of EC/DMC (1 : 1) doped with 1 M LiClO4 . A scan rate of 0.1 mV s−1 was chosen to investigate the potential range between 1.5 and 4 V. Voltammograms during the first few charging cycles are shown in Fig. 5a. A negative current corresponds to Li insertion into the thin film. Integrating the charge on respective half cycles, capacities are determined as shown in Fig. 6b. The initial charge capacity (Li insertion into V2 O5 ) amounts to 66 μA h cm−2 μm−1 . A significant decrease is noticed during the first few cycles. After 25 cycles the capacity has decreased to 59 μA h cm−2 μm−1 . However, a plateau value of 89% of the initial charge cap-

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Table 1. Optimum deposition conditions for ion beam sputter-deposition and comparison of achieved and theoretically expected capacities. Material

LiCoO 2 Li4 Ti5O 12 V2 O5 Borate glasses Silicate glasses

Beam Beam Growth voltage current rate [V] [mA] [nm/min]

O 2 during Substrate sputtering temp. [10−2 Pa] [◦ C]

Post annealing

Capacity [mA h/g]

500 800 600 500

20 20 20 25

1.2 0.9 1.5 2.0

2.4 2.7 0.7 0.0

550 600 R.T. R.T.

– – 250 C, 3 d –

127 131 210 n.a.

500

25

2.0

0.0

R.T.

–

n.a.

◦

Theo. capacity [mA h/g] 140 175 225

acity is reached for intermediate cycle numbers indicating sufficient stability to produce useful batteries. For the sake of completeness electrochemical data of the Li4 Ti5 O12 layers that could be deposited in this work should be also presented. A cyclovoltammogram of these films is shown in Fig. 6a as well as a measurement of voltage during charging and decharging at constant current (chronopotentiometry). Since currents were rather high in measurement of the cyclovoltammogram, the apparent capacity tends to be low. However, even when applying the slower charging of the chrono-potentiometry, we had to learn that the storage capacity of our layers is with 35–40 μA h cm−2 μm−1 still significantly less than the theoretically expected capacity of 60 μA h cm−2 μm−1 . With Li4 Ti5 O12 layers produced by magnetron sputtering, Wang and coworkers [9] have already demonstrated a storage capacity closer to the theoretical maximum. As will be explained in the next section, more sensitive characterization must be applied to detect even minor modifications when deposition conditions are varied. On the basis of all the experimental characterization performed in this work, including the microscopic analysis discussed below, optimum conditions for deposition of the respective materials were determined. The most important parameters are summarized in Table 1 together with the achieved capacity. Further details on the three deposited electrode films can be found in [10,11].

3. Structural and chemical analysis of layer materials A prerequisite for controlled experiments with thin films is a correct chemical analysis of sputter-deposited materials. Beside conventional X-ray diffractometry to check lattice structure and lattice spacings, electron energy loss spectroscopy and energy filtered imaging in a TEM is used to characterize the deposited material in detail. Furthermore, high resolution, 3D chemical

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analysis in atomic sensitivity by atom probe techniques was developed to be applied to battery materials. In the past, atom probe tomography was restricted to metals or at least materials of sufficient electronic conductivity, which excluded many interesting functional materials from being measured. However, recently this situation has markedly changed. The method has been opened to oxides and other ceramics by introduction of laser-assisted evaporation modes. In this chapter, the two analytical techniques are discussed in greater detail.

3.1 Electron energy loss spectroscopy Electron microscopy is a versatile tool to verify the microstructure, determine the thickness of deposited layers, and to characterize interfacial roughness. In the case of multilayers, usually preparation of an electron transparent cross section sample is required. Traditionally, this has been obtained by mechanical grinding and subsequent ion milling. Nowadays many complex process steps can be saved with a lift out procedure performed within a dual beam focused ion beam microscope (FIB). If cross section samples of the layer stack are already prepared, detailed additional chemical information can be obtained by electron energy loss spectroscopy, without any further required preparation step. Any inelastic interaction of the beam electrons with the sample causes energy loss to the primary electron beam which can be measured by an energy dispersive electron optics. If the interaction of the primary beam takes place with core electrons, the energy loss is characteristic for the chemical species that are contained inside the illuminated area. To first approximation, such energy loss spectrum may be interpreted as the reciprocal of an energy dispersive X-ray spectrum. At the energy positions of the characteristic X-ray peak, an absorption edge appears in the energy loss spectrum. An example of a sputter-deposited NaO2 -B2 O3 network glass is presented in Fig. 7. The spectrum has been obtained with a Zeiss Libra 200 FE microscope comprising an in-column energy filter of ΔE = 0.7 eV resolution. Absorption edges of all glass components are clearly seen. In addition, carbon is indicated which stems from surface contamination of the cross section sample. In order to evaluate the edge intensities, background is subtracted locally in a power law approximation as demonstrated at the oxygen edge in Fig. 7. The remaining intensity in a defined energy window (marked as grey area) is compared with theoretical interaction cross sections. For this, a standard algorithm [12] is used as provided in commercial programs [13]. In this way, even a calibration free analysis becomes possible. For sputter-deposition of borate and silicate glasses, targets were obtained by melting alkali carbonates and network formers exactly in the desired final composition. Sputter deposition was performed in pure Ar atmosphere at room temperature. In all cases, in which a direct analysis by EELS was possible, it turned out that the stoichiometry of the target glass is reasonably transferred

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Fig. 7. EELS spectrum of a sputter-deposited Na borate glass layer (Zero loss and low energy region were skipped).

to the deposited thin film. In the example presented in Fig. 7, a composition of (38.2 ± 3.8) at.% boron, (53.3 ± 5.3) at.% oxygen and (8.5 ± 0.9) at.% sodium is determined from the thin film, while the same analysis of the target material yields (34.4 ± 4.2) at.% boron, (58.5 ± 7.1) at.% oxygen and 7.3 ± 0.9) at.% sodium. Thus, agreement within the error bounds of measurement can be stated. It has to be admitted however that the error ranges are quite significant and furthermore, this analysis is not successful with all types of alkali. As a remarkable disappointment, just the most interesting case of Li cannot be treated with sufficient accuracy, since the absorption edge of Li is found at such low energies and its intensity is so low that it can hardly be separated from the background. However, the fine structures of the absorption edges bear additional information that can be utilized, if suitable calibration samples are available. Based on comparison to finger prints of calibration states, a quantitative evaluation becomes possible. In this way, a very sensitive probe is available to determine even minor deviations in composition, oxidation state, or even the character of chemical bounds. We demonstrated this possibility at two important examples, sputter deposited thin films of V2 O5 [10] and LiCoO2 [11]. Interestingly, based on the fine structure of absorption edges, also the Li intercalation state of LiCoO2 films can be quantified, although the absorption edge of Li cannot be evaluated. In the case of vanadium oxide, the interesting absorption edges V L and O K fall into a narrow energy window (see Fig. 8). Therefore, peak overlap hinders the direct quantitative evaluation of absorption intensities. However, spectra obtained from commercial VO2 and V2 O5 powders demonstrate that the rising edge of V L shifts to higher energy loss with increasing oxidation state [10,14] while the position of the O K edge is preserved. Therefore, the difference in energy loss between these two edges provides a reliable parameter to quantify the

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Fig. 8. EELS study at sputter-deposited vanadium oxide layers. Varying sputter conditions results in a shift of the V L edge (a). Comparing the relative shift to O K , the oxidation states of three different deposition conditions are quantified (b) (for further details see [10]).

oxidation state and thus oxygen content of deposited layers. Different sputter conditions were tested in order to optimize V2 O5 layers deposited from powder targets. As optimum conditions, sputtering under a gas mixture of 10−4 mbar Ar and 0.2–1.0 × 10−4 mbar O2 at room temperature followed by a post annealing at 250 ◦ C in ambient atmosphere were identified. In this way, a maximum oxidation state of 4.7 is achieved (see demonstration in Fig. 8b). Obviously, still a slight under-stoichiometry must be accepted. Nevertheless, these thin films fulfill the desired electrochemical function, as we demonstrated already in Fig. 5. The same experimental procedures were also applied to determine the optimum conditions for deposition of thin films of LiCoO2 . In this case, EELS spectra of standardized powders of LiCoO2 , Co3 O4 and CoO were used to derive suitable finger printings. Formally, these compounds represent the Co

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Fig. 9. EELS spectra of LiCoO2 films in comparison to different powder materials as stated: (a) absorption edge of Co L , (b) absorption edge of O K .

oxidation states of 3.0, 2.7, and 2.0, respectively. Again, a chemical shift of the Co L 3 edge is demonstrated (see Fig. 9a), but we found that in this case the O K edge is an even more sensitive tool to characterize the layers, as shown in Fig. 9b. The spacing between O K 1 and O K 2 as well as the relative intensities of these peaks change dramatically with the cobalt to oxygen ratio. Produced under the optimized conditions identified in this work (sputtering under a 3 : 2 mixture of Ar : O 2 and a deposition temperature of 500–600 ◦ C [11]), the EELS

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Table 2. Quantitative evaluation of O K edges in LiCoO2 and de-lithiated thin films. Material

LiCoO2 Li0.9 CoO2 Li0.25 CoO2

Intensity ratio (O K1 /O K2 )

ΔE (O K 2 − O K 1 ) [eV]

1:1 0.93 : 1 0.69 : 1

11.7 11.0 5.9

Fig. 10. EELS of sputter-deposited LiCoO2 films (oxygen absorption edge). Films were delithiated electrochemically to defined states of 0.9 and 0.25 relative Li fraction.

spectra of the thin films agree in all important characteristic features to that of the commercial LiCoO2 reference powder (see Fig. 9b). Depositing thin films on glass slides coated with ITO, battery cathodes were produced. They were inserted in a chemical cell comprising of a graphite anode and the LiCoO2 cathode, both dipped into a solution of LiClO4 in a mixture of DMC/EC. Controlled amounts of Li were withdrawn electrochemically. In Fig. 10, respective EELS spectra of intercalation states containing a remaining Li fraction of 0.9 and 0.25 are compared to that of the ideal LiCoO2 film. Clearly, the spectra vary significantly with the Li content. Based on the spacing ΔE between the O K 1 and O K 2 edges or the relative intensities of these edges (see quantitative data in Table 2), an accurate measure of the intercalation state is established [11]. It should be noted that an intercalation state of 0.9 Li falls into the reversible range of battery function, while de-intercalation to 0.25 Li leads already to irreversible transformation of the oxide. In consequence, the spectrum

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of this extreme de-intercalated state resembles already that of CoO, which indicates a decomposition process, probably accompanied by oxygen losses.

3.2 Atom probe tomography The previous discussion of electron energy loss spectroscopy pointed out that the microscopic analysis of Li is only possible in favorite situations based on finger printing methods. To study battery functions on the atomic scale in a versatile manner, alternative analysis techniques are therefore urgently needed. To this aim, we started to develop atom probe tomography towards the capability of analyzing complex oxides and also layer systems, as they are required for fabrication of thin film batteries. Atom probe tomography (APT) provides an interesting alternative in microscopic analysis [15]. Although it requires considerable experimental effort in sample preparation and instrumentation, the method promises direct threedimensional analytical data with single atom sensitivity and subnanometer resolution. In practical analysis of nanocrystalline metallic thin films, the spatial and chemical resolution of atom probe tomography can exceed that of analytical TEM considerably [16]. The technique is based on the controlled field-desorption of individual atoms or at least small molecules from needleshaped samples with apex radii below 50 nm. Desorbed species are detected by a position-sensitive detector setup and identified by time of flight spectroscopy. Subsequent to measurement, the origins of the atoms at the tip surface are calculated to better than 0.5 nm accuracy. By consecutive field desorption, a complete volume of the sample is analyzed and 3D maps of the atomic arrangement are reconstructed. In the last decade, the method was revolutionized by introducing fast delayline detector systems [17–19] and sample preparation based on focused ion beams (FIB) [20,21]. Nowadays, a reconstructed volume of typically 106 nm3 , comprising about 100 million atoms, can be regularly expected from a successful measurement. In the context of battery materials, the recent extension of atom probe tomography by short pulse laser systems is most remarkable. Traditionally, evaporation of atomic species from the sample was achieved by nanosecond high voltage pulses which restricted the method to metallic materials. In the new, laser-assisted evaporation mode, only a dc base voltage is supplied to the tip, which is slightly too low to initialize field evaporation. The final trigger to evaporate the molecules is provided by a low intensity short laser pulse. This technique has become possible only by the recent availability of reliable laser systems with sufficient repetition frequency. Meanwhile, the successful analysis of oxides and ceramics by atom probe tomography has been reported multiple times, see e.g. [22–26]. A new instrument dedicated to the analysis of complex materials was developed and put into operation at university of Muenster recently [27].

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Fig. 11. TEM micrograph of a coated tungsten substrate tip. The measured volume is indicated by dashed lines. Only the inner part (dotted) is presented by the volume reconstruction in Fig. 11b.

Meanwhile, first measurements of relevant battery materials were tested. For this purpose, needle-shaped substrate tips are produced by electro-polishing tungsten wire and shaped to 20–30 nm curvature radius by appropriate field evaporation in a field ion microscope. Subsequently, these prepared needles are coated with the layer system of interest. Since substrates are kept in field-free space, the described variant of particle beam sputtering is particularly suited to coat these sharp needles. By contrast, uncontrolled growth at the tip apex appears with conventional magnetron sputtering, since the tips function as tiny antennas in the plasma field so that they attract ions by coulomb interaction in an unpredictable way. For illustration, Fig. 11 shows a transmission electron micrograph of a prepared substrate tip coated with a LiCoO2 layer. The typical volume of analysis is marked by dashed lines. Owing to the hemispherical curvature of the substrate, also deposited layers become curved. On the lateral scale of 10–20 nm however, interfaces appear still approximately flat. In Fig. 12a, a mass spectrum of a sputter-deposited LiCoO2 layer including the first part of the W substrate is shown. Li and W are detected as atomic species in one-fold or three-fold charged state, respectively. The natural isotopes are clearly distinguished. By contrast, Co is frequently found as part of oxide molecules. In view of the difficulties of the TEM techniques EELS and EDX with Li, the simple counting of Li with the atom probe is remarkable. Most of the molecular mass peaks in the spectrum can be assigned unambigously to a certain combination of atomic species, so that the atomic fractions of all components can be correctly evaluated from the number of detected events. After assigning atomic species to mass peaks, the tomographic volume reconstruction is calculated as presented in Fig. 12b (only the volume indicated by the dotted line in Fig. 11 is presented in order to reveal the slightly curved interface clearly). The fact that molecular events are detected has important consequences to the accuracy of this volume reconstruction. Molecules have

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Fig. 12. Atom probe analysis of LiCoO2 deposited upon tungsten substrate tip: (a) mass spectrum determined by time of flight spectroscopy, (b) volume reconstruction representing the position of atomic species. Only the inner part (about 10 nm) of the analyzed total volume is plotted. Inside tungsten (110) lattice planes are resolved.

to be split into their components of which individual positions are unknown. This necessarily introduces some uncertainty in the evaluated atom positions on the distance of a lattice constant. Mostly for this reason, it is impossible to resolve individual lattice planes within the oxide layer in contrast to the metallic substrate, where the (110) lattice planes of the bcc structure of tungsten are frequently resolved (see inset). On a coarser scale of about 0.5 nm however, compositions can be evaluated accurately by counting the different species within given small sampling volumes. In this way, composition profiles are determined perpendicular to the layer interface as shown in Fig. 13. As average on the total layer, the composition is measured to be 28.4 at. % Li, 25.8 at. % Co, and 44.2 at. % O in reasonable agreement to the aimed stoichiometry of LiCoO2 , which confirms

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Fig. 13. Composition profiles across a LiCoO2 film of 30 nm thickness as determined by atom probe tomography.

that the peak assignment in the mass spectrum has been probably correct. Nevertheless, a slight under-stoichiometry of oxygen is indicated. It is not clear to which extent oxygen might be lost during the measurement. An accuracy up to a few at.% was proven in other experiments with oxides [23]. However, more significant are the unbalanced fractions of the metallic species Li and Co. In a subsurface layer of 5 nm thickness, the compound appears significantly enriched in Li while the Co content is suppressed. (Standard deviation of individual data points smaller than ±3 at. %.) This tendency of Li to segregate to the surface would be naturally understood by the higher stability of Li oxide in comparison to Co oxide and the availability of excess oxygen at the surface (it is remarkable that the oxygen content just at the surface amounts exactly to 50 at. %). If this selective oxidation of Li became too pronounced, the forming Li2 O layer would presumably hinder fast intercalation of Li+ . Therefore, this observation by atom probe tomography may have significant practical consequences. A similar analysis was performed with a Li-borate glass. In this case, tungsten tips were coated with a layer of about 20 nm thickness, sputtered from a target with a nominal composition of (Li2 O)0.2 (B2 O3 )0.8 . A mass spectrum determined from the layer region is shown in Fig. 14a. In comparison to the previous example of the intercalation compound, the spectrum appears much more complex. Most events must be assigned to larger molecules. Some peaks could not be identified unambiguously. Therefore, the total measurement of composition suffers from an uncertainty in the range of about 5 at. %. Averaged on the complete volume, the composition of the sputter-deposited glass is determined to be 12.1 at. % Li, 37.4 at. % B, and 48.2 at. % O, while the nominal composition of the target requires 8.7 at. % Li, 34.8 at. % B, and 56.5 at. % O, respectively.

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Fig. 14. Atom probe analysis of a 20 nm thick (Li2 O)0.2 -(B2 O3 )0.8 glass layer deposited on a tungsten substrate tip: (a) spectrum of detected masses, (b) composition profile determined across the layer, and (c) sketch illustrating the impact of the evaporation field on mobile ions within the sample.

As in the previous experiment, an understoichiometry of oxygen is stated. On the average furthermore, Li seems to be more abundant than expected. Such increase in Li content would have significant impact on the conductivity of the

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glass. But in view of the complex mass spectrum prone to misinterpretation, we are not sure whether this enrichment of the alkali is indeed significant. The fact that correct sodium composition has been proven by EELS in the case of sodium-borate glasses raises further doubts. While absolute quantities in a calibration free measurement remain a delicate matter, spatial variations in the relative number of detected elements are certainly significant. As before, a clear enrichment of Li is detected close to the surface. But in the case of the glass here, the enrichment of Li is balanced by a joint decrease of the second metal and oxygen as well, in obvious contrast to the previous case of LiCoO2 in which the increase of Li was accompanied by an increase of the oxygen content. Furthermore, the elevated Li content at the surface is counterbalanced by an obvious loss of Li at the opposite side of the glass layer towards the tungsten substrate, as marked by light and dark shaded areas in Fig. 14b. This enrichment of Li at the surface of the layer and the compensating lack of Li at the inner interface was well reproduced in more than 15 independent measurements and was observed furthermore for four different glass compositions (Li2 O)x -(B2 O3 )1−x with x = 0.15, 0.2, 0.25 and 0.3. Therefore, this observation strongly suggests that the glass layer becomes polarized by coherent shift of Li cations relative to the negatively charged network. As sketched in Fig. 14c, a strong field of the order of a few tens of volts per nanometer (!) is supplied to the tip during measurement. Since the glass layer is electronically insulating, a significant fraction of this field penetrates into the glass layer. The polarity is such that positive charges are driven towards the surface. Thus, provided sufficient mobility, Li ions will accumulate at the surface and will be measured predominantly while they are underrepresented in the last part of the layer measured just before reaching the metallic substrate. At present, details of the observed polarisation profiles are not understood. This understanding would require solving the Laplace–Boltzmann equation for the varying thickness during the measurement and taking into account the dynamic balance of evaporation from the sample surface and transport through the remaining layer. At present, the mobility of the ions can hardly be predicted, since the temperature is almost unknown. The base temperature during measurement amounts to 50 K. But the tips are heated for a very short duration (ps) by the laser pulse to about 300 K so that the effective temperature is unclear. Although still preliminary, the two presented examples of successful atom probe measurements demonstrate that important details on the local distribution of the Li ions may be detected by this microscopic technique in future. The direct proof of ionic migration in thin films of a few nanometers in thickness suggests studying the microscopic mechanisms of intercalation and ion transport. We are still just at the beginning of introducing this tool into battery research.

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Fig. 15. Specific conductivities of various material classes.

4. Ion conductivity of sputter-deposited oxide glasses In comparison to conventional secondary batteries, which all contain some ion conducting liquid or gel, all-solid-state batteries would offer the possibility of direct integration in microelectronic devices and would promise furthermore secure operation, as any leakage of a liquid is reliably prevented. Therefore, this kind of solid state devices has been a dream for decades but could not be realized apart from a few prototype models of which reversible function is still a delicate problem [28,29]. In such kind of battery, the conventional liquid ion conductors must be replaced by a solid material of sufficient ionic conductivity. In Fig. 15, a chart of typical specific conductivities of different material classes is presented. Solid ion conductors that are used in fuel cells, batteries, or sensor devices are found in the grey shaded parameter field. None of these can compete in terms of specific conductivity with aqueous solutions. Therefore, it is no surprise that the few superionic conductors, which reveal the highest specific conductivity close to 102 Ω−1 cm−2 , are primarily suggested to be used as membranes in solid state batteries. For practical function however, the absolute conductivity of a membrane is decisive, not the specific material property. Thus, a low specific conductivity may be compensated by small thickness. Applied in a thickness range down to only 10 nm, even borate and silicate glasses may become useful alternatives, although they are, according to the chart in Fig. 15, usually not considered for functional devices. In the following section, we will explore the particular behavior of these glasses, if they are deposited as thin films. The structure of the discussed glasses is primarily determined by the trigonal or tetrahedral building units of borate or silicate, respectively. In order to get the networks conductive, they are doped by alkali-oxides. Each molecule of these so-called network modifiers inserts a negatively charged defect into

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Fig. 16. Sample structure to measure very thin films. Top electrode is formed to a circular disk down to 0.3 mm ∅. At the right, the micro-contacting tool used in this work is shown.

the covalently bonded network and introduces a mobile cation as the vehicle of transport. By varying the alkali-content, the conductivity of bulk glasses (solidified from the melt) can be largely varied as marked by finite ranges in Fig. 15. For studies of conductivity, a variety of targets were produced. The experiments comprised a series of Li-borates with varying alkali content (15, 20, 25, 30, and 35%), borate glasses of the same doping level but of different cations (Li, Na, Rb, Ag), and for comparison, a borate and silicate glass with identical modifier content of 35% Li2 O. The thickness of the sputtered glass layers was varied between 7 nm and 700 nm. In order to measure the conductivity by impedance spectroscopy, multilayer systems were sputter-deposited on polished silicon substrates. The bottom planar electrode consisted of 15 nm of a Al-8.4 at. %-Li alloy, followed by the glass membrane. By using a suitable deposition mask, the top electrode was produced as an array of circular disks as indicated in Fig. 16 (left). This contact comprised of a triple layer of Al(Li), Ta, and Au, as already discussed with Fig. 3. In the case of very thin glasses, the lateral size of the contacts was decreased to 0.3 mm in diameter to reduce the risk of short circuit failures. A dedicated micromechanical device equipped with fine spring loaded Aucoated pins was designed (see Fig. 16 right) to contact the tiny electrodes without mechanical damage of the sensitive structures. This device comprised a ceramic heating plate that allowed heating samples up to about 400 ◦ C during electrical measurement. Further details on the technical equipment of the experiment may be found in [30]. In order to determine the dc conductivity from impedance data, plotting the real part of conductivity vs. frequency and identifying a low frequency plateau in conductivity would be the natural choice. With decreasing thickness however, electrode polarization becomes a serious problem so that not in all cases clear plateaus could be identified. To our experience, the interpretation of Nyquist diagrams based on appropriate equivalent circuits yielded

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Fig. 17. Impedance spectroscopy of a (Li2 O)0.2 -(B2 O3 )0.80 glass layer. (a) Nyquist diagrams of the borate film sandwiched between Al(Li) (•) and LiCoO2 (◦) electrodes. (b) Capacity determined from equivalent circuit vs. glass thickness. Solid line was calculated with ε r = 12.

more reliable results, in a broader range of temperatures and glass thicknesses. A Nyquist plot of a (Li2 O)0.2 -(B2 O3 )0.8 glass, 700 nm in thickness and measured at 180 ◦ C, is shown in Fig. 17a (solid symbols) for example. The right hand branch (low frequencies) is due to electrode polarization while the semicircle at the left (high frequencies) is assigned to the volume properties of the glass layer. This identification can be secured by two additional observations: (i) In addition to a sample with Al(Li) electrodes, also the impedance of an identical sample, except that the metallic electrodes were replaced by LiCoO2 , is shown in the figure (open symbols). Obviously, the exchange of the electrode material does only affect the low frequency part, while the semi-circle at high frequencies is preserved. (ii) Formally, we can describe this semi-circle by an equivalent circuit [31] that connects a resistor and a constant phase element in parallel, which yields

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as quantitative expression for the impedance Z(ω) =

R . 1 + Q R(iω)n

(1)

(ω denotes the angular frequency, R the ohmic resistor and Q and n prefactor and exponent of the constant phase element, respectively.) This model yields indeed a reasonable description of the measured impedance (see solid line in Fig. 17a), although a detailed understanding of the frequency dependence needs a physically sound model, such as for example the “Concept of Mismatch and Relaxation” (CMR) by Funke and coworkers [32]. Based on this latter concept, we have demonstrated [33] that the capacity across the glass layer can nevertheless be derived from the characteristic parameters of the constant phase element:  nπ  1−n . (2) C = R n Q n sin 2 ω=ωp In this way, the Nyquist diagrams of glass layers of varying thickness were evaluated. Determined capacities are plotted in Fig. 17b vs. the thickness of the glass layer. As it is expected, the capacity scales with the reciprocal of layer thickness and corresponds furthermore to a dielectric constant of ε r of 12, quite a reasonable value for the considered glass. Having checked in this way the nature of the high frequency semi-circle, detailed measurements of the dc conductivity were performed by fitting Eq. (1) to the respective data and determining the appropriate resistor R as a measure of dc conductivity. Taking into account the geometry of the layer, the specific conductivity of the glass was calculated. At first hand, it is interesting to compare the specific conductivity of sputter-deposited glass layers with that of corresponding bulk glasses. To make this comparison as direct as possible, the bulk conductivities were determined at the respective targets which were afterwards used for thin film deposition. A few selected examples of such comparison are shown in Fig. 18; data of sputter-deposited thin films are indicated by solid symbols, those of the bulk glasses, solidified from the melt, by corresponding open symbols. The conductivity of Na (circles) and Rb (squares) borate thin films is by one to two orders of magnitude higher than that of the respective bulk material. The same result was also found for Li and Ag borate glasses [30]. But in remarkable contrast, thin film Li silicate glasses (triangles) reveal exactly the same conductivity as the bulk glass of identical composition. This increase of conductivity in sputter-deposited alkali-borates was studied in detail for the network modifier Li2 O. The conductivities of glasses of varying modifier content are shown in Fig. 19a,b for bulk glasses and thin films, respectively. The isothermal data at a selected temperature of 180 ◦ C presented in Fig. 19c allow an easy comparison. As expected, the conductivity of the bulk

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Fig. 18. Comparison of conductivities of bulk glasses that were quenched from the melt (open symbols) and of sputter-deposited films (solid). Deposited films of borate reveal a significant increase in conductivity with respect to the bulk, while in the case of silicate glass both reveal identical conductivity.

glasses increases with concentration of mobile charge carriers. It should be noted, that this increase exceeds by far a simple proportionality with the number of charge carriers, which demonstrates that Li does not only function as mobile charge carrier but indeed also modifies the network. Furthermore, filling of existing sites with mobile species up to an atomic “Fermi-level” [34] may play an additional role. This concept would in particular explain the observed decrease of activation energy with increasing alkali content in natural way. Also the thin films reveal this general dependence on alkali content, even though significantly weaker than the bulk glasses. But more important here, thin films always reveal a higher conductivity than the corresponding bulk, although this surplus in conductivity tends to level out at higher compositions (see Fig. 19c). The fact that the effect of elevated conductivity is only observed in thin borate films but not in silicate made us suggesting a model that assumes an increased content of non-bridging oxygen (NBO) defects in the sputtered thin films. In order to compensate for the additional positive charge of the cations, negatively charged defects must be introduced into the covalent network. In the case of silicates, these are predominantly NBOs, while NBOs as well as tetrahedrically coordinated borons (BO4 ) both contribute in borate glasses. Recent molecular dynamics simulations [36,37] have indicated that the mobility of the alkali ions is significantly higher in the neighborhood of a NBO defect. This may by understood by the broken bond of the NBO that weakens the network

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Fig. 19. Ion conductivity of different Li-borate glasses: (a) bulk glasses quenched from the melt, (b) sputter-deposited films (thickness ranged from 160 to 360 nm), and (c) comparison of bulk and thin films at 180 ◦ C.

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Fig. 20. Density of charged network defects in borate glasses of varying Li2 O content: total density of defects (solid), partial densities in bulk glasses (dashed), estimated NBO density of sputter-deposited glass films (data points).

locally, while the BO4 units even strengthen the network in comparison to the trigonal units of the undoped borate network. Only the total content of both defects is fixed by stoichiometry, according to the relation x N def = , (3) 1−x in which x denotes the relative fraction Li2 O. The relative abundance of both defect types is variable and will presumably depend on the production route of the material. For bulk glasses the density of both defects was determined by NMR [35]. These literature data are reproduced by dashed lines in Fig. 20. Also the total content of network defects determined by Eq. (3) is shown (black solid line). If we assume conductivity as a unique function of the density of NBOs, this density can be estimated from the NBO content of the bulk glass of the same conductivity shown in Fig. 19c. This estimated density is plotted by the data points in Fig. 20. It becomes obvious that the surplus in conductivity of the thin films must naturally level out with increasing Li2 O content, since the number of NBOs approaches the maximum total number of network defects. (Further details of this study are discussed in [30].) As a second important aspect, a finite size effect was observed in samples with very thin glass layers sandwiched between Al(Li) electrodes. In samples that were annealed after deposition, the conductivity increases by up to three orders of magnitude, if thickness is reduced to below 10 nm. Experimental data of this remarkable effect are shown in Fig. 21 for Li-borate glasses of varying alkali content. In the extreme, a membrane of only 7 nm in thickness was successfully measured. It must be emphasized that the trivial increase of absolute conductivity with decreasing thickness is already taken into account by calcu-

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Fig. 21. Specific conductivity of sputtered thin films of Li borate glasses vs. thickness. Compositions of the glasses as indicated.

lating the specific conductivity. Therefore, the presented increase in specific conductivity reflects a real property change of very thin films. In two articles we published and discussed this observation in detail [6,7]. At first hand, one could make space charge zones responsible for a higher conductivity of very thin layers. Indeed, a space charge model, assuming diffusion of Li+ ions from the Al(Li) electrodes into the glass and a negative charge at the electrode surfaces for compensation, can formally describe the experimental data, as shown by the solid lines in Fig. 21. However, the Debye screening length, which is necessary to fit the data appears with 15–20 nm way too large in view of the high concentration of mobile charge carriers. Therefore, probably a space charge model must be ruled out. As an alternative approach, we suggested a cluster model. During the annealing treatment, NBOs tend to cluster and form local regions of higher conductivity. Below the percolation limit, these regions of high conductivity are still insulated within a matrix of poor conductivity. However, if the thickness of the glass film approaches the typical diameter of the conductive clusters, an increasing number of these clusters will bridge the two electrodes and thus will lead to a significant increase in conductivity. Using a model resistor network, this concept was evaluated and it was demonstrated that indeed a quantitative description of the observed conductivity increase can be achieved [7].

4.1 Measurement of ionic transport by optical transmission As we have seen in the presented study on ion conductivity of glasses, impedance spectroscopy is a powerful tool to determine specific transport prop-

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Fig. 22. Chrono-potentiometry of a sputtered LiCoO2 film, 120 nm in thickness. Voltage increases by de-intercalation of Li. Reversible battery function appears between about 4.1 and 3.8 V vs. Li reference (x = 0.5–1.0). Pictures in the left corner illustrate change of transparency of thin layers under variation of Li content.

erties also in thin films, even if interfaces prevent a permanent dc transport. However, just this real dc transport is required for many applications, which raises the question to which extent the data obtained by impedance spectroscopy are relevant for practical devices. We therefore developed a method of direct transport measurement in thin films based on the electro-chromatic properties of LiCoO2 which were recently reportet [38]. In Fig. 22, a chrono-potentiometry of the sputter-deposited LiCoO2 films is shown. Reversible Li intercalation is expected for relative Li contents between Li0.5 CoO2 and Li1.0 CoO2 . If thin films (120 nm) are deposited on transparent glass slides coated with ITO providing the necessary contact, it is seen that the transparency of the Li oxide films decreases with Li de-intercalation. (See three examples shown in the left corner of Fig. 22.) Based on this observation, a physical measurement of Li transport is designed. The required, quite simple instrumental setup is sketched in Fig. 23. A transparent substrate coated with ITO and Li-oxide is inserted into a transparent electrochemical cell filled with an electrolyte that provides mobile Li ions (mixture of DMC/EC and 1 M LiClO4 ). The counter-electrode (graphite) and a reference are placed so that they do not disturb light transmission through the LiCoO2 cathode and the cell. By illumination with a power-LED and intensity measurement by a sensitive photo-cell, the relative transparency of the LiCoO2 film is determined. After establishing the initial equilibrium potential between cathode and reference (about 3.8 V vs. Li), the cell voltage is suddenly increased by a small step (0.3 V) within the reversible range. The response in transparency is traced by relaxation curves as presented in Fig. 24. Curves of LiCoO2 films of differ-

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Fig. 23. Setup to measure degree of intercalation by optical transmission.

Fig. 24. Relaxation of optical transparency after abrupt change of cell potential for LiCoO2 films of different thickness. Inset shows the dependence of scaling time (time required to reach stable end value of transparency) on film thickness.

ent thickness (12.5–100 nm) are shown. By proper scaling in time, these curves reasonable merge on a common master curve. The required scaling times for this normalization fulfill a linear relation with the square of thickness, in remarkable accuracy (see inset of the Fig. 24). This demonstrates clearly that the observed process is controlled by diffusion within the LiCoO2 layer. If applied potential steps are sufficiently small, transparency varies proportionally to the Li content within the layer. The out-diffusion of Li after disturbing the equilibrium by the applied voltage can be described by solution of Ficks second law under consideration of appropriate boundary conditions [39]. Therefore in first approximation, the transparency is expected to vary with

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time t as

      π 2 D Li+ 1 9π 2 D Li+ ΔT(t) ≈ exp − t + t + . . . exp − T0 − T∞ 4d 2 9 4d 2

(4)

In this equation T0 , T∞ , d and D Li+ denote the transparencies at the beginning of the experiment and after very long waiting, the layer thickness, and the diffusion coefficient of Li+ within the oxide, respectively. By fitting the relation to the presented experimental curves, the diffusion coefficient of Li ions in the sputtered layers at room temperature is determined to D Li+ = (1.0 ± 0.2) × 10−14 cm2 /s. Since this value does not rely on electrical measurement but is determined from a property change related to the de-intercalation of Li, it certainly characterizes the practical achievable intercalation rate in thin film devices. Remarkably, the diffusion rate determined in this way is about two orders of magnitude lower than literature data [40,41] obtained by means of impedance spectroscopy. At present, it is not clear whether this discrepancy is due to a general physical difference of the applied different methods or whether it is due to the distinguished properties of the thin films produced in this work. By the applied particle beam sputtering technique, LiCoO2 layers grow in strong c-axis texture. In the non-cubic lattice structure, transport rate depends on lattice direction and is slowest in c-axis direction perpendicular to the oxygen planes. Thus, in our experiments transport appears in the slow direction while in other studies fast transport parallel to the layers may have been measured. Having demonstrated in this way that a variation of the Li content in sputtered LiCoO2 layers can be accurately traced by measurements of transparency, the same LiCoO2 layers can be used as a substrate to characterize the permeation through other interesting materials as long as these form transparent films and control the kinetics due to slower mobility. This strategy shall be demonstrated by a measurement of Li penetration through thin glass membranes of (Li2 O)0.35 -(SiO2 )0.65 as shown by the data curves in Fig. 25. In this case, the absolute variation of the voltage generated by the photocell is plotted, which represents a measure for the absorption coefficient in arbitrary units. The important measured parameter is the time lag between the change of the chemical potential and the onset of the response of the optical signal. As seen in the figure, in the case of a naked LiCoO2 layer of 100 nm thickness, the optical response appears practically instantaneously with the sudden increase in supplied cell voltage. By contrast, samples coated with an additional glass layer respond only after a characteristic time delay. This delay nicely scales with thickness of the added glass layers (see inset). Again a parabolic relation with the thickness is indicated. This justifies evaluation of the time lag by tdelay (d) =

d2 6D Li+

(5)

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Fig. 25. Measurement of Li transport through thin films of a (Li2 O)0.35 -(SiO2 )0.55 glass by optical transmission based on the time-lag method. Inset compares measured time lags with prediction from impedance data.

(see [39]). In this way, the diffusion coefficient of Li through the thin glasses is determined to be (1.6 ± 0.3) × 10−15 cm2 /s. The conductivity of the same silicate glass layers was determined independently by impedance spectroscopy [30]. From published conductivity data, the respective diffusion coefficient, is determined to 1.2 × 10−14 cm2 /s applying Nernst–Einstein relation. Obviously, the time lag method yields a diffusivity that is about an order of magnitude slower than that determined by impedance spectroscopy. It must be noted that the interpretation of conductivities in terms of diffusivity needs the concentration of mobile charge carriers. By contrast, the time lag method delivers the respective diffusion coefficient directly. Thus, at present, we cannot decide whether the stated discrepancy is due to an erroneously assumed density of charge carriers or can be interpreted as evidence for a considerable Haven ratio. It is noteworthy that the method described here closely corresponds to charging and decharging in thin film batteries, while impedance spectroscopy delivers only indirect evidence for the function of a battery. Therefore, the described time lag method is certainly of interest from a technical point of view. The essential prerequisite for the discussed time lag method is the fabrication of thin films in well controlled thickness. If such layers are available, the method is capable to determine the kinetic barrier formed by an additional glass layer of only 5 nm thickness. Even at this small thickness, the measured time

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lag still amounts to 27 s, which is easily resolved in measurement. Therefore, there is certainly a potential of obtaining meaningful measurements of even thinner layers. Although the preliminary data presented in Fig. 25 need further confirmation, it is already indicated that down to 5 nm thickness, the influence of the glass layer still obeys a parabolic dependence on thickness. From this observation, it can be concluded that the interface between the coating glass and the intercalation compound as well as the surface of the glass in contact to the electrolyte do not introduce further distinguished interfacial transport barriers. It will be a future point of interest to study this aspect with even thinner glass layers to reveal the particular impact of the interfaces.

5. Summary In this work, a study on several aspects of thin film battery materials is presented. The possiblity of depositing functional thin films by neutral particle beam sputter-deposition is explored. Suitable process parameters are derived for deposition of LiCoO2 , V2 O5 , Li4 Ti5 O12 electrodes as well as alkali borate and silicate glasses to be used in batteries as thin film ion-conductive membranes. The crystalline structure and chemical function of the electrode materials were verified by X-ray and cyclo-voltammetry. In the case of V2 O5 , also the long-term cycling stability was demonstrated. The chemical analysis of depositied layers, was demonstrated by EELS techniques. Beside the established procedure of comparing the height of characteristic absorption edges, also useful finger-printing methods were demonstrated. With finger-printing, the oxidation state of V in vanadium oxide can quantified. Also, the intercalation state of LiCoO2 becomes measurable without evaluating the Li absorption edge. The analysis of LiCoO2 and of borate glass films by laser-assisted atom probe tomography was introduced. The application of this innovative method, revealed a tendency of Li2 O segregation at the surface of LiCoO2 . In borate glass layers, polarization by migration of mobile Li+ ions was demonstrated, which probably appears during the atom probe analysis. The atom probe study demonstrates that migration of Li and polarisation can be microscopically observed and evaluated in thin glass layers of only 20 nm thickness. The ionic conductivity of thin, network glasses was characterized in detail by impedance spectroscopy in comparison to the corresponding bulk materials. It is shown that sputter-deposited borate glasses doped with various alkali network modifiers reveal a higher specific conductivity than bulk glasses which were solidified from the melt. The depencence of this effect on composition and the fact that thin silicate glasses do not show any modification of conductivity make plausible that the higher conductivity in sputter-deposited borates is related to an increased density of non-bridging oxygen defects.

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Finally, it has been demonstrated that by using thin LiCoO2 films of well controllable thickness, intercalation rates can directly be measurement based on the electrochromatic properties of this oxide. The intercalation in thin films of 100 nm thickness is still diffusion controlled. By measurement of light transmission through more complex layer stacks, even the permeation through silicate membranes of only 5–20 nm in thickness can be studied and permeation rates are quantified. It is seen that diffusional transport controls the permeation even in a membrane of to 5–20 nm in thickness while the influence of interfacial barriers is still negligible. Surprisingly, the absolute diffusivity is an order of magnitude slower than concluded from impedance measurements.

Acknowledgement Generous support by the German Research Foundation (DFG, SFB 458) is gratefully acknowledged. We are grateful to all members of the collaborative research center 458 for numerous fruitful discussions on the behaviour of network glasses. In particular, we would like to thank Cornelia Cramer-Kellers and Dirk Wilmer for their help with impedance and IR spectroscopy and Klaus Funke for introduction to his CMR concept.

References 1. J. M. McGraw, C. S. Bahn, A. Philip, P. A. Parill, J. D. Perkins, D. W. Readey, and D. S. Ginley, Electrochim. Acta 45 (1999) 187. 2. P. J. Bouwman, B. A. Boukamp, H. J. M. Bouwmeester, H. J. Wondergem, and P. H. L. Notten, J. Electrochem. Soc. 148 (2001) A311. 3. N. J. Dudney and Y.-I. Jang, Analysis of thin-film lithium batteries with cathodes of 50 nm to 4 μm thick LiCoO2 , J. Power Sources 119–121 (2003) 300–304. 4. X.-J. Zhu, L.-B. Cheng, C.-G. Wang, Z.-P. Guo, P. Zhang, G.-D. Du, and H.-K. Liu, J. Phys. Chem. C 113 (2009) 14518–14522. 5. J. Deng, Z. Lu, I. Belharouak, K. Amine, and C. Y. Chung, J. Power Sources 193 (2009) 816. 6. F. Berkemeier, M. S. Abouzari, and G. Schmitz, Thickness dependent ion conductivity of lithium borate network glasses, Appl. Phys. Lett. 90 (2007) 113110-1. 7. F. Berkemeier, M. S. Abouzari, and G. Schmitz, Thickness dependence of the dcconductivity in Li borate glasses, Phys. Rev. B 76 (2007) 024205. 8. G. Wei, T. E. Haas, and R. B. Goldner, Solid State Ion. 58 (1992) 115. 9. C.-L. Wang, Y. C. Liao, F. C. Hsu, N. H. Tai, and M. K. Wu, J. Electrochem. Soc. 152 (2005) A653. 10. T. Gallasch, T. Stockhoff, and G. Schmitz: Ion beam sputter deposition of V2 O5 thin films, J. Power Sources 196 (2011) 428–435. 11. T. Stockhoff, T. Gallasch, F. Berkemeier, and G. Schmitz: Ion beam sputter deposition of LiCoO2 films, J. Electrochem. Soc. (submitted) (2010). 12. D. Williams and C. Carter, Transmission Electron Microscopy, Springer, Berlin, (1996), p. 655. 13. Application program: “Digital Micrograph” by Gatan (2010). 14. L. Laffont, Micron 37 (2006) 459–464. 15. M. K. Miller, Atom Probe Tomography, Kluvwer Academic, New York (2000).

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16. P. Stender, T. Heil, H. Kohl, G. Schmitz: Quantitative comparison of energy-filtered transmission electron microscopy (EFTEM) and atom probe tomography (APT) measurements of Cr/Fe multilayers, Ultramicroscopy 109 (2009) 612. 17. T. F. Kelly and M. K. Miller, Rev. Sci. Instrum. 78 (2007) 031101. 18. A. Cerezo, et al., Mater. Today 10 (2007) 1. 19. G. Schmitz, Nanoanalysis by atom probe tomography, in: Nanotechnology (Vol. 6), H. Fuchs (ed.) Wiley-VCH, Weinheim (2009). 20. D. Larson, Microsc. Microanal. 7 (2001) 24. 21. M. K. Miller and K. F. Russel, Ultramicroscopy 107 (2007) 761. 22. B. Gault, F. Vurpillot, A. Bostel, A. Menand, and B. Deconihout, Appl. Phys. Lett. 86 (2005) 094101. 23. C. Oberdorfer, P. Stender, C. Reinke, and G. Schmitz, Laser-assisted atom probe tomography of oxide materials, Microsc. Microanal. 13 (2007) 342. 24. J. H. Bunton, J. D. Olson, D. R. Lenz, and T. F. Kelly, Microsc. Microanal. 13 (2007) 418. 25. Y. M. Chen, T. Ohkubo, M. Kodzuka, K. Morita, and K. Hono, Scr. Mater. 61 (2009) 693. 26. M. Gruber, C. Oberdorfer, P. Stender, and G. Schmitz, Laser-assisted atom probe analysis of sol-gel silica layers, Ultramicroscopy 109 (2009) 654. 27. R. Schlesiger, C. Oberdorfer, R. Würz, G. Greiwe, P. Stender, M. Artmeier, P. Pelka, F. Spaleck, and G. Schmitz, Design of a laser-assisted tomographic atom probe at Münster University, Rev. Sci. Instrum. 81 (2010) 043703. 28. R. Kanno and M. Murayama, J. Electrochem. Soc. 178 (2001) A742. 29. W. Weppner, Ionics 9 (2003) 444. 30. F. Berkemeier, M. S. Abouzari, and G. Schmitz: Sputter-deposited network glasses – structural and electrical properties, Ionics 15 (2009) 241–248. 31. J. R. Macdonald, Impedance Spectroscopy, Wiley, New York (1987). 32. K. Funke and R. D. Banhatti, Solid State Sci. 10 (2008) 790. 33. M. S. Abouzari, F. Berkemeier, G. Schmitz, and D. Wilmer, On the physical interpretation of constant phase elements, Solid State Ion. 180 (2009) 922–927. 34. R. Kirchheim and D. Paulmann, J. Non-Cryst. Solids 286 (2001) 210. 35. J. Zhong and P. Bray, J. Non-Cryst. Solids 111 (1989) 67. 36. C. P. E. Varsamis, A. Vegiri, and E. Kamitsos, Phys. Rev. B 65 (2002) 104203. 37. A. Vegiri and C. P. E. Varsamis, J. Chem. Phys. 120 (2004) 7689. 38. Y. Takahashi, N. Kijima, and J. Akimoto, Cryst. Growth Des. 7 (2007) 2491. 39. J. Crank, Mathematics of Diffusion, 2nd Edn., Oxford University Press, Oxford (1975). 40. H. Xia, L. Lu, and G. Ceder, J. Power Sources 159 (2006) 1422. 41. S.B. Tang, M.O. Lai, and L. Lu, J. Alloys Compd. 449 (2008) 300.

Ion Jump Dynamics in Nanoscopic Subvolumes Analyzed by Electrostatic Force Spectroscopy By Andr´e Schirmeisen1 , ∗, Ahmet Taskiran1 , Hartmut Bracht2 , and Bernhard Roling3 1 2 3

Institute of Physics and Center for Nanotechnology (CeNTech), Westf. Wilhelms-University of Münster, Wilhelm-Klemm-Str. 10, 48149 Münster, Germany Institute of Materials Physics, Westf. Wilhelms-University of Münster, Wilhelm-Klemm-Str. 10, 48149 Münster, Germany Department of Chemistry and Center for Materials Science (WZMW), University of Marburg, Hans-Meerwein-Strasse, 35039 Marburg, Germany (Received June 28, 2010; accepted in final form August 13, 2010)

Ion Dynamics / Electrostatic Atomic Force Microscopy A nanoscopic technique based on electrostatic force spectroscopy in the time domain is introduced. This technique is used to characterize the ion dynamics in nanoscale subvolumes of solid electrolytes. Nanoscopic polarization spots can be created and directly visualized, and their time evolution can be studied. In the case of partially crystallized glass ceramics, the dynamic processes in different phases (glassy, crystalline and interface) can be distinguished and their activation energies and pre-exponential factors can be quantified. By applying grid-type spectroscopic measurements, maps of the local relaxation strength are obtained, giving information about the spatial distribution of the glassy and crystalline phase.

1. Introduction Crystalline, glassy and polymeric ion conductors are used as solid electrolytes for various applications. Important examples are energy storage and conversion in batteries, super-capacitors, and fuel cells. Despite intensive research efforts, many of these applications are still lacking optimized materials, which satisfy the complex requirements of high ionic conductivity as well as chemical and electrochemical stability. Often changes in the chemical composition for enhancing ion conductivity lead in return to a reduction of the chemical and/or electrochemical stability [1]. * Corresponding author. E-mail: [email protected]. Z. Phys. Chem. 224 (2010) 1831–1852 © by Oldenbourg Wissenschaftsverlag, München

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Fig. 1. a) Mobile ions in a glass matrix move via discrete jumps between potential energy minima. b) In partially crystallized glass ceramics, interfaces between glassy matrix and crystallites are believed to act as fast conduction pathways for mobile ions.

Disorder plays an important role for ion transport processes. For instance, glassy ion conductors do not exhibit a periodic structure and, therefore, the distinction between ions on regular lattice sites and ionic defects, as in crystalline ion conductors, is not valid. Potentially, all ions possess a long-range mobility via thermally activated jump diffusion between the energetically favorable sites in the glass matrix, as depicted in Fig. 1a. In this example, the glass consists of a rigid silicate network with the alkali ions occupying voids in this network. A promising method for material optimization, which is increasingly used, is the preparation of nanostructured materials with a high amount of internal interfaces. For example, Indris and Heitjans observed a distinct enhancement of the ionic conductivity, when they added insulating B2 O3 nanocrystals to ion conducting Li2 O nanocrystals [2]. This apparent paradox can be resolved by assuming that the interfaces between the ionic conductor Li2 O and the insulator B2 O3 are preferred diffusion pathways for the mobile Li+ ions. For nanocrystalline materials, the volume fraction of the interfaces is high enough that the fast interfacial ionic conduction contributes significantly to the macroscopic conductivity. Similar effects were observed in hetero-structures consisting of ultra-thin films with different ionic conductivities. For example, Sata et al. investigated the ionic conductivity of hetero-structures with alternating layers of CaF2 and BaF2 [3], while varying the layer thickness. Layer thicknesses in the range from 100–400 nm resulted in a conductivity proportional to the number of interfaces in the hetero-structure. Below 100 nm, the conductivity increased even more strongly with the number of interfaces. Korte et al. studied the ion conductivity at interfaces between oxide ion conductors and insulating oxides [4]. They found that the interfacial ion conductivity is determined by the interfacial strain. Dilatative strain results in an increase of the conductivity, while compressive strain leads to a decrease of the conductivity. Another interesting example are partially crystallized glass ceramics containing nanocrystallites (see Fig. 1b). Adams et al. performed in-situ conductivity measurements on a 0.57AgI · 0.29Ag2 O · 0.14V2 O5 glass ceramic during partial crystallization and observed a clear enhancement of the conductivity in

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the early stages of crystallization [5]. A direct comparison of the conductivities with X-ray diffraction data indicated that the observed conductivity enhancement is proportional to the interfacial area between nanocrystallites and glassy matrix. Also polymer electrolytes show a strong influence of the internal interfaces on the ionic conduction. An example is the addition of oxide nanoparticles (Al2 O3 , TiO2 or SiO2 ) with diameters around 10 nm to salt-in-polymer electrolytes [6,7]. The addition inhibits the crystallization affinity of these electrolytes, while the lithium ion conductivity is enhanced. However, from a theoretical point of view, the influence of the interfaces on the ion transport mechanisms is not well understood so far. In nanocrystalline materials, the space charge concept of Maier [8] is often used to explain interfacial effects. In the case of a material with conducting and insulating nanocrystallites, the following scenario is envisaged in the framework of this concept: Mobile ions from the ion conducting crystallites accumulate at the internal interfaces due to the local gradients in their chemical potential. This generates space charge regions extending into the ion conducting crystallites, in which defects with opposite charge accumulate. Due to the higher defect density, the ion conductivity in the space charge regions is enhanced with respect to the bulk of the nanocrystals. For ion conducting glass ceramics and polymer electrolytes, space charge effects are presumably of subordinate significance, since the high number density of mobile ions leads to a Debye length which is of the order of atomic dimensions. Alternatively, a high mobility of ions at the interfaces might be responsible for the conductivity enhancement. In this case, the percolation of the interfaces would be an important prerequisite for a large conductivity enhancement. Unfortunately, such mobility effects at interfaces could not be directly verified by experiments so far. One of the main reasons for this is the traditional characterization of the ion dynamics by means of macroscopically averaging techniques, such as conductivity spectroscopy, tracer diffusion measurements, and NMR relaxation techniques. Therefore, an experimental method capable of probing ion transport on nanoscopic length scales would be highly desirable. In this paper, a recently developed approach based on electrostatic force microscopy is presented. The method utilizes an atomic microscope and is non-onvasive, while being sensitive to ion conduction properties in nanoscopic volumes below the surface. The atomic force microscope (AFM) [9] traditionally visualizes surface topography by scanning a sharp tip over a sample while simultaneously detecting the tip–sample forces. The tip is mounted on a soft cantilever which deflects due to the force between tip and surface. The cantilever deflection, which is proportional to the force, is typically measured by a laser beam deflection system. Electrical scanning force methods are characterized by an additional electrical potential difference between tip and substrate. Figure 2a shows the

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Fig. 2. a) Schematic illustration of the experimental setup for electrostatic force spectroscopy on solid electrolytes. b) Equivalent circuit for modeling the overall capacitance of the system.

principle setup of an electrical force microscope. In good approximation, the electric field around the tip decays radially, so that most of the potential drop in the sample occurs within a small subvolume underneath the tip, the subvolume being of the order of the cube of the tip diameter. This allows measuring the electric sample properties with a local resolution of the order of the tip diameter. In contrast to conventional force microscopy, which is sensitive to surface properties, here the signal is sensitive to subsurface properties. Typical tip radii are of the order of 20 nm, which results in a local resolution of about one hundred times better than methods using micro-electrodes [10]. In recent years the application of electrical scanning force methods for the investigation of different classes of materials has strongly increased. The method of scanning capacitance microscopy is used to investigate the electrical properties of semiconductors and semiconductor devices [11–13]. Here, the investigated semiconductor is covered with a thin insulating oxide layer, and the scanning tip is in contact with this layer. An alternating voltage is applied in order to obtain spatially resolved information about the electrical capacitance, which depends on the number density of mobile electrons and holes. This method has also allowed to investigate the nanoscopic conduction pathways for electrons and holes in electroluminescent polymers [14], in metal–insulatornanocomposites [15] and in networks of nanotubes [16]. The conventional AFM mode, where the tip is in contact with the surface during scanning (static or contact mode), has been refined by oscillating the cantilever some nanometers above the sample, avoiding direct contact (dynamic or non-contact mode) [17]. In this case, tip–sample forces induce a change in the resonant frequency of the cantilever oscillation. This detection technique features a higher sensitivity and is commonly used for atomic resolution imaging of surfaces (for a review see [18]). If a voltage is applied between tip and sample in non-contact mode AFM, the additional electrostatic forces will induce changes in the oscillation frequency of the system. On the one hand, this method can be used to perform pure electrical microscopy by measuring the local changes of the resonant frequency during scanning of the

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sample surface. This has been used, e.g., to investigate the electrical conductivity of nanotubes and biological molecules [19,20]. On the other hand, spatially resolved spectroscopy can be carried out by monitoring the resonant frequency as a function of time after applying the tip–sample voltage. Such time-domain spectroscopic techniques have successfully probed the dynamics of thin polymer films close to the glass transition temperature [21,22]. Despite the high potential and success of electrical force microscopy techniques, they have been rarely applied to ion conducting materials. Exceptions are measurements of the ion conductivity of CaF2 [23], and the dynamics of solute ions at the step edges of simple salts (e.g., NaCl, KCl, KBr, KI) [24]. Other examples are investigations of local ionic conductivites in salt-in-polymer electrolytes by Layson et al. [25] and by Bhattacharyya et al. [26]. Both groups used spreading resistance microscopy to measure separately the ion conductivity of amorphous and crystalline parts of the samples, while not exploiting the high spatial resolution of the method. O’Hayre et al. [27] reached a local resolution of about 100 nm for the investigation of proton transport in wet Nafion membranes allowing them to discriminate between hydrophobic and hydrophilic areas of the membranes.

2. Results and discussion 2.1 Time domain electrostatic force spectroscopy Figure 3 shows the working principle of the TD-EFS (time domain electrostatic force spectroscopy) technique. The tip of a conductive cantilever is oscillating with small amplitudes of 1–3 nm above the surface of the solid ionic conductor. While the sample is fixed on a metallic plate at ground potential, a potential in the range from −1 to −4 V is applied to the tip. The electrical field emanating from the tip penetrates the sample causing the positively charged ions to accumulate underneath the tip. In the case of small cantilever oscillations and long ranged electrostatic interactions, the frequency shift Δ f induced by the bias voltage U is given by [28]: Δ f(t) = −

f 0 2 ∂ 2 C(t) U 4c ∂z 2

(1)

where C(t) denotes the overall capacitance between biased tip and ground, c is the normal spring constant and f 0 is the free resonance frequency. C(t) can be modeled by an equivalent circuit illustrated in Fig. 2b. The probed nanoscopic subvolume of the sample is represented by a resistor R nano in parallel to a capacitor C nano . The resistor R nano models ionic conduction, while C nano models the capacitance due to electronic and vibrational polarization. The gap between tip and probed subvolume is represented by a vacuum capacitor

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Fig. 3. a) When a negative potential is applied to the tip, the positively charged ions within the probed subvolume move towards the tip, creating an electrical polarization in the subvolume. b) The polarization leads to an additional attractive force between tip and sample, which is reflected in a decrease of the resonance frequency.

C V in series to the R nano C nano element. Additionally, a capacitor C S in parallel to the other elements is introduced, which represents all stray capacitances between tip and ground. Upon application of the voltage U, all capacitors are instantaneously charged. Subsequently, the capacitor C nano is discharged through the resistor R nano . This leads to an increase of the overall capacitance C(t) and thus to a decrease of the resonant frequency f(t). In the framework of the equivalent circuit, the time dependence of C(t) is given by [29]:   CV · exp(−t/τ) + C S (2) C(t) = C V 1 − C nano + C V with τ = R nano (C nano + C V ). From a microscopic point of view, the discharge of the sample capacitor is due to mobile ions moving in the direction of the externally applied electric field, until the field in the probed subvolume becomes zero. Thus, the time-dependent drop of the cantilever resonance frequency reflects the time-dependent built-up of an electrical polarization in the subvolume caused by jump processes of mobile ions. One of the most important features of this measurement technique is the probed subvolume. Finite element simulations show that the electric potential drop occurs in a subvolume with a depth and width of the order of the tip diameter. With a typical tip diameter of 20 nm and a tip–sample distance of 10 nm, the approximate probed sample volume is (40 nm)3 . Taking into account the number densities of alkali ions in typical glass samples, which is of the order of 1022 cm−3 , the dynamic behaviour of an ensemble of less than 106 ions is measured, many orders of magnitude below the macroscopic level. In particular, at lower temperatures, only a fraction of these ions are expected to carry out hopping movements. This opens up the possibility to study movements of only a few ions.

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Fig. 4. Polarization spot experiment at T = 295 K: After switching off the tip potential at time t = 0, the polarization profile decays with time.

2.2 Creation and visualization of a nanoscopic polarization spot A direct experimental verification of the probed subvolume was obtained by the following procedure: A single nanoscopic polarization spot in a K 2 O · 2CaO · 4SiO2 glass sample was generated by biasing the tip with −4 V. At the sample temperature of 295 K the electrical relaxation time was τ = 30 s. Subsequently, the tip was set to ground potential, and a rapid line scan was performed above the electrical polarization spot. During the scan, the polarization induces mirror charges in the tip changing the forces between sample and tip. In Fig. 4, these force changes are visualized as apparent height profiles after subtracting the zero-voltage height profile, which was obtained before the creation of the polarization spot. The lateral width of the peak (FWHM) is about 160 nm. This corresponds roughly to twice the width of the polarization spot, since the peak width results from a convolution of the tip size and the polarization spot size. The peak height decreases with time, since the spot depolarizes after setting the tip to ground potential. As seen from Fig. 4, the relaxation time of the depolarization corresponds to the electrical relaxation time τ.

2.3 Homogeneous solid ion conductors Further proof-of-principle experiments were carried out by applying the TDEFS technique to two homogenous glass samples and by comparing the results to their macroscopic electrical properties [29]. The chemical compositions of these glasses were 0.25Na2 O · 0.75GeO2 (NG glass) and 0.143K 2 O · 0.286CaO · 0.571SiO2 (KCS glass), respectively. The activation energies of the dc conductivity, E dc A , reflecting the thermally activated long-range alkali ion

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Fig. 5. a) Time dependence of the cantilever resonance frequency after applying a voltage of U = −4 V to the 0.143K 2 O · 0.286CaO · 0.571SiO2 glass. The frequency axis is normalized by the modulus of the maximal frequency shift. The solid lines represent best fits of the experimental data to a stretched exponential function. b) Arrhenius plot of the TD-EFS relaxation times (symbols) and of the macroscopic relaxation times τmacro = R macro C macro (solid lines).

transport, are 0.74 eV in the case of the NG glass and 1.05 eV in the case of the KCS glass, respectively. Since the structure of both glasses is homogeneous on length scales of 10–30 nm, the differences in the mobility of the alkali ions should also manifest in the nanoscopic electrical properties as probed by TD-EFS. In the case of the KCS glass, the time-dependent resonance frequency change was measured at different temperatures in a range from 376–570 K. Figure 5a shows selected relaxation curves, which were normalized to their respective saturation frequency shift values Δ f saturation . This normalization was done for a better comparison of the relevant time scales. The relaxation curves were fitted by a stretched exponential function of the form [28]:   (3) Δ f(t) = (Δ f saturation − Δ f fast ) 1 − exp(t/τ)β + Δ f fast .

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Here, Δ f fast denotes the instantaneous frequency shift due to ultrafast relaxation processes (vibrational and electronic polarization). The same type of measurement was also performed on the NG glass sample in a temperature range from 253–296 K. The stretch factors are β = 0.65 for the KCS glass and β = 0.55 for the NG glass, respectively. In Fig. 5b we show an Arrhenius plot of the relaxation time τ. The data of both samples follow, to a good approximation, an Arrhenius law. This is expected, since the temperature dependence of τ is governed by the temperature dependence of R nano , which, in turn, is governed by the activation energy of ion transport in the probed subvolume. The solid lines in Fig. 5b represent the macroscopic relaxation times τ macro = R macro C macro , which we calculated from the macroscopic resistances R macro and capacitances C macro of the samples. Clearly, there is good agreement between the relaxation times τ and τ macro and their respective temperature dependencies. This shows that the same dynamic processes are probed by electrostatic force spectroscopy and by macroscopic electrical spectroscopy, namely the dynamics and transport of the mobile ions. On the other hand, a quantitative comparison between nanoscopic und macroscopic results is difficult for two reasons. First, the values of τ are influenced by the vacuum capacitor C V . Secondly, the description of the electrical properties of the probed subvolume by a parallel R C element is an approximation. The time-dependent electrical properties of the glass samples lead to a stretched exponential relaxation of the experimental C(t) data, in contrast to the simple exponential time dependence of C(t) in Eq. (2).

2.4 Nanostructured solid ion conductors An interesting class of nanostructured model materials for TD-EFS measurements are glass ceramics based on the system Li2 O · Al2 O3 · SiO2 . These glass ceramics are lithium ion conductors with a conductivity that depends on the degree of crystallinity. From macroscopic conductivity measurements [30] it is known that LiAlSiO4 glass is a moderate ion conductor with an activation energy of 0.72 eV, while a completely crystallized LiAlSiO 4 sample is a poor ion conductor with a high activation energy of 1.07 eV. At room temperature, the macroscopic electrical relaxation times of the pure glass and of the completely crystallized ceramic are about 10−2 s and 103 s, respectively. Investigations of the lithium ion conductivity of LiAlSiO4 glass ceramics with different degrees of crystallinity [30] revealed that in a range from χ = 0 to χ = 0.4, the conductivity increases with increasing χ, see Fig. 6. This effect is most likely related to fast ion conduction at the interfaces between crystallites and glassy phase, see the schematic illustration in Fig. 7a. On the other hand, at χ > 0.4, the ionic conductivity drops strongly with increasing χ, suggesting that lithium ion transport is blocked by the poorly conducting crystallites, see Fig. 7a.

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Fig. 6. a) Crystallization of a LiAlSiO4 glass can be achieved by an annealing process above the glass transition temperature. Depending on the annealing temperature and time, crystallites with diameters from a few 10 nm to a few μm are formed. b) Macroscopic conductivity measurements on the LiAlSiO4 glass ceramic as a function of crystallinity χ showing a conductivity maximum around χ = 40% (taken from [30]).

When the TD-EFS technique is applied to such a class of materials, the probed subvolume contains, depending on the position of the tip, different amounts of glassy phase, crystallites and interfacial areas. Since the ionic conductivity in these phases is different, we expect the electrostatic force spectra to vary with the position of the tip. But how can we separate the contributions from the three phases? At different temperatures, the contribution from the different phases should vary strongly, since their relaxation times exhibit different activation energies. The time scale of our TD-EFS measurements is limited to a range from 1 ms to 10 s and thus we have to adjust the sample temperature in a way that the contribution from one phase predominates. For example, at room temperature, movements of ions in the glassy phase govern the relaxation, while the ions in the crystalline phase are immobile on the TD-EFS time scale [31]. On the other hand, at elevated temperatures, the ions in the crystallites start to contribute significantly, while the relaxation of the ions in the glassy phase becomes so fast, that we can not resolve it anymore. Hence, the relaxation of the latter ions will simply add to the ultrafast relaxation processes caused by electronic and vibrational polarization. For each sample temperature we performed TD-EFS measurements at different positions of the tip above the surface of a glass with 42% crystallinity. As an example, Fig. 8 shows the frequency shift of the oscillating cantilever as a function of time for T = 506 K and at five different positions. As expected, we find a large instantaneous frequency shift Δ f fast , which results from both fast ion movements in the glassy phase and ultrafast electronic and vibrational

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Fig. 7. a) Schematic illustration of the ion transport in partially crystallized glass ceramics with low, medium and high degree of crystallinity χ. At low χ values, long-range ion transport takes place essentially in the glassy phase (light gray). At medium values, interfaces (medium gray) between glassy phase and crystallites may act as local electrical shorts, leading to a conductivity enhancement. At high χ values, the poorly conducting crystallites (red) block the ion transport. b) The contributions of ion movements in different phases to the overall relaxation curve is expected to depend on the position of the tip above the surface.

Fig. 8. a) Surface topography of a glass ceramic with χ = 0.42. b) TD-EFS relaxation curves at a temperature of T = 506 K, obtained at the positions indicated by the circled numbers. The right graph is a zoom of the curves in the left graph, where the black solid lines represent a fit with a stretched exponential decay function.

polarization. In addition, there is a slow relaxation process with τ = 1 s, which can be attributed to ionic movements in the crystalline phase. Qualitatively similar relaxation curves were obtained for a glass ceramic with 13% crystallinity, see Fig. 9. For a better comparison of curves obtained at different temperatures, the frequency shifts were normalized to unity. The

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Fig. 9. Relaxation curves on a glass ceramic with χ = 0.13 in three different temperature regimes shown in (a), (b) and (c). The frequency shift axis was normalized to unity for a better comparison of the relaxation times. (d) This graph shows raw data curves obtained at low temperatures for χ = 0.13 and for a pure glass sample under the same measurement conditions, ruling out possible measurement artifacts.

thermally activated ionic movement in the different phases leads to a decrease of the relaxation time τ with increasing temperature T . The curves in Fig. 9a in a temperature range from 231–275 K reflect ionic movements in the glassy phase, while the curves in Fig. 9b in a temperature range from 545–620 K reflect movements in the crystalline phase. However, from the macroscopic measurements we expect, in addition, the existence of a third phase, namely the interfacial areas between the glassy and crystalline phases. These areas should exhibit a high ionic conductivity and a low activation energy. In order to detect movements at the interfaces, the sample temperature should be below room temperature. Figure 9c shows relaxation measurements of a glass ceramic with 13% crystallinity in a temperature range from 127 to 162 K. Apart from the instantaneous offset in the frequency shift originating from the ultra-fast processes, we detect an additional relaxation process, which may originiate from movements of the ions in the interfacial regime. Similar to the relaxation curves at higher temperatures we find a systematic shift of the curves to the left with increasing temperature, indicative of

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Fig. 10. Arrhenius plot of the TD-EFS relaxation times for χ = 0.13 and 0.42, respectively. For both samples, three relaxation processes can be identified.

a thermally activated ion hopping process. In order to exclude possible artifacts we performed additional test measurements on a homogeneous glass sample without internal interfaces. Figure 9d shows representative relaxation curves of the pure glass sample and of the glass ceramic with 13% crystallinity in direct comparison at the same temperature T = 139 K. While the fast process due to the ubiquitous electronic and vibrational polarisation is seen in both curves, only the partially crystallized sample shows a clear slower relaxation process. In the Arrhenius plot in Fig. 10 we show a compilation of the TD-EFS relaxation times. We can identify three distinct regimes: Around room temperature, a fit with the Arrhenius law yields an activation energy of 0.58 and 0.61 eV for the samples with 13 and 42% crystallinity, respectively. Macroscopic conductivity measurements on a pure glass sample show an activation energy of 0.71 eV. Within the experimental error, which is mainly due to uncertainties in the sample temperature during the TD-EFS measurements, the nanoscopic and the macroscopic results are in reasonable agreement. At temperatures above 500 K, the TD-EFS relaxation processes exhibit activation energies of 1.03 and 1.11 eV for the two samples with 13 and 42% crystallinity, respectively (see red markers (data) and dashed red line (fit) in Fig. 10). These activation energies are very close to the activation energy of 1.07 eV determined from macroscopic conductivity measurements of a completely crystallised LiAlSiO4 ceramic. Below room temperature, the Arrhenius fit yields an activation energy of 0.04 and 0.08 eV for the two investigated samples, which indicate ionic movements at the interfaces with activation energies of only a few times k B T [32]. This result for the interfacial activation energy clearly shows that the interfacial areas do not form percolating pathways through the glass ceramic. If that was the case, the macroscopic activation energy would also be in the range from 0.04 to 0.08 eV. We conclude that the ions have to cross the glassy phase in order to find macroscopic diffusion pathways, as sketched in Fig. 7a. Con-

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sequently, the higher activation energy for ion transport through the glassy phase governs the activation energy for macroscopic transport. However, the interfacial regions act as local electrical shorts, thus leading to a moderate but significant increase of the ionic conductivity as compared to a pure LiAlSiO4 glass. The short relaxation times of the interfacial process together with the low relaxation strengths indicate that the mobility of the ions in the interfacial regions is high, but that their number density is not enhanced as compared to the bulk. This picture differs from space charge scenarios, but is consistent with the assumption that in the interfacial regions the chemical bonds between lithium ions and alumino-silicate network are weaker than in the bulk, resulting in a higher ionic mobility.

2.5 The pre-exponential factor When studying glass ceramics with different crystallinities, we find that the TD-EFS relaxation times of the glass ceramics averaged over many different positions of the tip above the sample surface deviate in a systematic fashion from the macroscopic electrical relaxation times. The pre-exponential factor of the TD-EFS relaxation time obtained for a specific phase increases with decreasing relative amount of this phase in the glass ceramic. For instance, when we decrease the degree of crystallinity from 42 to 13%, the pre-exponential factor obtained for the crystalline phase increases, while that obtained for the glassy phase decreases. This type of information is not obtainable from macroscopic impedance spectra. In Fig. 11, Arrhenius plots of the relaxation time for the glassy phase, τ g , and for the crystalline phase, τ c , are shown for the pure glass and for crystallinities of 13, 42, 80 and 97% crystallized samples, respectively. Here, the τ g data can be fitted with an activation energy of 0.63 eV, independent of the degree of crystallinity, χ, while the pre-exponential factor of τ g increases continuously with increasing degree of crystallinity, χ. On the other hand, all τ c data points can be reasonably well fitted with an average activation energy of 1.04 eV, again in good agreement with the activation energy of the macroscopic lithium conductivity of a completely crystallised LiAlSiO4 ceramic (E A = 1.07 eV). Here, the pre-exponential factor of τ c decreases continuously with increasing degree of crystallinity, χ. In Fig. 12, we plot the pre-exponential factors of τ g and τ c vs. the degree of crystallinity χ. An important result is that the relaxation times in the single-phase materials, i.e. in the pure glass and in the completely crystallised ceramic, are characterised by pre-exponential factors in the range from 10−13 to 10−14 s. These are typical values for the inverse attempt frequency of ion hopping in the bulk of homogeneous solid electrolytes [33]. However, with increasing amount of a second phase, the relaxation time of the first phase increases, mainly due to an increase of its pre-exponential factor. Or in other

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Fig. 11. Arrhenius plots of relaxation times τg (left plot) and τc (right plot) for glass ceramics with different degrees of crystallinity χ. The symbols are experimental data, and the dashed lines are drawn to guide the eye.

Fig. 12. Pre-exponential factors of the relaxation times for the glassy phase, τg (red), and for the crystalline phase, τc (blue), vs. degree of crystallinity χ. The dashed lines are drawn to guide the eye.

words: With increasing amount of glassy phase, the relaxation due to ion movements in the crystallites becomes slower, and with increasing amount of crystallites, the relaxation due to ion movements in the glassy phase becomes slower. This observation can be explained in a qualitative fashion by considering simple equivalent circuit models of the tip/gap/sample system [34]. In these models, the glassy phase is represented by a resistor R G in parallel with a capacitor C G , and the crystallites by a resistor R C in parallel with a capacitor C C . From macroscopic measurements on a pure glass and on an almost completely

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Fig. 13. Equivalent circuit models for the capacitance relaxation of the tip/gap/sample system on the time scale of ionic movements in the glassy phase. a) Surface resistance is higher than bulk resistance. b) Surface resistance is lower than bulk resistance.

crystallised ceramic [30], we conclude that R C  R G and C G ∼ = C G . In the probed subvolume, these R G C G and R C C C elements are connected in parallel and in series, so that ion conduction pathways exist depending on the amount and the spatial distribution of the elements. Here, we simplify these models by considering only representative ion conduction pathways. On the time scales of τ g , the capacitors C G are discharged through the resistors R G . Since R C  R G , the capacitors C C are not discharged on this time scale, so that the crystalline phase can be represented exclusively by a capacitance C C . A simplified equivalent circuit for one such conduction pathway is shown in Fig. 13a. We assume that mainly R G C G elements are present, some of which are replaced by C C elements, thus representing the case of a small degree of crystallinity. At the sample surface two scenarios are considered: In the first scenario we assume that the surface resistance is higher than the bulk resistance. In this case, we can model the displacement currents by individual vacuum capacitances C V representing the gaps between the end point of the ion current at the surface and the tip. This is shown in Fig. 13a, where we sketch two representative pathways, the right one containing exclusively R G C G elements and the left one containing in addition a C C element. The second scenario includes a highly

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conductive surface region with a resistance lower than R G . On the time scale of τ G , this leads to a short-circuiting of the surface, as illustrated in Fig. 13b. In the first scenario we must realize that in the left pathway the length of the current pathway through the crystallite is small compared to the distance between sample surface and tip. Therefore, the capacitance C C is much larger than the vacuum capacitance C V . In this case, the larger capacitance C C acting in series to C V can be neglected. As derived in [34] the effective relaxation time of NP parallel current pathways is then: τ G = (R G NG ) ((C G /NG ) + C V ) .

(4)

We assume that C G  C V and C G /NG > C V , due to the much higher permittivity of the glassy phase and of the crystalline phase (ε ∼ = 10) as compared to vacuum. Therefore, the relaxation time τ G can be approximated by τ G ∼ = RG C G . Hence in this scenario τ G is roughly independent of the number of crystallites along the pathways, in clear disagreement with our experiments in Fig. 12. In the second scenario the highly conductive surface region causes the whole subvolume to be connected in series to a single vacuum capacitance C V . This single vacuum capacitance is much larger than the individual vacuum capacitances in Fig. 13a. Therefore, in this alternative model, we have to assume that C C < C V and that C G /NG < C V . This results in a blocking of the ion current pathways by the crystalline phase, so that only pathways leading completely through the glassy phase contribute significantly to the overall capacitance relaxation. The number of such parallel pathways, NP , decreases with increasing number of C C -elements. Following a similar approach as above, the effective relaxation time is [34]: τ G = R G (C G + C V NG /NP ) ∼ = R G C V NG /NP .

(5)

This equation implies that the decreasing number of parallel current pathways, NP , with increasing crystallinity results in an increase of the relaxation time τ G . This result is in qualitative agreement with our experiments. An equivalent analysis can be performed on the time scale of the ion relaxation in the crystallites [34]. Again, only for the second scenario, where the surface resistance is lower than R c , we find that the reduction of the number of parallel pathways through the crystalline phase, N P , with decreasing crystallinity χ leads to an increase of τ C , in qualitative agreement with our experimental results. Thus only the equivalent circuit models with a low surface resistance lead to a qualitative agreement with our experimental results for τ G (χ) and τ C (χ) in Fig. 12. The assumption of a reduced surface resistance as compared to the bulk resistance seems plausible, since (i) ions close the surface may be more loosely bound to the glass ceramic matrix and therefore more mobile, and (ii) differences in chemical composition between surface and bulk may lead to a higher surface mobility.

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Of course, the simplified equivalent circuit models presented here can only be considered as a first approach for modeling the TD-EFS results of heterogeneous ion conductors and can, therefore, only be used for a qualitative comparison between experimental and theoretical results. The next step will be the numerical simulation of three-dimensional networks of R G C G and R C C C elements and a comparison of the obtained complex capacitances with our simplified circuits and with our experimental results. In any case, the model analysis shows that the surface conductivity of the studied materials has to be taken into account for the interpretation of TD-EFS measurements.

2.6 The exponential stretch factor β So far we have not given an interpretation of the second fit parameter used to describe the time dependence of the relaxation curves, that is the exponential stretch-factor β in Eq. (3). This factor is typically in a range from 0.5 to 1.0, and not correlated to the time constant τ. In Fig. 14a we show a systematic analysis of the β factors for the same partially crystallized glass ceramic LiAlSiO4 as before, for different levels of crystallization, χ = 13, 42 and 80%. Depending on the sample temperature, either the relaxation processes in the glassy phase (T < 380 K) or the crystalline phase (T > 380 K) are probed. The β factor for each individual relaxation curve is shown, indicating a comparably large variation of around 20% for each sample. Despite the considerable spread of the individual data points we observe consistently that the β factors are systematically lower for the relaxation curves in the glassy phase (with an average value of β glassy = 0.6) than in the crystalline phase (average value of β cryst = 0.8). Figure 14b shows the β factors as averages for each sample system separated for the glassy and crystalline phases. Independent of the level of crystallization, the β factors are always systematically lower for the ion dynamics in the glass phase. Values of β below unity imply that the ion movements in the probed subvolume cannot be described by a simple resistor as shown in the equivalent circuit in Fig. 2. The reason is that on short time scales, the ion dynamics is characterised by correlated back-and-forth hops leading to a frequency dependence of the macroscopic ionic conductivity [33]. This frequency dependence can approximately be described by a Jonscher power law:   ν p σ (ν) = σ dc 1 + ∗ (6) ν with σ dc and ν ∗ denoting the dc conductivity and a characteristic frequency, respectively. The Jonscher exponent p can be considered as a measure for the strength of the backward correlation effects in the ion motion. In our TD-EFS experiments, the back-and-forth hopping motion leads to a stretched exponential relaxation with the stretch parameter β being the lower, the larger

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Fig. 14. a) Individual β values as obtained from the fits for single relaxation curves for glass ceramics with 13, 42 and 80% crystallinity. b) Average β values from relaxation curves of crystalline (red bars) and glassy (blue bars) phase for 0, 13, 42 and 80% crystallinity.

is the Jonscher exponent p. For ionic glasses, typical p values are in the range from 0.6–0.7 [35]. For crystals, the exponent p depends strongly on the d´ımensionality of the conduction pathways [35]. For three-dimensional pathways, the exponent is typically in the range p = 0.6–0.7, for two-dimensional pathways in the range p = 0.5–0.6, and for one-dimensional pathways in the range p = 0.3–0.5. In the case of LiAlSiO4 , the crystalline β-eucryptate phase has the same chemical composition as the glassy phase. In β-eucryptate, the ionic conductivity parallel to the c-axis is much higher than in the perpendicular directions [36]. Thus, the ion conduction pathways are essentially one-dimensional. In contrast, in the disordered glassy phase, the ion conduction pathways are expected to have a higher dimensionality [37]. Consequently, the crystalline phase should be characterised by lower Jonscher exponents p and by

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higher stretch parameters β than the glassy phase, as found in our TD-EFS experiments.

2.7 Local analysis of relaxation dynamics In Fig. 8 we have shown that the observed relaxation strengths depend strongly on the position of the tip. This implies that our TD-EFS measurements provide spatially resolved information on the distribution of glassy phase and of crystallites, i.e. information on the nano- and mesostructure of the material. In order to study this in more detail, we combine the technique of grid spectroscopy with the TD-EFS method. As shown in the sketch in Fig. 15a, an equally spaced grid is predefined, and at each position the time evolution of the frequency shift signal is acquired. In general, the signal will be a mixture of the partial volumes of the two phases, glassy and crystalline, within the probed subvolume (here we neglect the comparably minute contributions from the interfacial areas). Since at room temperature, the ions in the crystalline phase are too slow to contribute to the TD-EFS signal, the relaxation strength should increase with increasing volume fraction of the glassy phase in the probed subvolume. A corresponding measurement is shown in Fig. 15b on a glass ceramic with 42% crystallinity. The 1000 × 1000 nm2 surface area was divided into a 40 × 40 grid of 1600 spectroscopy points. At each position of the tip, the relaxation strength is shown, color coded from red (lower relaxation strength) to blue (higher relaxation strength). The positions of lower relaxation strength are coherent red areas, indicating the presence of a large amount of crystalline phase. The resulting picture in Fig. 15b suggests that crystallites with diameters in the

Fig. 15. a) Schematic of TD-EFS grid measurements, where a time-dependent relaxation curve is obtained at each point of a pre-defined grid on a previously scanned surface topography. b) Experimental data from a TD-EFS grid measurement with a color coding of the relaxation strength. Red areas denote positions with lower relaxation strength, indicating a larger amount of crystalline phase in the probed subvolume.

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range from 100–200 nm exist in the glassy matrix, in good agreement with the results of TEM studies [38]. Thus, for nanomaterials with strong correlations between structural and dynamic heterogeneities, TD-EFS is an ideal tool for obtaining structural information on nanoscopic length scales.

3. Conclusions and outlook We have demonstrated that the TD-EFS technique is capable of extracting quantitative information about the ion dynamics in solid electrolytes. In the case of nanostructured electrolytes, ionic movements in different phases and at interfaces can be much more easily distinguished than in macroscopic electrical spectra. Grid spectroscopy allow visualizations of the time evolution of a nanoscopic polarization as well as of structural features on nanoscopic length scales. We anticipate that the spatial resolution of the TD-EFS method can be further improved by using ultrasharp tips. Thereby, it should be possible to investigate the ion dynamics at individual interfaces. Furthermore, the high sensitivity of the method opens up the possibility to investigate movements of only few ions within the probed subvolume. In particular at low sample temperatures, only a small fraction of all ions in the subvolume carry out hopping movements, and we anticipate that our new method should be capable to detect individual ion hopping events.

Acknowledgement We are grateful to Prof. Harald Fuchs for providing access to the variabletemperature atomic force microscope. Furthermore, we acknowledge financial support of our work by the German Science Foundation (DFG, SFB 458).

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

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Theoretical Description of Ion Conduction in Disordered Systems: From Linear to Nonlinear Response By Steffen Röthel1 , Rudolf Friedrich1 , Lars Lühning2 , and Andreas Heuer2 , ∗ 1 2

Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Str. 9, 48149 Münster, Germany Institut für Physikalische Chemie, Westfälische Wilhelms-Universität Münster, Corrensstr. 30, 48149 Münster, Germany (Received July 2, 2010; accepted in final form September 16, 2010)

Ionic Conductivity / Non-Linear Effects / Fokker–Planck Equation / Computer Simulations The modelling of the ion conduction in disordered systems is analysed from two different perspectives. First, molecular dynamics simulations are employed to extract some basic properties of the hopping dynamics. It turns out that the dynamical processes can be described to a very good approximation as vacancy hopping processes. Second, the information content of nonlinear conductivity experiments, using high electric fields, is elucidated. For this purpose the single-particle dynamics on 1D and 2D model energy landscapes is elucidated numerically and partly analytically. The approaches encompass discrete as well as continuous energy landscapes, yielding complementary results about the dynamics. The impact for the interpretation of experimental data is discussed.

1. Introduction The dynamics of ions in disordered inorganic ion-conductors is a complex multi-particle problem [1–3]. The ion dynamics is strongly correlated with the network properties, first, because it supplies a persistent disordered potential energy landscape for the cations [4] and second, because their local fluctuations promote cationic jumps [5,6]. Another contribution to the multi-particle behavior stems from the interaction among the mobile cations [7–9]. Ion conducting glasses are investigated by various experimental methods, including EXAFS [10,11], NMR [12,13], neutron scattering [14,15] and conductivity spectroscopy [16,17]. Already very early consensus has been reached * Corresponding author. E-mail: [email protected] Z. Phys. Chem. 224 (2010) 1855–1889 © by Oldenbourg Wissenschaftsverlag, München

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that the dynamics occurs via hopping processes and corresponding models have been developed [1,2,18,19]. For example, a quantitative description of conductivity spectra has been formulated considering correlated back jumps [20,21]. This approach is used for the characterization of possible conduction mechanisms [22,23]. However, beyond phenomenological models no first-principles theory of ion conduction is available. Alternatively, one may study simplified models like the random barrier or random energy models [24–27], where the dynamics of a single particle in a disordered energy landscape is discussed. It turns out that, e.g., the frequency-dependence of the conductivity can be nicely reproduced from these models. A further approach to gain knowledge about ion conductors, partially discussed in this work, is via molecular dynamics simulations [5,28–33]. The additional microscopic information can be used to elucidate the mechanism of ion conduction somewhat closer. Thin films of solid ionic conductors are of special importance for practical applications. For thin films and ambient voltage the electric fields become very large so that nonlinear effects become relevant, see e.g. [34–37]. One typically observes an increase of the ionic conductivity with increasing field strength. From the theoretical side previous activities have mainly concentrated on the analysis of 1D systems because to a large extent their properties can be treated analytically [38–41]. The theoretical understanding of nonlinear effects in higher-dimensional systems, however, is still rather limited. In this paper we focus on two key problems related to the theoretical understanding of ion conduction in disordered systems. First, we review some results on molecular dynamics simulations of alkali silicate systems and summarize the new insight about the mechanism of the transport mechanism. Second, we analyze the nonlinear effects for some model systems and elucidate the question about the information content of nonlinear experiments. We concentrate on 2D systems which are mainly analyzed by numerical simulations. Discrete as well as continuous systems are analyzed using rate equations and Fokker– Planck-equations, respectively.

2. Molecular dynamics simulations 2.1 Model systems During the last decade we have analyzed lithium silicate, potassium silicate and mixed alkali glasses. Here we concentrate on (Li2 O) x (SiO 2 ) (1−x) with x = 0.33 and x = 0.5. The temperatures of the simulations are chosen such that the ionic subsystem is in quasi-equilibrium with respect to the disordered frozen silicate matrix. This sets a lower limit for the temperature range accessible to simulations. We have used a BKS-type potential developed by J. Habasaki [42]. Previous studies with this potential have shown good agreement with experimental results for static and dynamic quantities [28,30,43,44]. The molecular dynamics simulations have been performed with a modified version of Moldy [45].

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Fig. 1. A slice of the simulation box for (Li2 O) 0.5 (SiO2 ) 0.5 for T = 640 K. Taken from Ref. [46].

2.2 Site properties Ionic sites have been identified based on the long-time trajectories of the ions. First the simulation box is divided into small cubic cells. Then the probability that a cell is occupied by a lithium ion is extracted from long trajectories. The cells with nonzero counts describe the portion of the system that has been visited by lithium ions. These cells include the lithium sites as well as the connecting paths between them. To identify the ionic sites and to eliminate the paths between the sites, cells with only a few counts are dismissed. With these cells a cluster analysis is performed. Only cells which share a face are grouped into one cluster. The result is illustrated in Fig. 1, showing a snapshot of a slice through the system, reflecting the silicate network and the lithium ions. The white objects are the clusters obtained from the analysis described above, with each small white sphere representing one of the cells. Most of the lithium ions reside inside of one of these clusters, and most of the clusters are occupied by exactly one lithium ion. Of key relevance is the observation that less than 10% of all sites are not occupied, i.e. may serve as new sites for a hopping ion [33,46,47]. Furthermore, it has been observed that an ion belongs to a site

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most of the time (99.5%) [46]. Thus, the transitions between adjacent sites are very fast and long interstitial dynamics between distant clusters can be excluded.

2.3 Vacancy picture Due to the small number of vacant sites the ion dynamics is restricted by the fact that adjacent unoccupied sites are rarely available. Based on these observations it has been speculated that the multi-particle cation dynamics might be also interpreted as a single-particle vacancy dynamics [31,46,48]. Very recently, this hypothesis has been analyzed in great detail [49,50]. Mapping the ion dynamics on a vacancy dynamics one may analyse the individual dynamical processes either in terms of ion hops or vacancy hops. Here we briefly discuss two observables which clearly reveal that the vacancy picture is by far superior to the particle picture. First, we have checked whether the average waiting time of an ion (or vacancy) in a specific site is correlated to the potential energy E tot of that site. E tot is defined as the average energy of an ion residing in this site. As shown in Fig. 2 the correlation is much stronger in the vacancy picture as compared to the ion picture. As argued in [50] the strong interaction of the ions render the site energy unimportant for the dynamics, interpreted in the ion picture. In contrast, in a single-particle picture for vacancies the site energy indeed has a strong influence on the dynamics. This observation is a strong motivation to consider single-particle hopping models in disordered systems where the particle corresponds to a vacancy. Note that the broad distribution of waiting times reflects the strong dynamic heterogeneities in disordered ion conductors.

Fig. 2. The average waiting time for a site in the ion and vacancy picture in comparison to the average potential energy, related to this site. Taken from [50].

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Fig. 3. Electric field strength dependency of the current density j(E) for lithium silicate. Four different trajectories, using cubic as well as tetragonal simulation boxes, are analyzed. Taken from Ref. [51].

Similarly, we have shown [50] that the waiting time distribution for a single site in the vacancy picture is nearly exponential. This is expected for a single particle hopping between different sites. In contrast, the waiting time distribution for the ions is significantly broadened. This reflects the shortage of free sites and thus the multi-particle aspects of the ion picture. The jump probability at a given time not only depends on the height of the barriers but also on the availability of adjacent sites. The latter will fluctuate with time, giving rise to a fluctuating jump rate and thus to a non-exponential local waiting time distribution.

2.4 Field dependence In the linear response regime the diffusion constant is directly related to the response to an external electric field. Of course, by varying the field strength one crosses over to the nonlinear regime with a corresponding change in conductivity. Simulated data for two different temperatures are shown in Fig. 3 [51]. One observes an increase of the nonlinear effects with decreasing temperature and increasing electric field. Interestingly, the comparison of different samples shows that in the weakly nonlinear regime there are major fluctuations between the different samples. They disappear for larger fields. For a broad range the field-dependent current can be written as a power-law with exponents close to 2 (1.8 and 2.2, respectively). Furthermore we checked [51] that the structure of the system slightly starts to change when the conductivity has increased by a factor of approx. 2. Therefore for the fields, which are of experimental relevance, no structural modifications have to be taken into account.

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3. Nonlinear conductivity For sufficiently large samples it is expected that the conductivity remains the same if the sign of the electric field is inverted. Of course, in general this does not need to be the case and has been studied in the context of rectification effects, see [52] and references therein. A very simple model to rationalize the emergence of nonlinear effects is a hopping model with identical sites. In this case the current can be written as [52–54] j(w) ∝ sinh(w)

(1)

Here we have introduced the dimensionless variable w ≡ qβa E/2, denoting the normalized electric field. q is the charge of the ion, a the hopping distance, k B β the inverse temperature and E the applied electric field. The Taylor expansion of Eq. (1) reads j(w) = jlin (w)[1 + σ r w2 /6 + O(w4 )]

(2)

with σ r = 1. jlin (w) is defined as [lim w→0 j(w)/w] · w and denotes the linear contribution to the current. Analogously Eq. (2) can be expressed in terms of the corresponding conductivities σ(w) = σ lin [1 + σ r w2 /6 + O(w4 )] .

(3)

Equation (3) may be also taken as the general ansatz to characterize nonlinear effects in disordered systems. The value of σ r contains the relevant information about the nonlinear conductivity. σ r > 0 indicates an increase of conductivity with electric field and σ r > 1 implies that this increase is even larger than for the case of a homogeneous energy landscape. Note that in the limit of large w one would obtain an exponential increase of the conductivity with field. Typical experiments are performed in the range w = 0.01–0.05 [55]. Thus relevant information is contained in the first nonlinear correction characterized by σ r . Experimentally, σ r is of the order of 102 with a minor temperature dependence (lower temperatures corresponding to larger σ r ), see [55] and references therein. For reasons of comparison we just mention that in our previous work σ r has been denoted a2app /a2 .

4. 1D hopping model 4.1 Model formulation Motivated by the detailed analysis of the energy landscape of disordered ion conductors one may be tempted to describe the ion dynamics as a singleparticle hopping model where the elementary particles are vacancies rather than ions. A sketch of the 1D hopping model with site and barrier disorder is

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Fig. 4. Sketch of the underlying hopping model with binding energies Ui and N sites.

shown in Fig. 4. The binding energy of site i is denoted U i (large U i denotes a deep well). Without an external field the individual hopping rates are denoted γi,± where by definition the plus sign corresponds to a hop from site i to site i + 1 and the minus sign to site i − 1. These rates have to fulfill the detailed balance relation γi,+ /γi+1,− = exp(−β(U i+1 − U i )). Specifically, we use the standard choice γi,± = 1 if U i±1 > U i and γi,± = exp[β(U i±1 − U i )] otherwise. This choice implies the presence of constant barriers between adjacent sites. Furthermore periodic boundary conditions are used. With an additional external field the rates are modified and are denoted Γi,± where by definition the field is oriented such that jumps along the plus direction are favored. Thus, one can write Γi,± (w) = exp(±w)γi,± . The time evolution is characterized by the rate equations (d/dt) p i (t) = − Γi,+ (w) p i (t) − Γi,− (w) p i (t) + Γi−1,+ (w) p i−1 (t) + Γi+1,− (w) p i+1 (t).

(4)

Starting from an arbitrary initial condition the p i (t) will approach a stationary value. In general, the current can be calculated via j(t, w) ∝ p i (t)Γi,+ (w) − p i+1 (t)Γi+1,− (w) .

(5)

Note that the current between all pairs of sites is identical so that the result does not depend on the index i. This property can be used to solve the system of equations analytically for arbitrary N; see Refs. [38–41] for detailed information.

4.2 Key results In Ref. [54] the first nonlinear term of the Taylor expansion of j(w) has been calculated. Here we restrict ourselves to the special case of a point-symmetric energy landscape, corresponding to a strictly antisymmetric dependence of the

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current on the electric field (see Ref. [52] for results without intrinsic pointsymmetry). One obtains for large N (in practice N > 40) σ lin ∝

1 Nb 1

(6)

and σr = N

4(b 1 − b 0 ) b1

using the quantities   1 b0 = γi,−

(7)

(8)

and 

γi,+ b1 = γi,− γi+1,−

 (9)

b 0 and b 1 can be calculated for different types of disorder. In case of a random barrier model for which all U i are constant one obtains b 0 = b 1 and thus σ r = 1. In case of pure energetic disorder we choose a model for which U i is either e/2 or −e/2, both with 50% probability. Then after a short calculation one finds b 0 = (1/4) exp(βe) and b 1 = (3/8) exp(βe). For large N one thus gets σ r = 4N/3 [54]. Most importantly, for large N one has σ r ∝ N independent of the underlying distribution. Therefore, in the 1D case the nonlinear contribution of a finite system can become very large, extending any reasonable value suggested by the experiment. Thus, an important step in the understanding of nonlinear effects is to check the effect of going beyond 1D systems. For a single realization of the disorder the specific value of σ r can be largely different to the average value, calculated above. The reason is that the standard deviation of σ r is of the same order as σ r itself. In practice 1000 up 10 000 different realizations have been used. Mathematically the divergence of σ r suggests that in the thermodynamic limit the function j(w) is not analytical. When is the thermodynamic limit relevant? The third order term σ r w2 /6 is proportional to Nw2 . For a typical experimental value of w ≈ 0.05 the Taylor expansion thus breaks down if N significantly exceeds the very small scale of 102 sites. As explicitly derived in [55] (see also [52] for similar work) one obtains j(w) ∝

1 sinh(w) . N b 0 + (b 1 − b 0 ) exp(−2|w|)

(10)

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Fig. 5. Plot of the ion conduction j(w)/w for different system sizes (varied by powers of 2) for the model with purely energetic disorder and a bimodal distribution of energies, differing by 6k B T . A disorder average has been performed.

Interestingly, this result factorizes into a term, reflecting the homogeneous system without any disorder (sinh(w)) and one term, reflecting the disorder. Without disorder one has b 0 = b 1 = 1. For small w Eq. (10) can be rewritten as   (11) j(w) = jlin 1 + σ 2 |w| + O(w2 ) 0) . with jlin given again by Eq. (6) and σ 2 = 2(b1b−b 1 In Fig. 5 we show j(w)/w vs. w for different values of N, as obtained from an disorder average of the exact solution. Of course, for finite N j(w)/w is analytic around w = 0. One can clearly see how for large N the non-analytic behavior emerges. Qualitatively speaking, for finite N three distinct regimes can be identified. For very small w the ratio j(w)/w has a quadratic w-dependence as expressed by Eq. (3). For somewhat larger w (with an N-dependent crossover) the quadratic behavior transfers into a linear behavior. Finally, for very large w one has an exponential increase, see Eq. (10). On a qualitative level this behavior will be also seen for the 2D systems.

5. 2D hopping model 5.1 Model formulation The 2D case is a straightforward generalization of the 1D model as formulated in Eq. (4). Here we concentrate again on uncorrelated energies on a square lat-

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tice which do not possess any additional symmetry properties. We use the same convention for the definition of rates between adjacent sites as before. Periodic boundary conditions are used for both dimensions. In the general case energy as well as barrier disorder have to be taken into account. However, for reasons of simplicity we restrict ourselves to the case of energy disorder. In particular we concentrate on the case of a Gaussian distribution with standard deviation  g . This choice is motivated by the numerical result that the energies of the different sites are distributed according to a Gaussian [49]. The energy units are always given with respect to k B T . For example  g = 5.5 corresponds to a macroscopic (apparent) activation energy of 18 for k B T = 1, derived from the slope of the linear current in an Arrhenius diagram. The 1D case can be solved analytically because the current is identical for all bonds. This is longer the case for the 2D case. As a consequence, no analytic solution is available so far. Thus we have to resort to numerical solutions. The rate equation can be written in the general form (d/dt)P(t) = T · P(t)

(12)

where the vector P(t) contains the population of all sites at time t and the matrix T the transition elements. In the stationary long-time limit the values for the site population result from the solution of the homogeneous linear system of equations T · Pstat = 0 together with the normalization condition for P. The maximum system size was 1024 × 1024. This corresponds to the solution of more than one million linear equations. Due to the large number of averages over different disorder realizations this problems requires very efficient numerical approaches. Fortunately the matrix T is only sparsely populated. This allows us to use the application of optimized numerical approaches, as implemented in the PARADISO code [56].

5.2 Conductivity In [54] we have reported 2D simulations of a N × No system with a bimodal distribution of energies. No denotes the number of sites in the direction orthogonal to the field. Of course, No = 1 corresponds to the 1D case. After independent variation of N and No the following conclusions were drawn in Ref. [54]: (i) The scaling σ r ∝ N also holds for two-dimensional systems as long as N is significantly larger than No , i.e. one has a quasi 1D system. (ii) σ r strongly decreases with increasing No . Comparing No = 10 and No = 20 one may speculate that at least in this range of parameters σ r ∝ N/No . As a consequence one can expect that for No ≈ N the typical 1D effects, giving rise to the non-analytic behavior, will disappear. In what follows we restrict ourselves to the 2D square system with N o = N. Results for j(w)/w(= σ(w)) are shown for the box distribution with width  b k B T (Fig. 6) and the Gaussian distribution (Fig. 7). In all cases averages have

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Fig. 6. The conductivity j(w)/w vs. w for a box distribution of energies and for system size N = 128. Different values of the box width are used.

Fig. 7. The conductivity j(w)/w vs. w for a Gaussian distribution of energies and for system size N = 128. Different values of the standard deviation are used.

been performed over 105 different realizations. Any w-dependence can be directly regarded as a nonlinear effect. Whereas the linear response value of the conductivity is obtained for small fields, in the large-field regime one observes an exponential increase of the conductivity. In this regime simulations clearly show (see below) that the current is mainly restricted to 1D paths oriented parallel to the electric field. Therefore

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Fig. 8. The same as in Fig. 7 but restricted to a small range of w-values. The standard deviation is  g = 5.5.

one can expect that the 1D result j(w) ∝ exp(w) should hold for large w. In any event, this limit cannot be accessed experimentally. In the range of intermediate w nonlinear effects are much more pronounced for the box distribution. The monotonous decrease for small w directly translates into σ r < 0. Therefore the hopping model with a box distribution of energies does not display the phenomena, seen in experiments. We checked that the same holds for the Cauchy distribution and the Laplace distribution. For the Gaussian distribution the nonlinear effects are hardly visible on the scale, chosen in Fig. 7. In Fig. 8 the data are replotted for the specific case  g = 5.5. Now one can clearly see the increase of conductivity with field. For w ≈ 0.01 this effect is very small (0.2%) whereas for w ≈ 0.1 the nonlinear effects is already significant (4%). Generally speaking three different w-regimes can be identified for the Gaussian distribution. First, for very small w a parabola can be fitted to σ(w) vs. w. Here this corresponds to roughly w < 0.01. For somewhat larger w a weaker w-dependence (approximately linear) is observed. Finally, as already mentioned the large w-regime seems to display an exponential dependence. Interestingly, this scenario is fully compatible with the previous 1D results. Note that in the experimental situation reliable nonlinear data are typically obtained above w = 0.01. This renders a precise determination of σ r somewhat difficult and only some effective value can be obtained. In practice it turns out that for an upper limit of w = 0.05 the apparent value of σ r (obtained from a brute force fitting of a parabola) is roughly smaller by a factor of 2 as compared to the theoretical values at small w.

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Fig. 9. σlin, i vs. σnonlin, i for  g = 5.5 and N = 128, using 100 different energy realizations.

5.3 Linear vs. nonlinear contributions Numerical simulations of disordered systems typically involve a disorder average. Conceptually, this average can be split into two parts. First, for a fixed realization of N 2 energies, drawn from a Gaussian distribution, one can elucidate the effect of a spatial redistribution of these energies on the sites of the lattice. The resulting observable will be marked by an index i. This step corresponds to a pre-averaging procedure. Second, one can redraw N 2 energy values from the same distribution several times. Before analysing the nature and the information content of this preaveraging concept we ask about the relevant observable to capture the nonlinear contribution. For the description of the disordered 1D system the observable σ r has been employed as a convenient dimensionless number. Here we argue that from some other considerations the variable σ nonlin ≡ σ r σ lin

(13)

has a more relevant meaning. This product just corresponds to the total nonlinear contribution, as seen in Eq. (3). To analyse the question about the respective relevance we have calculated j(w) for 100 different energy realizations, using  g = 5.5 and N = 128. For each realization we have calculated 1000 different spatial rearrangements and calculated the average function j(w)i . The resulting values σ lin, i and σ nonlin, i are correlated in Fig. 9. It turns out that apart from the region of very small σ lin, i the data are basically uncorrelated despite a large variation of σ lin, i . As a direct consequence of the relation σr,i = σ nonlin, i /σ lin, i the value of σr,i is strongly correlated with σ lin, i . We have checked that a similar scenario also holds for the 1D case.

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5.4 Spatial and size aspects In the thermodynamic limit one expects for self-averaging observables such as the current that a spatial rearrangement of a fixed set of energies yields the same properties as a set of realizations with randomly chosen new energies (for a fixed distribution). Stated differently, the pre-averaging as defined above is equivalent to the disorder average. We note in passing that this is not true for open boundary conditions as discussed in [52]. However, for finite N deviations are possible so that one can identify the relevance of a specifically chosen set of energies as compared to the spatial arrangement. This is one example where finite-size systems contain important information about the underlying mechanisms. The subsequent analysis is performed for N = 128 and a Gaussian distribution of  g = 5.5. This energy scale is relevant for two reasons. First, as discussed further below for this set of energies the apparent activation energy of the linear conductivity (relative to k B T ) is close to typical experimental values. Second, on a more qualitative level the distribution of energies of the different sites, as determined numerically, is also significantly larger than k B T [49]. For general grounds one expects significant sample-to-sample fluctuations which requires an extensive averaging over different realizations and a finitesize analysis. One can estimate the system size Nlarge for which finite-size effects would disappear. For a Gaussian distribution the average energy, dominating the thermodynamic properties, can be easily calculated to be −β2g [57]. The number of sites within an interval k B T around this energy is given by 2 Nlarge (2π2g )−1/2 exp(−2g /2). This number should be significantly larger than unity to avoid sample-to-sample fluctuations and corresponding finite-size effects. For  g = 5.5 this would result in Nlarge ≥ 104 . This value is significantly beyond the system size which can be treated numerically. The resulting values for σ r for one arbitrarily chosen distribution of energies is shown in Fig. 10. Whereas the average is positive one can clearly see the enormous distribution of σ r -values which emerges by rearranging the energies on the lattice. A more quantitative analysis shows that the average variance upon pre-averaging is 99% of the variance after a complete disorder average. Thus, pre-averaging of one fixed set of energies is enough to cover all relevant values of σ r . Interestingly, the opposite behavior is observed for σ lin . The respective distribution for four randomly chosen energy realizations upon spatial rearrangement is shown in Fig. 11. One can directly see that the variation of σ lin upon spatial rearrangement is very small as compared to the variation when changing the specific realization of energies. Here the average variance after pre-averaging is only 1% of the total variance after the complete disorder average. Thus, for the linear conductivity a system size of N = 128 is still far away from the thermodynamic limit.

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Fig. 10. Distribution of σr for structures with an identical set of energies, drawn from a Gaussian distribution with  g = 5.5, but different spatial arrangements. The average is indicated by the broken line.

Fig. 11. Distribution of σlin for structures with an identical set of energies, drawn from a Gaussian distribution with  g = 5.5, but different spatial arrangements. Four different examples are shown, each characterized by a different set of energies.

One possible interpretation of these results is that the value of the linear conductivity is mainly governed by properties of individual sites whereas the nonlinear conductivity results from a complex interplay of many sites. This argument can be made more explicit. In Fig. 12 one observes a strong correlation of the linear conductivity with the lowest of the N 2 energies, denoted U 0 . Thus, this single site has a strong impact on the linear conductivity. This is consistent with the observation that the rearrangement of energies only has a minor influence on the resulting conductivity. We checked that for N = 512 the de-

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Fig. 12. Correlation of σlin, i with the lowest energy U0 . Parameters as in Fig. 9.

Fig. 13. Correlation of σnonlin, i with the lowest energy U0 . Parameters as in Fig. 9. Included is a cubic fit.

pendence on U 0 is significantly weaker. Of course, in the thermodynamic limit this correlation has to disappear. In contrast, σ nonlin only has a very weak dependence of U 0 ; see Fig. 13. This is also in agreement with the above interpretation that the nonlinear effect is based on cooperative effects rather than on single-site properties. Interestingly, as also shown in Fig. 13, one observes a significant decrease of σ nonlin to negative values for small values of U 0 . This is compatible with the observation that for box-like distributions where no particular low-energy sites are present, the nonlinear effects are negative. The opposite limit just reflects the fact that there the factor σ lin approaches zero for large U 0 . Simulations also have the advantage to visualize the resulting conductivity patterns. More specifically in Fig. 14 we highlight those bonds for which a significant current is observed. Already for a value of w as small as 0.01 the

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Fig. 14. Visualization of the current pattern of the individual bonds for a N = 128 system with w = 0.01 and w = 5, respectively, and  g = 5.5. The dots denote the 10 lowest energies.

Fig. 15. Visualization of the nonlinear contribution to the conductivity, as found on the individual bonds. Positive contributions are scaled (black arrows), negative contributions are unscaled (grey arrows). The parameters are the same as in Fig. 14. The dots denote the 10 lowest energies.

directions of these bonds show a preferred alignment parallel to the external field. Of course, for very large values of w one observes quasi 1D paths. Obviously the regions of maximum current are not explicitly related to near-by sites with a low energy. In a next step we identify the nonlinear contributions via j(2w) − 2 j(w) for w = 0.01. For small w this is proportional to σ nonlin . Interestingly, the patterns of the positive contributions are basically identical to that of the linear current (which is close to the total current); see Fig. 15. This is in agreement with the 1D scenario where the nonlinear contribution is always positive and propor-

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Fig. 16. σr vs. 1/N after disorder average for  g = 6.

Fig. 17. σr vs.  g after disorder average for N = 128.

tional to the total current. A very different structure is observed for the negative contributions. They are arranged in clusters which again are not correlated with the positions of the lowest energies in this energy realization. The above results have been obtained for a system size of N = 128. For comparison with the experimental system it is important to study the system size dependence as shown in Fig. 16. One can see that σ r strongly depends on N but remains positive also in the thermodynamic limit. The reason for the emergence of the finite-size effects have been already discussed further above.

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Finally, we study the dependence of σ r on  g ; see Fig. 17. Of course, for small  g one has to approach the limit σ r = 1. After some intermediate energy range the nonlinear effects exponentially increase with increasing value of  g . This observation is very promising because in the experimental case σ r is of the order 102 . This order of magnitude is realized for the largest values of  g which, as discussed above, is a reasonable choice to reproduce experimental apparent activation energies.

6. 2D model for continuous dynamics In the following we shall consider the model of a charged particle moving under the influence of thermal noise in a disordered force field in a time dependent electric field. Our treatment is based on a derivation of an exact expression for the nonlinear conductivity and approximate evaluation of this nonlinear conductivity based on perturbation theory. Perturbation theory becomes exact in the case of high external electric fields. Furthermore, a numerical solution of the Fokker–Planck equation is presented. A comparison between analytical and numerical results shows that the results based on the lowest order perturbation theory already yields a correct qualitative description of the nonlinear conductivity. The treatment of the diffusive motion of particles in a disorder potential and an external field by means of a Fokker–Planck equation has been adressed by various authors (for a review we refer the reader to [40]). An exact expression for the static conductivity for the one-dimensional case has been given by Scheidl [58], and Le Doussal and Vinokur [59], using the well-known expression for the current density for particles moving in a potential U(x) (see e.g. Risken [60]). The instanton method has been applied to the one-dimensional problem by Lopatin and Vinokur [61]. However, it seems that the case of a time dependent external field has not yet been considered so far.

6.1 Fokker–Planck Equation We consider the diffusive motion of an overdamped particle of charge q in a force field, which is composed of several parts. The first part is a disordered force field F(x), the second part is a force due to a time dependent external electrical field E(t) and the third part are thermal fluctuations η(t). The corresponding Langevin equation takes the form α x˙ = F(x) + q E(t) + η(t) ,

(14)

where α denotes the viscous friction constant. Thermal fluctuations are taken into account by δ-correlated Gaussian white noise η(t) η i (t) = 0,

η i (t)η j (t ) = 2α2 D δ ij δ(t − t ) ,

(15)

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where the diffusion constant D is connected via the Einstein relation D = k B T/α to the temperature. The statistical treatment of the Langevin equation (14) is based on the corresponding Fokker–Planck equation ∂ f(x, t) = [L 0 + L D ] f(x, t) . ∂t

(16)

Thereby, the Fokker–Planck operators are defined as follows: 1 L 0 = − q, E · ∇ x + DΔ x , α 1 L D = − ∇ x · F(x) . α

(17)

The statistical properties of the disorder field F(x) are contained in the characteristic functional   d  Z[u] = ei d x u(x)·F(x) . (18) Basically, this means that the correlation matrices

Fi (x + r)Fj (x) = −

δ2 Z[u] δu j (x) δu i (x + r)

(19)

are known (d denotes the dimension of the space). In the case of homogeneous statistics the correlation matrices only depend on the distance r. For Gaussian statistics all statistical information is encoded in the two-point correlation function.

6.2 Disorder averaged generalized Fokker–Planck equation Following the well-known Mori–Zwanzig projection operator formalism [62], we introduce a decomposition of the probability density into a disorder averaged quantity h(x, t) and deviations g(x, t): f(x, t) = h(x, t) + g(x, t),

h(x, t) =  f(x, t) .

(20)

Using the standard projection methods (see e.g. [62]) one arrives at the following time-evolution equation for the disorder averaged probability density h(x, t): 1 ∂ h(x, t) = L 0 h(x, t) + 2 ∂t α



t d

dx



dt

−∞

× ∇ x · F(x)G(x, t; x , t )∇ x · F(x ) h(x , t ),

(21)

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where the kernel G(x, t; x , t ) is the solution of the inhomogeneous equation

∂ − L 0 − L D G(x, t; x , t ) + L D G(x, t; x , t ) = δ(x − x )δ(t − t ). ∂t (22) We are interested in the disorder averaged current density  j(x, t) . From the disorder averaged Fokker–Planck equation (21) we see that the current density is given by 1 h  j(x, t) = q E(t)h − 2 α α



t d

dx





dt F(x)G(x, t; x , t )∇ x · F(x ) .

−∞

(23) We remind the reader that the kernel G is a fluctuating quantity. Furthermore, we can assume that the disorder averaged probability density h(x, t) is constant with respect to space and time. The central quantity, which determines the electric current, is the propagator G(x, t; x , t ). It can be calculated by perturbation theory in powers of the disorder field F(x, t).

6.3 Weak disorder In the following we shall consider the limiting case of weak disorder. In this approximation the kernel G 0 (x, t; x , t ) can be approximated by the propagator of the Fokker–Planck operator L 0 :

∂ (24) − L 0 G 0 (x, t; x , t ) = δ(x − x )δ(t − t ) ∂t In this case, G 0 can be calculated explicitly since it is the propagator of a Brownian particle moving in a time dependent, spatially homogeneous external field. For the case of periodic boundary conditions it takes the form G 0 (x, t; x , t ) = G 0 (x − x , t − t )  1



2

eik·[x−x −e(t,t )] e−Dk (t−t ) , = Θ(t − t ) V k 2π nj, Lj

kj =

n j = 0, ±1, ±2, . . . ,

V = L 1 L 2 L 3,

(25)

where we have defined q e(t, t ) = α

t



dt

E(t

). t

(26)

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As a consequence of our approximation only the correlation matrix of the disorder force field enters the expression for the disorder averages.

6.4 Statistical characterization of the disorder potential In the following we shall assume that the disorder force field can be obtained from a disorder potential U(x), F(x) = −q∇ x U(x), which can be represented by a Fourier series in a region with periodic boundary conditions  U k eik·x , U k* = U −k . (27) U(x) = k

Now we specify the statistics of the coefficients A k in the disorder ensemble. The coefficients A k are statistically independent for each wave vector k and obey Gaussian statistics with zero mean and standard deviation D k : U k = 0,

U kU k = C 2k δ k −k .

(28)

This leads to a homogeneous and isotropic statistic where the correlation function of the disorder potential  C 2k eik·r , (29) C (U ) (r) = U(x + r)U(x) = k

only depends on the distance r = |r| due to the spherical symmetry. Furthermore, the correlation function (29) is periodic with respect to n · L with n j = 0, 1 and the relationship C (U ) (n · L − r) = C (U ) (r) reveals a mirror symmetry with respect to r = n · L/2. Its Fourier transform reads ) ˜ U(k ˜ ) = C 2k δ k −k . C˜ (U = U(k) k

(30)

C(r) Furthermore, we can define the quantity R(r) = C(0) . This function is exhibited in Fig. 19 for a typical one dimensional potential, shown in Fig. 18. The correlation matrix of the disorder force field is defined as

C ij(F) (r) = Fi (x + r)Fj (x) = −

∂ ∂ (U ) C (r) , ∂r i ∂r j

yielding for (29)  C ij(F) (r) = k i k j C 2k eik·r .

(31)

(32)

k

It has the general form C ij(F) (r) =

2 (U )

1 ∂C (U ) (r) ∂ C (r) 1 ∂C (U )(r) x i x j − , δ ij − r ∂r ∂r 2 r ∂r r2

(33)

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Fig. 18. Realization of a one dimensional disorder potential U(x) with the Fourieramplitudes 2 Uk = C 0 e−βk /2+iϕk for two different values of β (β = 10−5 , 5 × 10−4 ). The disorder is generated by the randomly choosen phase ϕk .

Fig. 19. Correlation function R(x) for β = 10−5 , 5 × 10−4 obtained as an ensemble average and a spatial average.

where the last fraction of the second term is the projection operator r r /r2 . For the further calculations it is convenient to introduce the correlation vector

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∂ C i(F) (r) = Fi (x + r)∇ x · F(x) = − C ij(F) (r) , ∂r j  (F) 2 2 ik·r C j (r) = −ik j k D k e .

(34)

k

Its general expression reads 3 (U )

∂ C (r) (d − 1) ∂ 2 C (U ) (r) 1 ∂C (U ) (r) x i + − . C i(F) (r) = ∂r 3 r ∂r 2 r d−1 ∂r r

(35)

Due to symmetry, it is a vector pointing in the direction of r.

6.5 Numerical solutions We have performed direct numerical solutions of the Fokker–Planck equation (16) based on a pseudo-spectral method. Thereby, we have considered the onedimensional case (2048 grid points) as well as the two dimensional case (256 × 256 grid points). These calculations allow us to directly evaluate the local current q j(x, t) = [F(x) + E(t) f(x, t) − D∇ f(x, t)] (36) α For alternating fields E(t) = E0 cos ωt,

E0 = [E 0 , 0]

(37)

we are able to determine the spatially resolved spectral decomposition of the current   jn (x, ω) cos(nωt) + jn

(x, ω) sin(nωt) j(x, t) = (38) n=1

A spatial average yields the nonlinear conductivities   σ n (ν) cos(nωt) + σ n

(ν) sin(nωt) E 0 jx =  n=1

=

σ n (ν) cos (nωt − ϕn (ν))

(39)

n=1

We have evaluated the contirbutions to the sums up to the order n = 3. It is important to note that the oscillations with 2ω do not contribute to the spatially averaged current, due to the presence of the reflection symmetry. The angular frequency is denoted by ω = 2πν. It is interesting to compare the results of direct numerical solutions with the results obtained in lowest order perturbation theory. We remind the reader that the results of perturbation theory become valid for the limiting case of weak

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disorder. Additionally, the higher order contribution to the lowest order perturbation theory tend to zero for the limiting case of infinite frequency ω of the electric field. In the following we shall summarize our results obtained for time constant and time periodic external fields.

7. Results for the continuous dynamics 7.1 Time constant electric field Let us first consider results obtained for the two dimensional case in the presence of a constant external electric field. Figure 20 exhibits a comparison between direct numerical simulations and lowest order perturbation theory. We observe that the perturbation theory yields qualitatively similar results, which become exact in the high field case. Let us consider the results of perturbation theory in some more details. According to the time-evolution Eq. (21) the disorder averaged probability density h(x, t) can be considered as stationary and homogeneous for homogeneous statistics of the disorder force field. This yields for the disorder averaged cur-

Fig. 20. Nonlinear conductivity σ(E) obtained from a direct numerical simulation of the Fokker– Planck equation for the two dimensional disorder potential. The exact nonlinear conductivity is compared to the one obtained from the approximate expression in lowest order perturbation theory. The approximation becomes exact for the high elecric field case.

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Fig. 21. Electric currents in a disorder potential for various values of the electric field strength. The field strength increases from top to bottom. The current is located near the minima of the potential field U(x) and becomes delocalized with increasing external field.

rent density h qh E− 2  j = α α



t dt G 0 (r, t − t )C (F) (r).

d

dr

(40)

−∞

Using the Fourier transform we obtain with an arbitrary C (F) (r) for the current density  j =

h C˜ k(F) qh , E+ α α k iqk · E − αDk2

(41)

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and specifically with (34)  j =

h  k2 D2k qh E. E− D2 k4 α qαE2 k 1 + qα22(k·E) 2

(42)

As is well known, in the one dimensional case there exist an exact expression for the current I [60] I = σ(E) E,

(43)

where the conductivity depends in a nonlinear way on the electric field: L L 0 dr e−qEr/(αD) qh σ(E) = .   α L dr L dx e[U(x+r)−U(x)−qEr]/(αD) 0

(44)

0

The trivial case with a constant potential U(x) yields the ohmic conductivity σ Ohm = qh/α,

(45)

where the corresponding current I = σ Ohm E = hqE/α according to the Ohm’s law behaves linearly to the electric field E. Here the current density can be viewed as the product of the density ρ = h and the speed v = qE/α of the charged particles. Figure 20 exhibits a comparison between numerical results (2d case) and the results of the perturbation theory. As expected the nonlinear conductivity σ(E) shows significant differences at low values of the electric field. However, the qualitative behaviour is reproduced by lowest order perturbation theory. The electric currents in a disorder potential for increasing values of the electric field strength are exhibited in Fig. 21.

7.2 Time dependent field We consider now the case where we have homogeneous and isotropic statistics of the disorder force field as well as an oscillating electric field E(t) = E0 cos ωt. Lowest order perturbation theory yields the disorder averaged current density qh h  j(t) = E(t) − 2 α α



t dt G 0 (r, t, t )C (F) (r).

d

dr

(46)

−∞

Fourier transform with an arbitrary C (F) (r) again leads us to  qh h 

2

 j(t) = dt eik·e(t,t ) e−Dk (t−t ) C˜ k(F) , E(t) − 2 α α k t

−∞

(47)

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Fig. 22. Frequency dependent conductivities σ1 (ω), σ1

(ω), σ1 (ω) and the phase shift ϕ1 (ω) obtained from direct numerical simulations (solid lines) and from lowest order perturbation theory (dotted lines). The approximation becomes exact for high frequencies.

where the antiderivative (26) of the electric field becomes oscillating e(t, t ) =

 q E0  sin ωt − sin ωt . αω

(48)

We can obtain a further representation of the current density by taking into account that the Green’s function has the form   (49) G 0 r − e(t, t ); t − t

which leads us to qh h  j(t) = E(t) − 2 α α



t dt G 0 (r; t − t )C (F) (r + e(t, t )).

3

dr

(50)

−∞

Next, we perform a power series expansion with respect to the electric field. To this end we notice that e(t, t ) = q

   E0  sin ωt 1 − cos ω(t − t ) + cos ωt sin ω(t − t ) αω

(51)

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Fig. 23. Frequency dependent conductivities σ3 (ω), σ3

(ω), σ3 (ω) and the phase shift ϕ3 (ω) obtained from direct numerical simulations (solid lines) and from lowest order perturbation theory (dotted lines). The approximation becomes exact for high frequencies. A qualitatively similar behaviour is observed in experiments [55].

This formula explicitly shows the presence of the higher harmonics to the mean current. Again, we compare the above defined nonlinear, frequency dependent conductivities σ n (ν), σ n

(ν) obtained from the direct numerical simulation and the results of the perturbation theory. Figures 22 and 23 summarize the behaviour of σ 1 (ν), σ 1

(ν) and σ 3 (ν), σ 3

(ν). Again, the frequency behaviour is qualitatively reproduced by perturbation theory and coincides with direct numerical solutions in the case of high frequencies. We want to point out that a similar quantitative behaviour of the nonlinear conductivities are well-known from experiments [55]. Especially, one observes the existence of negative conductivities σ 3 (ν) < 0 for certain ω-regimes. It is instructive, to consider the current distribution j(x, t). We have performed numerical simulations for a square of length L = 1 with 256 × 256 grid points. The maximum of the potential U(x) has been taken to be |U(x)|/k B T = 5, the frequency of the alternating field was ν = 10−1 Hz, whereas the field strenght has been taken to be E x L/k B T = 200. We have performed a spectral decomposition of the field of the electric current, according to Eq. (38). Figures 24–26 exhibits the first three components ji (x), i = 1, ..., 3. It is obvious that the higher harmonics contain contributions, which predominantly stem

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Fig. 24. Spatially resolved current distribution j1 (x), defined in Eq. (38). The background field is a contour plot of the disorder potential.

Fig. 25. Spatially resolved current distribution j2 (x), defined in Eq. (38).

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Fig. 26. Spatially resolved current distribution j3 (x), defined in Eq. (38).

Fig. 27. The distribution ∇ · j1 (x). The dipolar structures are the sources of the current j1 (x).

from regions, where the potential has a local minimum. Further information is obtained from contour maps of the quantity ∇ · ji (x), see Figs. 27–29. These maps indicate the existence of dipolar sources generating the currents. A com-

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Fig. 28. The distribution ∇ · j2 (x).

Fig. 29. The distribution ∇ · j3 (x). The dipolar sources have an opposite orientation as compared to the sources of the current j1 (x). This is related to the fact that the conductivity σ3 (ω) is negative.

parison of the dipols corresponding to the currents j1 (x) and j3 (x) shows that the dipols may have opposite orientations. This is related to the existence of the negative conductivities mentioned above, which are also well-known from experimental investigations [55].

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8. Conclusions In the main part of this work we have presented a thorough analysis of dynamic processes under large electric fields. Analytic expressions can be derived for the stationary 1D case, for high frequencies as well as for large fields (corresponding to small disorder). The other regimes have to be described via numerical simulations. The type of current paths in the discrete and continuous description looks quite similar. Whereas at low fields paths form which are only slightly oriented along the field one observes a dramatic elongation for larger fields. It is an open question whether one can identify simple markers predicting the location of these paths. Interestingly they tend to cluster in space. In contrast, the nature of the nonlinear contributions seem to be different between both approaches. In the discrete case the main positive contributions are still related to the current paths whereas the negative ones form larger clusters. In contrast, in the continuous case the nonlinear current j3 (x) is less localized and less correlated with linear current j1 (x). The simulations of the Gaussian discrete energy landscape suggest under which conditions σ r  1 can indeed be observed. First, the temperature has to be quite small, corresponding to a large value of  g . Second, the site energy disorder should be, on the one hand, sufficiently Gaussian-type (and not boxtype), i.e. have a few energetically low-lying sites, in order to observe σ r  1. On the other hand, the lowest value should not be too small because otherwise σ r approaches again a very small value. One could image that in a real energy landscape very low sites cannot exist due to the well-defined local order in glasses. The experimental observation σ r  1 just points into this direction. In any event, this work has shown that the mere experimental observation σ r  1 already excludes many models of disorder. Several interesting questions remain open for future work. What is the possible effect of interaction if more than one particle is present? What is the model which is closest to the experimental situation? What happens if the disorder is no longer static? Are there other regimes where analytical predictions are possible? Interestingly, there is one region where additional analytical results can be obtained, namely for very high (but finite dimensions) where corrections to the trivial mean-field limit (σ r = 1) can be obtained. Work along this line will be published elsewhere.

Acknowledgement We greatly acknowledge the support of the DFG via the SFB 458 as well as fruitful and helpful discussions with many of its members. In particular we would like to thank K. Funke for his outstanding personal engagement to keep this SFB running for one decade and B. Roling for the stimulating scientific interaction during this time. Furthermore A. H. thanks R. D. Banhatti, D. Did-

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dens, M. Kunow, H. Lammert, and A. Maitra for the fruitful collaborations on the field of ion conduction during this time.

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42. J. Habasaki and I. Okada, Mol. Simul. 9 (1992) 319. 43. J. Habasaki and I. Okada, Mol. Simul. 10 (1993) 19. 44. A. Heuer, M. Kunow, M. Vogel, and R. D. Banhatti, Phys. Chem. Chem. Phys. 4 (2002) 3185. 45. K. Refson, Comput. Phys. Commun. 126 (2000) 310. 46. H. Lammert, M. Kunow, and A. Heuer, Phys. Rev. Lett. 90 (2003) 215901. 47. J. Habasaki and Y. Hiwatari, Phys. Rev. B 69 (2004) 144207. 48. J. C. Dyre, J. Non-Cryst. Solids 324 (2003) 192. 49. H. Lammert, R. D. Banhatti, and A. Heuer, J. Chem. Phys. 131 (2009) 224708. 50. H. Lammert and A. Heuer, Phys. Rev. Lett. 104 (2010) 125901. 51. M. Kunow and A. Heuer, J. Chem. Phys. 124 (2006) 214703. 52. M. Eimax, M. Koerner, P. Maass, and A. Nitzan, Phys. Chem. Chem. Phys. 12 (2010) 645. 53. N. Mott and E. Davis, Electronic Processes in Non-Crystalline Materials, Clarendon, London (1979). 54. A. Heuer, S. Murugavel, and B. Roling, Phys. Rev. B 72 (2005) 174304. 55. B. Roling, S. Murugavel, A. Heuer, L. Luehning, R. Friedrich, and S. Röthel, Phys. Chem. Chem. Phys. 10 (2008) 4211. 56. O. Schenk and K. Gaertner, J. Future Gener. Comput. Sys. 20 (2004) 475. 57. A. Heuer, J. Phys.: Condens. Matter 20 (2008) 373101. 58. S. Scheidl, Z. Phys. B 97 (1995) 345. 59. P. L. Doussal and V. Vinokur, Physica C 254 (1995) 63. 60. H. Risken, The Fokker–Planck Equation, Springer, Berlin (1989). 61. A. V. Lopatin and V. M. Vinokur, Phys. Rev. Lett. 86 (2001) 1817. 62. R. Zwanzig, Nonequilibrium Statistical Mechanics, Oxford University Press, New York (2001).

First and Second Universalities: Expeditions Towards and Beyond By K. Funke1 , ∗, R. D. Banhatti1 , D. M. Laughman1 , L. G. Badr1 , 2 , ˇ M. Mutke1 , 2 , A. Santi´ c1 , 3 , W. Wrobel1 , 4 , E. M. Fellberg1 , and C. Biermann1 1 2 3 4

Institute of Physical Chemistry, University of Muenster, Corrensstrasse 28–30, 48149 Muenster, Germany NRW Graduate School of Chemistry, University of Muenster, Corrensstrasse 28–30, 48149 Muenster, Germany NMR Center, Ruder Boˇskovi´c Institute, Bijeniˇcka c. 54, Zagreb, Croatia Faculty of Physics, Warsaw University of Technology, Koszykowa 75, Warsaw, Poland (Received July 8, 2010; accepted in final form September 14, 2010)

Ion Conduction / Self-Diffusion / Model / Materials Science Understanding the mechanisms of translational and localised ionic movements in disordered materials has seen intense activity spanning several decades. This article attempts to convey a concise overview of our contribution to this field over the period from 2005 to 2010 and to place it in its broad context.

1. Introduction In condensed matter, disorder is a prerequisite for atomic and ionic transport. Different levels of disorder have been considered in the evolving scheme presented in the preface to this issue. In that scheme, level two is characterised by individual point defects moving randomly from site to site, thus serving as vehicles for translational motion of atoms or ions. By contrast, structural disorder is the key feature of level three, where mobile ions no longer experience a static energy landscape, but rather single-particle potentials that are time-dependent and non-periodic. As a consequence, their movement is not random, but highly correlated, see Refs. [1–6]. In the field of solid state ionics, which comprises both basic science and high-tech applications, materials with high ionic conductivities are of prime interest. As ionic conduction is facilitated by structural disorder, these materials generally belong to level three. The striking difference between ion conductors * Corresponding author. E-mail: [email protected] Z. Phys. Chem. 224 (2010) 1891–1950 © by Oldenbourg Wissenschaftsverlag, München

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Fig. 1. Scaled representation of experimental and model conductivities (circles and solid line, respectively) of 0.45 LiBr·0.56 Li2 O·B2 O3 glass [10,18]. Here, σ (0) ≡ σDC . This curve is representative of a large number of disordered ion conductors which largely differ in their structures (first universality). Its slope increases continuously, slowly tending towards unity.

belonging to levels two and three is best seen in their frequency-dependent conductivities. At level two, the random walk of the mobile ions or ionic defects results in a conductivity which is constant up to microwave frequencies. An example is AgBr at 200 ◦ C [7]. On the other hand, level-three materials display conductivity spectra that are not constant, but increase with frequency, often over many decades on the conductivity and frequency scales [8–10]. Ion-conducting materials with quite different kinds of disordered structures, see below, have been found to show an unexpected degree of similarity in their conductivity spectra. In particular, two surprising “universalities” have been detected. One of them, the “first universality”, is considered a fingerprint of activated ionic hopping along interconnected sites [11–13], while the other, the “second universality”, seems to reflect non-activated, strictly localised movements of the ions [14–17]. The former is observed at sufficiently high temperatures, while the latter is found at sufficiently low ones, e.g., in the cryogenic temperature regime. The two universalities are defined on the basis of specific properties of experimental data that are usually taken in the so-called impedance frequency regime, below a few MHz. The meaning of the term “first universality” is visualised in Fig. 1 [10,18]. In most ion conducting materials with disordered structures, the scaled ionic conductivity, σ S = σ/σ DC , when measured in the impedance regime at different temperatures, is found to be a unique function of scaled angular frequency,

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Fig. 2. Low-temperature conductivity isotherms of 0.30 Na2 O·0.70 B2 O3 glass, exhibiting the linear frequency dependence and the essentially absent temperature dependence that are characteristic of the second universality.

ωS = ω/ωO . Here σ DC denotes the low-frequency limit of the conductivity, and ωO is the angular frequency marking the onset point of the dispersion, see Sect. 3.2. The surprising statement made by the first universality is that very much the same function or “master curve” is obtained for a large number of ion conductors which largely differ in their structures. These include level-three crystals [10,19], glassy [10,18,20] and polymer electrolytes [21,22], and even ionic melts [23] and liquids [24,25]. Figure 2 exemplifies the meaning of “second universality”. This term expresses the ubiquity of an a priori unexpected low-temperature phenomenon, also known as Nearly Constant Loss (NCL) behaviour, in ionic materials with disordered structures. In the NCL effect, which was detected by Nowick et al. in 1991 [26], the dielectric loss function, ε , is virtually independent of both frequency and temperature. Note that this constant dielectric loss corresponds to an electric conductivity that is directly proportional to frequency and temperature-independent, as seen in Fig. 2. In 2005, our understanding of the phenomena was as follows. A model formalism had been created, providing explanations for the first universality, as in Fig. 1, and for the NCL-type effect as observed at radio and microwave frequencies [10,21,27]. However, our attempt at modelling the second universality, as in Fig. 2, was still in its infancy. Carefully designed expeditions towards and beyond the universalities have, therefore, been undertaken in the period from 2005 to 2010. New manifestations of the first universality have been discovered, for instance in frequency-dependent fluidities of ionic liquids and in correlated rotational diffusion.

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In the other terra incognita, our endeavour was to find a connection that would take us from the known high-frequency NCL-type effect to the lowtemperature second universality. A new understanding has now emerged on the basis of information obtained from a systematic analysis of conductivity spectra measured at cryogenic temperatures. These findings are reported in the following sections.

2. Broadband conductivity spectroscopy 2.1 Basics The two universalities pose questions regarding their physical origin in terms of the movements of the mobile ions. To find answers to these questions it is important to trace the development of the ion dynamics from very short times to long ones, at different temperatures. This corresponds to a broad range of frequencies. Therefore, it is imperative not to restrict conductivity measurements to the usual impedance regime, but to include radio, microwave and infrared frequencies as well. Conductivity spectroscopy thus spans more than 17 decades on the frequency scale, from less than 1 mHz up to more than 100 THz, see Fig. 3 [28–30]. Electrodes are used only in the impedance regime. Guided electromagnetic waves are employed at radio and microwave frequencies, and free ones in the (far) infrared. Although a number of different experimental set-ups are required to cover the entire frequency range, the general procedure towards the electrical characterisation of materials is essentially the same in any part of the spectrum. In fact, the frequency-dependent complex conductivity, σˆ (ν), is always determined by measurement of amplitudes and phases of quantities related to the field-induced current in the sample. These are voltages and currents, if electrodes are employed. If the technique is electrode-free, the relevant quantities are, instead, complex field amplitudes of electromagnetic waves transmitted or reflected by the sample. In the latter case, σˆ (ν) is obtained from the measured data by means of Maxwell’s equations and the boundary conditions at the interfaces, which guarantee the continuity of the electric and magnetic field components. Linear response theory provides a link between frequency-dependent conductivities and time correlation functions describing the ion dynamics [31]. It is, therefore, indispensable for interpreting experimental conductivity spectra in terms of processes happening on the time scale. In the following sections, correlation effects of ionic hops or displacements will be studied by the combined use of conductivity spectroscopy and linear response theory. The following aspects of linear response theory are relevant in the present context [30,32]. Suppose a constant electric field is switched on at time t = 0. In the sample under consideration this will cause a (vectorial) current density, ι (t), which is (experimentally) found to decay from its initial value, ι (0),

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Fig. 3. Schematic overview of different techniques for the measurement of frequency-dependent complex conductivities.

to a final value, ι (∞). A normalised function, Ψ (t), is introduced by writing ι (t) = ι (0) Ψ (t). The time derivative of Ψ (t), denoted by Ψ˙ (t), turns  out to be  proportional to the autocorrelation function of the current density, ι (0) ι (t) . As a consequence, the frequency-dependent  complex current density can be re lated to the Fourier transform of ι (0) ι (t) [30,32], and thus an expression is obtained for the frequency-dependent complex conductivity that is caused by the hopping or displacive motion of the mobile ions, V σˆ hop (ω) = 3k B T

∞

  ι (0) ι (t) hop exp (−iωt) dt .

(1)

0−

Here V is the volume of the sample, and the current density is the inverse volume times the sum over the products of the velocities, υ i , and charges, q i , of the mobile ions. Equation (1) is considerably simplified, if all N mobile ions are of the same kind, with charge q, and if we may assume that the cross terms in the velocity correlation function do not contribute significantly to the conductivity. This yields q2 N σˆ hop (ω) = 3Vk B T

∞

  υ (0) υ (t) hop exp (−iωt) dt,

(2)

0−

where υ (0) υ (t) hop denotes the velocity autocorrelation function of the hopping ions. Indeed, Monte Carlo simulations have shown that neglecting the cross terms is often well justified [33].

2.2 Conductivity components at low and high frequencies Before analysing the information on translational and localised ionic movements that is contained in frequency-dependent conductivities, we need to mention additional components which contribute to the experimental spectra at

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Fig. 4. Log–log representation of the real part of the relative permittivity of 0.30 Na2 O·0.70 B2 O3 glass vs. angular frequency, at 698 K. The solid line is a model fit based on the linearised Poisson–Boltzmann equation.

low and high frequencies, but will not be discussed in the following sections. They are due to electrode polarisation and excitation of vibrational motion, respectively. The effects of electrode polarisation show up in conductivity spectra, when ionic charge carriers are piled up or depleted at the blocking electrodes used in the experiment. Typically, the conductivity is found to decrease rapidly with decreasing frequency, while the (relative) permittivity increases tremendously, often attaining low-frequency values in excess of a hundred million, see, e.g., Ref. [34]. An example is presented in Fig. 4 [35]. The figure shows experimental permittivity data of a sodium borate glass at 698 K, complemented by a Lorentzian model line for the permittivity component caused by the blocking electrodes, ε be . The line has been obtained from the simplest possible model, which is based on the linearised Poisson–Boltzmann equation. The only parameters required are the (known) volume permittivity of the glass, the (known) overall number density of the sodium ions in it, n 0 , as well as the thickness of the sample between the electrodes. Here, our main purpose is to emphasise that this kind of model treatment, though repeatedly presented in the literature [36–38], is entirely inadequate. The reason is the following. Evidently, the chemical potential of the sodium ions close to the electrodes, μ, will be non-ideal, for reasons of stress and strain. The electrochemical potential per sodium ion, η = μ + eϕ, with e and

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ϕ denoting the elementary charge and the electrical potential, respectively, will thus also differ from the ideal case. This has two consequences. In the first place, once a gradient of η is created by an applied electric field, the rate of its decay will not be proportional to the current density, which is itself proportional to the gradient. Therefore, the gradient will not approach its equilibrium value of zero in an exponential fashion. This is different from the ideal case, where the exponential decay yields a simple Lorentzian for the frequency-dependent permittivity, cf. the line in Fig. 4. Secondly, the Boltzmann equation is no longer applicable. This is seen from the condition for electrochemical equilibrium, η(x) − μo ≡ 0, which may be written as k B T ln a + (x) + eϕ(x) = 0. Here, the normalised local number density of the sodium ions, n + (x)/n 0 , has been replaced by their activity, a + (x) = γ+ (x)n + (x) /n 0 . The resulting identity, a + (x) = exp (−eϕ (x) /k B T ), may be used to express n + (x) in terms of the electrical potential, ϕ (x), and the activity coefficient, γ+ (x). The local charge density, ρ (x) = e (n + (x) − n 0 ), thus becomes     1 eϕ (x) −1 . (3) exp − ρ (x) = en 0 γ+ (x) kB T This differs from the expression for the ideal case by the term γ+ (x). Close to either electrode, the effect of γ+ (x) is to reduce the absolute value of ρ (x). Combining Eq. (3) with Poisson’s equation yields a suitably modified version of the Poisson–Boltzmann equation. As γ+ (x) is unknown, this equation is not readily solved. In fact, the term replacing the usual expression for the square of the inverse Debye length, ξ 2 = e2 n 0 / (ε 0 εk B T ), becomes not only positiondependent, but also much smaller, especially in the charge layers in front of the electrodes. This results in an “effective Debye length” that is, besides being position-dependent, larger than in the ideal case, in particular in the vicinity of the electrodes. This had to be expected, since the Debye length (and thus the thickness of the charge layers in front of the electrodes) as derived from the model line in Fig. 4 is only about 1 Å, which is of course by far too small. The conductivity component caused by the excitation of vibrational motion, σ vib (ν), constitutes the main high-frequency contribution to experimental conductivity spectra, to be observed in the (far) infrared. Disordered materials such as glasses differ from ionic crystals by the absence of lattice periodicity and of Brillouin zones in which optical phonon modes can be identified. Compared with an ionic crystal, the “vibrational” conductivity component observed in a glass is, therefore, much broader, extending to lower frequencies. Figure 5 shows the example of a silver thiogermanate glass [39,40]. In this log–log representation of the total (ionic) conductivity, σ (ν), the contribution due to vibrations is seen to have a slope of two on its low-frequency flank, which is known to be the usual behaviour in ionic solids [41]. This component, σ vib (ν), can be carefully removed from the experimental spectrum. In spite of the scatter in the data thus obtained, a short plateau is seen to emerge at

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Fig. 5. Frequency-dependent conductivity, σ (ν), of Ag2 S·GeS2 glass at 273 K [32,40]. Below 10 GHz, σ (ν) is essentially caused by the hopping motion of the silver ions, while above 20 GHz it is essentially due to the low-frequency flank of the vibrational component. For the model curve, see Sect. 3.2.

the high-frequency end of σ hop (ν) = σ (ν) − σ vib (ν), i.e., of the conductivity caused by the hopping motion of the mobile ions. Here we want to mention that the removal of σ vib (ν) from the total spectrum is experimentally demanding and, therefore, non-trivial. The detection of a high-frequency limiting value of σ hop (ν), called σ HF , is an important result. It is in agreement with expectations from simulations and modelling.

3. Translational ionic motion and first universality 3.1 Experimental spectra revisited The characteristics of frequency-dependent electric and dielectric data of disordered solid electrolytes are briefly recalled in Fig. 6. In the figure, conductivities are presented in panels a) and b), permittivities in panels c) and d), with d) being a scaled version of c), and, finally, real and imaginary parts of the dielectric modulus in panels e) and f), respectively. The hopping motion of mobile ions via interconnected sites and passageways is reflected in broadband conductivity spectra, σ hop (ν), such as the one shown in Fig. 5 for a silver thiogermanate glass at 273 K. In Fig. 6 a), which is a log–log plot of σT vs. frequency, this isotherm is complemented by others for the same glass [39,40]. An analysis of the temperature dependences of the limiting conductivities at low and high frequencies, σ DC and σ HF , shows that both are Arrhenius activated, the activation energy for σ DC being larger than the one

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Fig. 6. Frequency-dependent conductivities, permittivities and dielectric moduli of binary ion-conducting glasses. a) Complete conductivity isotherms with HF plateau resolved, for Ag2 S·GeS2 glass [39,40]; b) high-precision conductivity isotherms of 0.3 Li2 O·0.7 B2 O3 glass in the impedance frequency regime [10]; c) permittivity isotherms in 0.2 Na2 O·0.8 GeO2 glass in the impedance regime [20,27]; d) scaled permittivities for the same glass as in c), as per Eq. (4) [20,27]; e) and f) are isotherms of the real and imaginary part of the dielectric modulus for the same glass as in c). The solid curves have been obtained from the MIGRATION concept formalism, see Sect. 3.2.

for σ HF . The dispersion thus becomes increasingly pronounced with decreasing temperature. In contrast to the data shown in panel a), those of panel b) are restricted to the impedance regime, where conductivity data are measured with highest precision. The set of isotherms shown in the figure are for a lithium borate

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glass [10]. They superimpose when shifted along a straight line with a slope of one, i.e., they show the property of time temperature superposition [11,12,42– 44] in the form of Summerfield scaling [45]. The “master curve” thus obtained and the first-universality curve as presented in the Introduction are, indeed, identical. Because of the Kramers–Kronig relations [46,47], scaled conductivities, σ S (ωS ) = Reσˆ S (ωS ), are accompanied by scaled permittivities, ε S (ωS ) = Reˆε S (ωS ) = Imσˆ S (ωS ) /ωS . As before, σˆ S is meant to be σˆ S = σˆ hop /σ DC = iωε 0 εˆ hop /σ DC , where ωS = ω/ωO is scaled angular frequency. Note that εˆ hop = εˆ − ε (∞) does not contain the high-frequency permittivity caused by faster processes, ε (∞). For simplicity, primes are omitted in the real parts of conductivity and permittivity. The scaled permittivity may now be written as ε S (ωS ) =

ωO ε 0 (ε (ωS ) − ε (∞)) . σ DC

(4)

The permittivity spectra shown in Fig. 6c have been measured on a sodium germanate glass, at different temperatures. As expected, the scaled data points, ε S (ωS ), are found to superimpose, see Fig. 6d [20,27]. The complex dielectric modulus, defined as Mˆ = 1/ˆε [48–50], has been frequently used in the literature. In panels e) and f) we present the real and imaginary parts of this function, denoted by M  (ω) and M  (ω), respectively, for the sodium germanate glass already considered in panels c) and d). Here it is important to note that, in contrast to σ hop and ε hop , M  and M  do not display the property of scaling. In fact, scaling is destroyed by inclusion of the term ε (∞) in the definition of Mˆ = 1/ˆε [51]. Figure 6a–f includes solid lines that reproduce the experimental data. These have been obtained from the MIGRATION concept, to be outlined in Sect. 3.2. It is worth mentioning that the permittivity data in panels c) and d) deviate from the model curves at low frequencies, which is due to the effect of electrode polarisation, cf. Sect. 2.2.

3.2 The MIGRATION concept Over the years, various concepts, models and computer simulations have been published, all of them aiming at a description and/or modelling of broadband conductivity spectra. Here we note in particular Jonscher’s power law [8,52], Ngai’s Coupling Model (CM) [1,53–55], Monte-Carlo studies by Bunde et al. [33,56–58], Dieterich’s Counter-Ion Model [3,59], the Random Barrier/Random Energy Models [13,60–64] and the Jump Relaxation Models [2,10,65,66]. In the group of Jump Relaxation Models, the most recent version is the MIGRATION concept (MC) [20,21,23,27]. As the MC forms the basis for a number of further findings, to be reviewed in the following sections, it will be briefly outlined.

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Fig. 7. Schematic representation of the leitmotif of the MIGRATION concept. The backward hop of an ion signifies the possible relaxation along the single-particle route, while the shift of the caged potential indicates the possible relaxation along the many-particle route. The effective potential experienced by the ion is sketched by the red solid line.

The acronym MIGRATION stands for MIsmatch Generated Relaxation for the Accommodation and Transport of IONs. In the physical picture conveyed by the acronym, we emphasise the mismatch introduced by any hop of an ion and the ensuing relaxation (rearrangement) of the ionic neighbourhood, which may or may not result in an accommodation (stabilisation) of the ion at its new position. When this is achieved, an elementary step of macroscopic transport has been completed by the ion. The leitmotif of the model is shown in Fig. 7. The figure demonstrates that, after a hop, the mismatch thus created can be reduced along two competing routes. One of them is the “single-particle route”, with the ion hopping backwards, while the other is the “many-particle route” with the neighbours rearranging. For constructing complex conductivity spectra from time-dependent functions, the MIGRATION concept starts out from Eq. (2). At this stage, it is useful to introduce the time-dependent correlation factor, W (t), which is by definition the normalised time derivative of the mean square displacement due to the hopping motion,   d  2   d  2  . (5) r (t) hop r (t) hop  W (t) = dt dt t→0   Note that the slope of r 2 (t) hop is finite at short times, since this function considers instantaneous hopping, disregarding the ballistic regime. Because of Eq. (5), W (0) is unity and W (t) is the normalised integral  of the velocity auto correlation function for the hopping motion, υ (0) υ (t) hop , as used in Eq. (2). Equation (2) thus becomes σˆ hop (ω) = 1+ σ HF

∞ 0+

˙ (t) exp (−iωt) dt W

(6)

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and, after integration by parts, σˆ hop (ω) = 1 + iω σˆ S (ω) = σ DC

∞ [WS (t) − 1] exp (−iωt) dt .

(7)

0

In Eq. (7), the function WS (t) = W (t) /W (∞) is the scaled time-dependent correlation factor. Equation (7) is more useful than Eq. (6), since, in contrast to σ HF , the value of σ DC is normally known from experiment. The following rules, which constitute the MIGRATION concept, determine the shape of W (t): ˙ (t) W = −B g˙ (t) W (t) g˙ (t) − = Γ0 W (t) N (t) g (t) N (t) − N (∞) = [Bg (t)] K −1 −

(8) (9) (10)

Here, g (t) is a normalised mismatch function. It describes the decay of the (normalised) local electric dipole moment that is created by the hop, under the assumption that at time t the ion is (still or again) at its new site. Therefore, g (t) decays from g (0) = 1 to g (∞) = 0. Equation (8) claims that the rates of relaxation along the two competing routes are proportional to each other. Integration of Eq. (8) yields the important result W (∞) =

σ DC = exp (−B) . σ HF

(11)

The fraction of hops that eventually prove successful is thus given by W (∞). Likewise, W (∞) is also the ratio of the rate of successful hops, ω O , by the elementary hopping rate, Γ0 . In conductivity spectra, ωO = Γ0 W(∞) marks the onset of the dispersion. Figure 8 shows a model spectrum including the angular frequencies marking the onset and end of the dispersion, denoted by ω O and ωE , respectively, as well as formal expressions for them. According to Eq. (9), the rate of mismatch decay is proportional to the driving force, which is mismatch itself, to the elementary hopping rate, Γ0 , and to a number function, N (t), see below. It is also proportional to W (t), since the mobile neighbours perform correlated forward-backward hops in the same fashion as the “central” ion does. N (t) is an effective number of mobile neighbours that, in spite of ongoing shielding, still notice the dipole at time t. For the decay of N (t), see Eq. (10), an empirical function has been chosen that yields excellent agreement with experimental conductivities and permittivities. In Eq. (10), N (∞) is the effective number of nearest neighbours, i.e., of those ions that never get shielded. This

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Fig. 8. A model conductivity spectrum derived from the MIGRATION concept. For details, see text and Refs. [5,10].

number determines the low-frequency limiting value of the permittivity and turns out to be temperature-independent, as expected. The value of the parameter K , see Eq. (10), that has been used in Fig. 8 (and also in Fig. 1) is 2.0. This value of K is, indeed, typical of experimental conductivity spectra not only of structurally disordered crystalline electrolytes such as α- and β-RbAg 4 I 5 , but also of simple binary glasses, of polymer electrolytes and ionic liquids. The value of K has an effect on the shape of the conductivity spectra close to the onset of the dispersion. Values of K that are smaller or larger than 2.0 reflect an onset of the dispersion which is less or more gradual than in the first universality plot of Fig. 1, respectively. The value of the parameter K helps gain insight into the factors that result in a particular shape of the conductivity spectra for particular systems. For example, a value of K ≈ 1.2, as found in a very fluid simple ionic melt [21,67,68] would imply that N (t) does not change much with time, and that long-range interactions are unimportant. If the parameter K is larger than 2.0, N(t) − N(∞) decays faster than g (t). Solid electrolytes show this behaviour, if the absolute density of mobile ions is low or if their pathways are restricted [5,10,69–72].

3.3 Mutual mapping of MC and CM Since Jonscher’s times, two complementary routes have been pursued to formulate the time-dependence of the electric and dielectric properties of ionconducting materials.

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Fig. 9. Macroscopic route: electric field decay in an ion-conducting material, schematic, cf . [48–50].

Fig. 10. Microscopic route: mean square displacement, time-dependent correlation factor, velocity autocorrelation function, schematic, cf. [5,10,23].

One of them was macroscopic by construction, focusing on the decay of an electric field inside the sample, caused by the motion of the mobile ions. With the conductivity being frequency-dependent, the decay function, Φ (t), cf. Fig. 9, was not a single exponential, thus calling for a description in terms of a distribution of relaxation times [48–50]. The other route is based on linear response theory [31] describing the movements of the mobile ions in terms of time-dependent functions such as those in Fig. 10, thus opening a door towards a microscopic understanding. Considering the two routes in a model-free fashion, we find that they have an important feature in common. In both cases, a real function of time is used to describe the phenomena, viz., the field-decay function, Φ (t), and the time-dependent correlation factor, W (t). Remarkably, the analogy goes much farther. The two time-dependent functions are related to their corresponding frequency-dependent complex quantities, which are Mˆ (ω) and σˆ hop (ω), respectively, in a formally identical manner: Mˆ (ω) ˆ˙ (ω) = 1+Φ M  (∞) σˆ hop (ω) ˆ˙ (ω) = 1+W σ HF

(12) (13)

Equation (13) and Eq. (6) are the same, with the one-sided Fourier transform ˆ˙ (ω). The analogous Eq. (12) was introduced much ˙ (t) now written as W of W earlier by Moynihan et al. [48–50].

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Fig. 11. Coupling model (CM): There is a crossover from a single exponential behaviour of the field-decay function at short times to a stretched exponential behaviour at long times. The nonexponentiality parameter is denoted by β.

A mutual mapping, i.e., a transformation of the time-dependent functions into each other, is now straightforward. According to Eqs. (12) and (13), respectively, Φ (t) and W (t) may be obtained from the frequency-dependent experimental data by (backward) Fourier transformation and subsequent integration, the only remaining missing link being the one between Mˆ (ω) and σˆ hop (ω). This is immediately provided by the equations Mˆ (ω) = 1/ˆε (ω), εˆ (ω) = εˆ hop (ω) + ε (∞) and εˆ hop (ω) = σˆ hop (ω) / (iωε 0 ). The decay function, Φ (t), may thus be expressed in terms of W (t) and vice versa. Almost 30 years ago, Ngai showed that a microscopically derived ionhopping correlation function, which has since been the basic ingredient of his coupling model (CM) [53,54], could be isomorphically mapped onto the field decay function introduced by Moynihan. Ngai’s function, ΦCM (t), is single exponential at short times and stretched exponential at long ones. A plot of − ln ln (1/Φ (t)) vs. ln t is given in Fig. 11. In terms of the frequency-dependent conductivity, the single-exponential part corresponds to the high-frequency plateau, while the “slowed-down” stretched exponential part corresponds to a dispersive conductivity of the power-law type. For a direct comparison of the time-dependent functions obtained from the MIGRATION concept (MC) and from the CM, let us consider the case of glassy silver thiogermanate at 183 K, cf. Fig. 6a. In Fig. 12, the functions W (t) and ΦMC (t), as obtained for this glass from the MC, as well as the function ΦCM (t) are plotted linearly vs. ln t [73]. At sufficiently long times, ΦMC (t) and ΦCM (t) are seen to superimpose. At shorter times, however, ΦCM (t) deviates from ΦMC (t). This had to be expected, since the CM predicts a dispersive conductivity of the power-law type, while the MC is in agreement with the first universality and hence with the frequency dependence of the experimental data. Figure 13 is a representation of − ln ln (1/Φ (t)) vs. ln t, with ΦMC (t) and ΦCM (t) for the same system. Again, it is evident that the stretched exponential

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Fig. 12. Model function W (t) corresponding to the 183 K spectrum of Fig. 6a). ΦMC (t) is derived from W (t) (for details, see text and Refs. [23,73]). ΦCM (t) is obtained from the CM model.

Fig. 13. Functions ΦMC (t) and ΦCM (t) for the same system as in Fig. 12, in the representation of Fig. 11.

ansatz of the CM is excellent at long times, while the agreement with the more realistic MC curve is less favourable at short ones. Another example is the supercooled molten salt, 0.4 Ca(NO3 )2 ·0.6 KNO3 , often abbreviated as CKN. Frequency-dependent data of CKN, taken at 353 K, are shown in Fig. 14 [23,32].

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Fig. 14. Broadband spectra of the ionic conductivity, the dielectric loss and the imaginary part of the dielectric modulus of supercooled 0.4 Ca(NO3 ) 2 ·0.6 KNO3 at 353 K, in log–log representations. The open squares are experimental data points, the solid lines are MC model spectra. The vibrational component is indicated by dashed lines with slopes 2 and 1 in the panels for σ and ε , respectively.

In the panels for the conductivity and the loss function, dashed straight lines are included with slopes of two and one, respectively, representing the contribution due to vibrational motion, while the solid lines have been derived from the MIGRATION concept. The figure clearly shows that the experimental data are well reproduced by these two components, without invoking any additional nearly-constant-loss effect. The latter becomes necessary, if the CM is used. Our results for CKN are thus similar to those for the silver thiogermanate glass. The remarkable observation of a first-universality-type conductivity dispersion in a supercooled molten salt will be discussed further in Sect. 3.5.

3.4 First universality in rotational diffusion In the field of dielectric loss spectroscopy of dipolar liquids such as, for instance, glycerol, a host of experimental data and model concepts has been published over the past years and decades [74]. Yet a basic question has remained unsolved, viz., the physical origin of what has been called the “excess wing” feature [75]. In the following, this feature is interpreted as just another fingerprint of the first universality, now becoming evident in non-random rotational diffusion of charged entities. Figure 15 summarises dielectric loss spectra that have recently been taken in our laboratory on dithioglycerol at different temperatures above the

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Fig. 15. Dielectric loss spectra of dithioglycerol in a log–log representation. The so-called “excess-wing” feature refers to the long tail with continually changing slope beyond the loss peak.

glass transition. The corresponding spectra of glycerol itself, which have also been measured, display similar features and are in agreement with literature data [76]. In the log–log representation of Fig. 15, the shape of the loss function, ε (ν), is not symmetric, thus excluding a Debye model. The slope on the highfrequency side has two properties that need to be explained. In the first place, its absolute value is smaller than the one at frequencies below the maximum. Secondly, this value is not constant, but becomes smaller with increasing frequency, thus exhibiting the so-called “excess wing”. The first of these properties may be accounted for by a stretched exponential function replacing the Debye ansatz, but this will not explain the second. In Fig. 16, the experimental data on dithioglycerol have been plotted in a log–log representation of scaled conductivity vs. scaled frequency. This provides a number of remarkable clues. (i) There is a DC conductivity [77]. (ii) Summerfield scaling is found to be valid for the entire spectrum. (iii) The high-frequency part of the scaled spectrum, which corresponds to the “excess wing” feature, is reminiscent of the first-universality master curve as shown in Fig. 1. The validity of Summerfield scaling implies that all processes that cause the loss and conductivity spectra are closely related, in the sense that any variation in temperature will have no other effect but to change all the rates involved by the same factor. Evidently, this includes macroscopic charge transport, which is however beyond the scope of our present discussion.

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Fig. 16. The loss spectra of dithioglycerol replotted as scaled conductivity vs. scaled frequency. Summerfield scaling, νs = ν/(σDC T ) [45], has been used for scaling the frequency axis.

Those features in Fig. 16 that correspond to the maximum of the loss function, ε (ν), may be explained in terms of a MIGRATION-type model for rotational diffusion. In the model, the coefficient of rotational diffusion (on a ring) is supposed to be time-dependent, due to the interactions between different molecular dipoles of the same kind, which result in correlated forwardbackward steps of their rotational movements. At short times/high frequencies, this should yield a behaviour that closely resembles the translational case. The high-frequency parts of Figs. 16 and 1 are, therefore, quite similar, both of them reflecting the characteristic shape of the first universality. In the case of rotational diffusion, this shape does of course not merge into a constant lowfrequency conductivity, σ DC , since the mobile charge is unable to leave the ring. Figure 17 is a scaled representation of the frequency-dependent loss function as obtained experimentally for dithioglycerol and for glycerol. Along with the data points, the figure also shows those solid lines that have been obtained from our MIGRATION-type model. In this model, we start with an equation for diffusion on a ring that contains a time-dependent rotational diffusion coefficient, D (t): ∂2c ∂c = D (t) 2 ∂t ∂ϕ

(14)

D (t) is then related to the first-universality-type time-dependent correlation factor, as derived in Sect. 3.2, which is now called W0 (t). By writing

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Fig. 17. Scaled loss function of dithioglycerol (top) and glycerol (bottom). These are log–log plots of Tε vs. scaled frequency, νs . Note that εs ∝ σs /νs is proportional to Tε .

D (t) dt = D 0 W0 (t) dt = D 0 d t˜ ,

(15)

we introduce a new time scale, t˜. In passing we note that the initial value of D (t), i.e. D (0) = D 0 , is supposed to be much larger than its long-time limiting value, D (∞) = D 0 W0 (∞). Figure 18 is meant to visualise the construction of t˜ (t) by forming the integral over W0 (t  ) for times from 0 to t, and also gives a sketch of the resulting function, t˜ (t). Equation (14) is now solved in a way that is analogous to the case where the diffusion coefficient is a constant, D • . In this simpler case one finds, with

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Fig. 18. Visualisation of the construction and shape of t˜ (t).

the initial condition given by c (ϕ, 0) = δ (ϕ), c (ϕ, t) =

∞

 1 cos (mϕ) exp −m 2 D • t . 2π m=−∞

(16)

Once the angle-dependent function, c (ϕ, t), is transformed into its distancedependent analogue, c (r, t) = 2c (ϕ, t) dϕ/dr, the mean square displacement becomes  2  r (t) =

2R c (r, t)r 2 dr = 2R2 (1 − exp (−D • t)) ,

(17)

where R denotes the radius of the ring. Quite analogously, Eq. (14) yields the expression



  2  r (t) = 2R2 1 − exp −D 0 t˜ , t

 with t˜ = W0 t  dt  .

(18)

0

0

Forming the normalised time derivative of this mean square displacement one finds ⎡ ⎤ t

  Wrot (t) = W0 (t) exp ⎣−D 0 W0 t dt ⎦ , (19) 0

which now replaces the Debye-type decaying exponential, exp(−D • t), that one would have obtained from Eq. (17). In Fig. 17, the results of our model treatment have been derived from this function, Wrot (t), by εrot (ω) = Re σˆ rot (ω) / (ε 0 ω), where σˆ rot (ω) is proporˆ˙ (ω). tional to 1 + W rot Parameters are D 0 , W0 (∞) and the elementary rotational jump rate, Γ0 . Introducing a characteristic jump angle, Δϕ, we may consider the relation

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2D 0 = Γ0 (Δϕ)2 . This expression contains a factor of 2, since a diffusive motion on a ring is one-dimensional. Multiplying it by W0 (∞) on either side we find 2D (∞) = ωO (Δϕ)2 . Here, ωO , is meant to be the onset angular frequency in the hypothetical spectrum, σ 0 (ω), that would be valid, had we considered diffusion on a straight line instead of a ring. The ratio, D (∞) /ωO = (Δϕ)2 /2, and hence Δϕ itself are temperature-independent. In constructing the model curves of Fig. 17, the values of Δϕ were estimated to be about 35◦ for dithioglycerol and about 20◦ for glycerol. From the above discussion it would follow that, for dithioglycerol, D (∞) should be smaller than ωO by less than one decade. In view of Fig. 16, this appears quite realistic. In the figure, the suitably normalised value of the hypothetical onset angular frequency, ω O , would be less than a decade above the normalised value of D (∞), which marks the position of the bend in Fig. 16 corresponding to the loss maximum in Fig. 17. This latter statement, ωmax = D(∞), is analogous to the Debye case, see Eq. (17), where the loss maximum is at ω max = D • .

3.5 First universality in fragile liquids DC conductivities of molten salts, ionic liquids and polymer electrolytes typically do not follow the Arrhenius law [78]. The empirical Vogel–Tammann– Fulcher (VTF) relation [79–81], which is often used to describe such nonArrhenius temperature dependences, has the disadvantage of not relating directly to the ion dynamics in the system. In this section, we show that a more coherent view is provided by considering the information contained in the conductivity spectra, which display the features of the first universality [82,83]. Even more remarkably, the same line of reasoning has been found to be valid for shear fluidities (inverse shear viscosities), which have many properties in common with ionic conductivities, not only regarding their temperature dependence at low frequencies, but also regarding their first-universality-type frequency dependence at fixed temperatures [24,25]. The distinguishing feature of the ion dynamics in fragile liquids is outlined in Fig. 19. In panel a), consider the frequency-dependent conductivity of an ionic liquid, LiCl · 7H2 O, at 253 K [5,10,84]. Here, the vibrational component has already been removed, and the data points are well reproduced by the MIGRATION concept. A set of model isotherms are then presented in panel b), reproducing experimental data taken at different temperatures. As in a solid electrolyte, one observes Summerfield scaling at low frequencies as well as Arrhenius activation at high frequencies. However, there is no Arrhenius behaviour in the DC regime, i.e., no constant DC activation energy. Rather, it is the end frequency of the dispersion, ν E = ωE /2π, which is now, within experimental uncertainty, independent of temperature. The presence of an Arrhenius activated high-frequency conductivity implies that the melt behaves like a solid, if the time window is smaller than

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Fig. 19. a) Non-vibrational conductivity spectrum of the ionic liquid LiCl·7 H2 O at 253 K (open circles). The solid line is a MIGRATION model spectrum. b) Set of MIGRATION model spectra for the ionic liquid LiCl·7 H2 O in a log–log representation of σ(ν)T vs. frequency. The onset and end frequencies of the dispersion, νO and νE , are marked as filled squares and by a dash-dotted line, respectively. c) Sketch of local potential experienced by an ion in an ionic liquid at a given time. Note that there exists no preformed neighbouring site. With the neighbouring ions rearranging suitably, the shoulder may develop to a new potential minimum. Or else, the ion “rolls back” without activation.

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a picosecond. In this time regime, individual activated displacements of individual ions may be visualised [5]. The temperature independence of ωE has two implications. One of them regards the local potential the ion is moving in, see panel c). Evidently, after an elementary displacement, the backward motion is not activated, but may be viewed as a roll-back process [82,83]. This means that the ion has not entered a pre-existing empty site as would be the case in a glassy network. Instead, the structure does not seem to provide such sites [83], and an accommodation of the ion at a new position is only possible, if the immediate neighbours move accordingly. The second implication concerns the positions of the isotherms relative to each other and hence the temperature dependence of σ DC . To find a closed expression for σ DC (T ), we first consider the relation ωE = Γ0 B K , see Sect. 3.2 and Fig. 8. As the rate of the elementary displacements is thermally activated, Γ0 ∝ exp (−E ∗ /k B T ), and as ωE is not temperature-dependent, this implies B ∝ (1/Γ0 )1/K ∝ exp (E ∗ / (Kk B T )). Secondly, we consider the identity, Tσ DC = Tσ HF W (∞), with Tσ HF being proportional to Γ0 , and W (∞) = exp (−B) being the fraction of successful displacements. The identity may thus be rewritten as [82,83]   ∗  E∗ E /K Tσ DC (T ) = α exp − − γ exp , (20) kB T kB T with



E∗ Tσ HF (T ) = α exp − kB T

 .

(21)

Equation (20), in which α and γ are constants, replaces the empirical VTF relation. Once K is fixed by modelling the conductivity spectra, the elementary activation energy, E ∗ , is the essential remaining parameter. In this case, the value of E ∗ perfectly determines the shape of log (Tσ DC ) vs. 1/T in the Arrhenius plot, while any vertical or horizontal shifting of this curve can be achieved by suitably varying α and γ . In the following, Eq. (20) is used to describe the temperature-dependent DC conductivity of the room-temperature ionic liquid BMIm-BF 4 , which is short for 1-butyl-3-methyl-imidazolium tetrafluoroborate. The experimental data are presented in Fig. 20, and the solid line for σ DC has been obtained from Eq. (20). The value used for K is 1.9. Indeed, the first-universality-type scaled conductivity, see Fig. 21, is very well reproduced with this value [24,25]. In Fig. 20, the high-frequency conductivity, cf . Eq. (21), is marked by a straight line, corresponding to an activation energy of E ∗ = (0.20 ± 0.02) eV. Vertical lines indicate the effect of multiplying the high-frequency data with W (∞). The figure also contains temperature-dependent values of the DC shear fluidity of BMIm-BF 4 , f DC (T ), which have been measured in our labora-

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Fig. 20. Arrhenius plot of log10 (TσDC (T )) and log10 (T f DC (T )) vs. 1/T . The data points are represented by open symbols, their size indicating the experimental error. Note that different 1/T scales apply for conductivity and fluidity. For the ionic conductivity, the bent and straight lines have been calculated from Eqs. (20) and (21), respectively, the lengths of the vertical lines representing log10 (1/W(∞, T ). After suitably shifting the coordinates, the same pair of lines is obtained for shear fluidity via Eqs. (22) and (23).

Fig. 21. Conductivity spectra of BMIm-BF4 in a scaled representation. The solid line is a MIGRATION model spectrum.

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tory recently. Similar to the DC conductivities, these data are represented as log 10 (T f DC (T )) vs. 1/T , since both Tσ DC and T f DC are related to diffusivities, via the Nernst–Einstein and Stokes–Einstein relations, respectively. From Fig. 20 it is evident that DC conductivity and DC fluidity have the same shape and superimpose, if the axes of the plot are suitably shifted. This suggests that equations with the same formal structure as Eq. (20) and Eq. (21) may be valid for the fluidity as well, i.e.,   ∗  E∗ E /K T f DC (T ) = α˜ exp − − γ˜ exp (22) kB T kB T and

  E∗ T f HF (T ) = α˜ exp − . kB T

(23)

In Eqs. (22) and (23), the tilde notation is used for quantities related to fluidity. In the first place, α˜ = α trivially results from the different units for fluidity and conductivity. The effect of γ˜ = γ is equivalent to the slight shift of the 1/T axis that is required in the Arrhenius representation to make conductivities and fluidities superimpose. This shift is Δ ≈ 8 × 10−5 K−1 . DC fluidities measured at temperatures T thus superimpose with DC conductivities measured at slightly lower temperatures, T ⊗ = 1/ (Δ + 1/T ), and the same is expected to hold for the respective HF transport coefficients, both being represented by the same straight line in Fig. 20. As an important consequence, the validity of Eqs. (22) and (23) would imply the existence of a pronounced frequency dependence of the fluidity, similar to that of the conductivity, spanning the entire distance from the DC to the HF data in Fig. 20. To detect the expected frequency dependence of the shear fluidity, we have performed measurements of f (ν), at different temperatures. Spectra taken on BMIm-BF 4 at 193 K and 198 K are presented in Fig. 22. They are, indeed, found to display the first-universality-type frequency dependence as described by the MIGRATION concept, with K = 1.9. Their shape is thus in perfect agreement with the conductivity spectra taken on the same ionic liquid. Therefore, Eq. (22) and Eq. (23) are considered valid, and the same value, E ∗ ≈ 0.20 eV, results for the activation energy for elementary displacements when derived from DC conductivities and DC fluidities [24,25]. However, the analogy goes even further. Inspection of the frequencydependent functions, f (ν, T ) and σ (ν, T ⊗ ), shows that they superimpose in a log–log plot without requiring any shift of the frequency axis. The respective scaled functions, f S (ν, T ) =

 f (ν, T ) σ (ν, T ⊗) = = σ S ν, T ⊗ , ⊗ f DC (T ) σ DC (T )

are thus identical for all frequencies.

(24)

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Fig. 22. Frequency-dependent fluidity of BMIm-BF4 as measured at 193 K and 198 K, in a log– log plot. The solid lines are MIGRATION model spectra calculated with K = 1.9, as for the conductivity isotherms. Fluidity and conductivity thus display the same first-universality-type frequency dependence.

In particular, this identity includes the temperature-independent end frequency of the dispersion, ν E = ωE /2π, which is thus found to be the same for ionic conductivity and shear fluidity. Further measurements of both DC and dynamic conductivities [22,25,85, 86] and shear fluidities [25], on other ionic liquids [25,82,85] and on polymer electrolytes [22,85,86], have meanwhile corroborated the findings exemplified above for BMIm-BF 4 . The validity of Eq. (24) is considered the most remarkable result, with both scaled functions showing the same first-universality-type frequency dependence. While, to the best of our knowledge, the time correlation function related with the shear fluidity is so far unknown (in contrast to that for the shear viscosity), we may conclude from our results that its time dependence at temperature T will closely resemble that of the current density autocorrelation function at the slightly lower temperature T ⊗ .

3.6 Structure and ion dynamics in Na-β -alumina This section and the next will take us on expeditions beyond the first universality. More specifically, we will discuss experimental conductivities that are caused by ionic transport in disordered materials, but do not just reflect the universal master curve of Fig. 1. Therefore, pertinent explanations require additional information on the ion dynamics.

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Fig. 23. Arrhenius plot of the DC conductivity of single-crystalline Na-β  -alumina of composition Na1.68 Mg0.67 Al10.4 O17.1 along the conduction plane, after Ref. [87]. For details regarding the change of slope, see main text.

The phenomena to be presented come into view with increasing temperature (Sect. 3.6) or pressure (Sect. 3.7). Here, we consider the two-dimensional sodium-ion conductor Na-β  alumina. Figure 23 is an Arrhenius plot of the DC ionic conductivity, as published by Farrington and Briant [87]. The conductivities were measured along the conduction planes of a single crystal of Na1.68 Mg 0.67 Al10.4 O 17.1 . A remarkable property of the data in Fig. 23 is the change of slope observed at about 465 K, which needs to be explained in terms of the ion dynamics. To this end, two further pieces of experimental information are required. One of them concerns the structure and the other the frequency dependence of the conductivity at different temperatures. Figure 24 is a sketch of part of the conduction plane, according to the single-crystal structural refinement done on Na1.70 Li0.32 Al10.66 O 17 by Dunn et al. [88]. This plane contains only sodium and oxygen ions. Quite importantly, the ratio of the numbers of the sodium and oxygen ions in the plane is very similar to the sample used by Farrington and Briant. The oxygen ions form a regular hexagonal structure, leaving approximately triangular areas between them for the sodium ions. On an average, about one out of six of these areas are unoccupied. One may, therefore, envisage a hopping motion of the sodium ions via a “vacancy mechanism”, with the number density of the “vacancies” being well defined and unusually high. Ionic conductivities have been measured on a sample of Na1.70 Li0.32 Al10.66 O 17 , at frequencies ranging from 100 MHz to 1 THz [32,89]. In the following, we do not discuss conductivity components that must be at-

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Fig. 24. Section of the conduction plane in Na-β  -alumina. The hexagonal arrangement of the oxygen ions provides regimes of residence for the sodium ions, one out of six remaining unoccupied [88].

Fig. 25. Conductivity isotherms according to the MIGRATION concept, schematic. For details, see main text.

tributed to the excitation of phonons and to the localised movements of the sodium ions in their roughly triangular regimes of residence. Rather, we report on the contribution to the conductivity that is caused by the translational hopping of the sodium ions via “vacancies”. The picture that emerges for the frequency dependence of this contribution, at different temperatures, is sketched in Fig. 25. The activation energy

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for an elementary hop, Δ hop , is seen to determine the temperature dependence of the conductivity at high frequencies, σ HF . At sufficiently high temperatures, the rearrangement of the neighbourhood is so fast that an ensuing backward hop is not preferred energetically. In this case, the DC and HF conductivities are identical, and the temperature dependence of σ DC (T ) is also determined by Δ hop . With decreasing temperature, however, correlated backward hops become more and more frequent. Now only a fraction, W∞ (T ) = exp (−B (T )) = σ DC (T ) /σ HF (T ), of the elementary hops are “successful” in the sense that a random sequence of them will constitute macroscopic transport. The activation energy for σ DC is now Δ DC , and the difference, Δ DC − Δ hop , is regarded as the energy required for the rearrangement of the ionic neighbourhood [89]. Although the frequency-dependent conductivities were measured on crushed polycrystalline samples, with grain-boundary polarisation effects extending into the experimental frequency window, the “DC” plateau could be determined, and the values of σ DC (T ) thus obtained were found to be in agreement with those of Fig. 23. In particular, the transition from a dispersive to a non-dispersive frequency dependence could be seen around 465 K. Therefore, the interpretation given in Fig. 23, which is based on the sketch of Fig. 25, is experimentally sound. In terms of the MIGRATION concept, Fig. 25 implies that the parameter B, see Eq. (8), has an interesting temperature dependence. When plotted vs. inverse temperature, B is close to zero at small 1/T but takes on a linear dependence at some crossover point. In the case of Na-β  -alumina, the crossover temperature is about 465 K. In view of the suggested vacancy-type model for the self-diffusion of the mobile sodium ions, the most important parameter provided by the conductivity spectra is the temperature-dependent onset angular frequency, ω O (T ), to be interpreted as the average rate of uncorrelated hops of an individual mobile defect, i.e., of a “vacancy”. If d is the mutual distance between centres of neighbouring sodium residence areas, then the coefficient of self-diffusion of the “vacancies” is given by ω O d 2 /4, where the factor of four indicates that in this system transport is confined to two dimensions. Denoting the DC conductivity of a polycrystalline material without surface polarisation by σ DC,POLY (T ), this quantity must be two-thirds the value σ DC (T ) of a single crystal in the conduction plane. The Nernst–Einstein relation then yields n VAC d 2 =

6k B Tσ DC,POLY (T ) = const. ωO (T ) e 2

(25)

Here, n VAC is the number density of vacant local regimes of residence, see Fig. 24. From Ref. [88], n VAC and d 2 are known and hence also the virtually temperature-independent value of their product, 1.65 × 105 cm−1 . Data for σ DC,POLY (T ) and ωO (T ) have been obtained from the experimental spectra

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taken at temperatures below 465 K, ωO (T ) being determined with the help of the MIGRATION concept [89]. As a result, the right hand side of Eq. (25) is, indeed, found to have the same temperature-independent value as the left hand side. This corroborates the validity of the vacancy-type diffusion model for the sodium ions [89].

3.7 Pressure-dependent ion dynamics in a glass This section is concerned with the cross terms in the velocity correlation function, which are included in the autocorrelation function of the current density, but not in the autocorrelation function of the velocity of the mobile ions. Here we recall that the cross terms have been omitted in the MIGRATION concept, where Eq. (1) is approximated by Eq. (2). We also recall that this procedure has been successful in reproducing the features of the first universality as presented in Fig. 1. The above approximation amounts to assuming that the Haven ratio, which is defined as H R = D ∗ /D σ , does not deviate from unity. Here, D ∗ denotes the coefficient of self-diffusion as measured in a tracer experiment, while D σ is obtained from the DC conductivity by means of the Nernst–Einstein relation. This means that the  right hand side of Eq. (2), taken at ω = 0 and multiplied with Vk B T/ q 2 N , yields D ∗ , while the same procedure, starting from the right hand side of Eq. (1), yields D σ . In most ionic systems with disordered structures, the Haven ratio, when determined at ambient pressure, is of the order of unity. However, recent measurements on a rubidium borate glass, of composition 0.3 Rb2 O · 0.7 B2 O3 , prove that pressure has a different effect on diffusion than it has on conduction [90]. Experimental data are presented in Fig. 26. They show that, under pressure, the transport of tagged ions (tracer ions) is much more reduced than the transport of charge. Indeed, the Haven ratio becomes as low as 0.02 at 6 kbar, while it is about 1/4 at ambient pressure. For an interpretation of the effect, it is useful to write HR explicitly as [15,90]  ∞  υ (0) υ (t) dt

HR =

D∗ 0 < 1. = ∞    ∞ 1...N  Dσ υ (0) υ (t) dt + N1 υ i (0) υ j (t) dt 0

(26)

0 i = j

The following conclusions can be drawn immediately. (i) Since the Haven ratio differs from unity, the cross terms (with i = j) are essential, implying that correlated processes involving different mobile ions play an important role in the hopping dynamics. (ii) The Haven ratio being smaller than unity, the cross terms are essentially positive, implying forward directional correlations between hops of different ions.

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Fig. 26. Log-linear plot of normalised tracer diffusivities (filled squares) and conductivities (open circles) vs. hydrostatic pressure. The dashed and solid lines are linear fits.

Fig. 27. Schematic view of a section of a pathway of an individual vacant site, exchanging its position with successive ions designated as “# j ” and “# j + 1”, where the angle Θ describes the departure from collinearity. Note that the arrows refer to the vacant site, not to the ions.

(iii) As D σ is much less pressure-dependent than D ∗ , the cross terms must have a particularly weak pressure dependence. The activation volume required is, therefore, particularly small, if the directional correlation between hopping ions is positive. The picture that emerges from these conclusions is sketched in Fig. 27, with ions performing successive hops in a directionally correlated fashion [15,90]. Note that this picture fits in well with results obtained recently, stating that in ion-conducting glasses the number of available sites may be not much larger than the number of mobile ions [91,92].

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In view of Fig. 27, the process may be described in terms of a vacant site exchanging positions with successive ions, say # j and # j + 1, with an average angle, Θ < π/2. This description has the advantage that we may dispense with the cross terms. Rather, the motion of either “species”, ions and vacant positions, is described by using only the self terms of their respective velocity correlation functions. In the expression HR =

D ∗ /D random , D σ /D random

(27)

the numerator may now be expressed in terms of randomly moving tagged ions and the denominator in terms of non-randomly moving individual vacant sites. This equation is valid at any pressure. The successive hops of the tagged ions are not directionally correlated, because successive vacant sites can approach the ions from random directions. The numerator is thus unity. On the other hand, the successive hops of a given vacant site are directionally correlated, and hence the denominator may then be expressed in the form of the series, 1 + 2 cos Θ + 2 cos Θ2 + 2 cos Θ3 + .... The Haven ratio itself thus takes the form H R ( p) =

1 − cos Θ ( p) . 1 + cos Θ ( p)

(28)

If the Haven ratio is known, a root mean square (rms) value of the angle Θ may be obtained from Eq. (28). Haven ratios of 1/4 and 0.02 as found at ambient pressure and at 6 kbar, respectively, correspond to rms Θ values of 53◦ and 16◦ . The huge decrease in H R ( p) is thus explicable in terms of a changed pathway topology. There is a striking similarity between Eq. (28) and the corresponding equation used by Le Claire and Manning [93] for the random motion of vacancies in crystal lattices, where the ions themselves move in a correlated fashion. In that case, the plus and minus signs are interchanged, and the correlations are backward instead of forward. As a consequence, the Haven ratios are again smaller than unity. They are very well defined, depending exclusively on the geometry of the crystal lattice. Obviously, the effects observed in glass, including their pressure dependence, have a quite different physical origin. To explain the directional correlation sketched in Fig. 27, we need to focus attention on those hops that eventually prove successful. With increasing pressure, their number per unit time decreases rapidly, and the ones remaining must at least be assisted by processes with small activation volumes. For example, to stabilise an ion at its new position, the vacant site has to be filled by another incoming ion in a way that exerts a minimal deformation on the surrounding matrix. The amount of this deformation will strongly depend on the angle Θ, decreasing with decreasing Θ. With increasing pressure, the remain-

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ing successful hops are, therefore, along conduction pathways that become increasingly linear in aspect.

4. Localised ionic motion and second universality 4.1 Separating first and second universalities At low temperatures, conductivity spectra of disordered ionic materials no longer conform to the first universality [14–17,26,94–98]. Rather, a gradual crossover from the first to the second universality is observed as temperature is reduced, e.g., from 300 K to 4 K. Quite generally, a problem arises when the attempt is made to define a demarcation between the respective phenomena. Yet there is a practicable solution, which will be outlined below. In the following, we consider the example of a sodium borate glass of composition 0.3 Na2 O · 0.7 B2 O3 . A set of conductivity isotherms measured in the impedance frequency regime at temperatures from 475 K down to 4 K is presented in Fig. 28 [98]. Clearly, the first universality is the dominant part at high temperatures, where the onset of the conductivity dispersion is within the experimental frequency window. On the other hand, the second universality is seen to govern

Fig. 28. Log–log plot of the frequency-dependent conductivity of 0.3 Na2 O · 0.7 B2 O3 glass at different temperatures. The filled diamonds mark the transition from the first to the second universality at three different frequencies. For the construction of these points, see Fig. 29.

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Fig. 29. Iso-frequency representation of log σ vs. 1/T for 0.3 Na2 O · 0.7 B2 O3 glass. The dashdotted lines indicate the Arrhenius temperature dependence of σDC and σHF . The filled diamonds represent the crossover from the first to the second universality. For details, see main text and Ref. [101].

the spectra at low temperatures, where the conductivity increases linearly with frequency and, most remarkably, becomes independent of temperature. Those isotherms that have been taken below 75 K, see Fig. 2 [99], are indeed indistinguishable. Here it is worth noting that any conductivity caused by the hopping motion of the ions must vanish at cryogenic temperatures, since both σ DC (T ) and σ HF (T ) are thermally activated [100]. Consequently, the second universality must have a different origin. The search for its underlying processes will be the main topic of the following sections. In Fig. 28, it is impossible to separate the first from the second universality merely by inspection. Nevertheless, crossover points can be determined, cf. the three points marked in the figure as filled diamonds. This has been achieved with the help of a different representation of the data, see Fig. 29 [101]. In this figure, log σ is plotted vs. inverse temperature, 1/T , for three fixed frequencies. At high values of 1/T , the conductivity displays the distinguishing marks of the second universality, i.e., it is independent of temperature and varies linearly with frequency. At low values of 1/T , however, we observe the characteristics of the first universality, although in a yet unfamiliar shape. At each given frequency, a reduction of temperature, i.e., an increase of 1/T , causes two transitions, the first from the DC plateau into the dispersive regime and the second from the dispersive regime into the HF plateau. The solid lines, which show these transitions, have been obtained from the equations of the MIGRATION concept. The two plateaus are indicated by lines that would be

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straight, if log (σT ) had been plotted vs. 1/T , the activation energies being E DC ≈ 0.80 eV and E HF ≈ 0.32 eV, respectively [101]. In Fig. 29, the crossover points (filled diamonds) have been chosen to be those data points, where the contributions of the two conductivity components are identical, at a distance of log 2 above the horizontal dashed seconduniversality lines. The positions where the MIGRATION-concept model lines merge into their HF plateaus have been chosen accordingly. The crossover points marked in Fig. 28 are the same as those in Fig. 29. Two further details are worth mentioning. (i) At each frequency, the data points at inverse temperatures slightly above the crossover are higher than the sum of the components indicated by the solid and dashed lines. In other words, the NCL-type component itself seems to show an increase with increasing temperature. This effect is seen in other glasses as well and will be reconsidered in Sect. 4.3, where it is explained in terms of activated localised movements of the ions, coming into view with increasing temperature [102]. (ii) At fixed temperatures, the frequency dependence of the conductivity is smaller in the first than in the second universality regime. As soon as frequencies are sufficiently high, the NCL-type component is, therefore, expected to swamp the low-temperature part of the MIGRATION-type component including the position where it merges into its HF plateau. Such examples will be presented in the next section. Note that, at this point, the origin of the second universality is still unexplained.

4.2 High-frequency data and localised motion For understanding the NCL phenomenon, it is helpful to consider also dispersive spectra taken at higher frequencies and higher temperatures. In such cases, the MIGRATION-type component is expected to determine the low-frequency part of the dispersion, while the NCL-type component may have swamped its high-frequency part. This would imply that the linear frequency dependence of the conductivity, i.e., the “NCL fingerprint”, becomes visible at frequencies at which the swamped part of the MIGRATION-type component has not yet attained its HF plateau. A beautiful example is provided by the low-temperature γ -phase of the crystalline silver-ion conductor RbAg 4 I 5 , see Fig. 30 [19,21,30,70,97]. In marked contrast to the non-vibrational conductivity of glassy Ag2 S·GeS 2 , cf . Figs. 5 and 6a, the spectrum of γ -RbAg 4 I 5 cannot be explained in terms of the MIGRATION concept alone. Rather, a nearly linear, i.e., NCL-type conductivity component appears to be superimposed at frequencies above some 10 MHz, attaining a high-frequency plateau of its own at some 10 GHz. When the spectrum of Fig. 30 was measured, different explanations had already been put forward for NCL-type effects, including the second universality

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Fig. 30. Non-vibrational conductivity spectrum of γ -RbAg4 I 5 at 113 K, along with a model spectrum (solid line) which is composed of two components resulting from “ordinary” and strictly localised hopping, respectively. To model the “ordinary” hopping, which is via structural vacancies [69,70], the MIGRATION concept has been used, with K having a value as high as 2.6. Note that the values of the parameters ω 1 = 2πν1 and Γ0 are indistinguishable.

in the strict sense of the word, cf . Fig. 2. There was and still is a broad consensus that these effects reflect a collective phenomenon, with a large number of ions moving locally in a cooperative fashion. In ionic crystals, the effect had been seen to develop from ordinary Debye relaxation, as the number density of locally mobile ionic defects was drastically increased [95]. Even before the NCL effect itself was discovered, static distributions of asymmetric doublewell potentials (ADWP) had been assumed to exist for locally mobile ions, providing a wide range of relaxation times [103]. Later, the ADWP model was often used to fit experimental NCL data. To visualise the collective nature of the phenomenon, Jain introduced the term “jellyfish” effect [14]. A big step forward was made when Dieterich et al. used Monte Carlo simulations to study random distributions of reorienting and interacting dipoles, as in Fig. 31 [4,104–107]. In the conductivity spectra thus obtained, the slope of log σ vs. log ν was found to change with increasing frequency, first from two to one and later from one to zero. As the ratio of Coulomb energy by thermal energy was increased, the frequency range of the NCL regime, between the two crossover points, was found to increase as well. In the Monte Carlo simulations, the static distributions of the ADWP model were thus successfully replaced by variations of local potentials in time, experienced by any locally mobile ion. Shortly later, the same idea was used to formulate a simple rate equation that again resulted in the same frequency

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Fig. 31. A collection of interacting electric dipoles which are randomly distributed on a lattice. From Ref. [4], by courtesy of W. Dieterich.

dependence, see Fig. 32 [5,10,21,32,98]. Now, the function considered was WL (t), i.e., the normalised first derivative of the mean square displacement of the locally mobile ions. If an ion is confined in a double-well potential as in Fig. 32, then WL (t) is the difference between the probabilities of finding it, say, on the right and on the left. In the absence of any Coulomb interactions, see Fig. 32a, WL (t) is an exponentially decaying Debye-type function, WDebye (t) = exp (−ω1 t), the ˙ Debye /WDebye = ω1 . In the presence of the Coulomb inrate equation being −W teractions, however, the shape of the potential is time-dependent, implying a time dependence of the hopping rate, see Fig. 32b. The rate equation now becomes [5,10] −

˙ L (t) W = ω1 + ω2 WL (t) , WL (t)

(29)

and its solution is WL (t) =

ω1 . (ω1 + ω2 ) exp (ω1 t) − ω2

(30)

Inserting this function into Eq. (6), one obtains conductivity spectra with consecutive slopes of two, one and zero in log–log representations, in perfect

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Fig. 32. a) An ion hops between the sites of a symmetric double-well potential. Interactions with other ions are not considered. b) As in a), but with interactions included. c) The frequencydependent conductivity resulting from the situation sketched in b).

agreement with the results of Dieterich et al., the crossover angular frequencies being ω1 and ω2 , cf . Fig. 32c. In Fig. 30, superposition of a MIGRATION-type component and an NCLtype component yields an excellent fit to the experimental data [21,32]. Remarkably, the rates Γ0 and ω1 may be taken to be the same, with the same temperature dependence, suggesting that the two hopping processes are across energy barriers of similar height, one of them along interconnected pathways and the other between sites with no further connectivity.

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Fig. 33. Perspective view of the crystal structure of γ -RbAg4 I 5 along the c-axis. White balls mark partially occupied silver positions contained in structural pockets between strongly distorted RbI 6 octahedra, cf. Ref. [70].

This interpretation triggered an investigation of the so far unknown crystal structure of γ -RbAg 4 I 5 , which resulted in the detection of isolated “pockets”, each of them containing a number of only partially occupied silver sites, see Fig. 33 [70]. The fraction of silver ions located in the pockets is about 56%. Within the pockets, they hop from site to site, while the other silver ions are able to traverse the crystal along pathways for translational motion [69,70]. The locally mobile silver ions have thus been identified, and the predictions made on the basis of the conductivity spectrum of Fig. 30 have been corroborated.

4.3 From NCL-type behaviour to second universality The second universality, cf . Fig. 2, is characterised by both ω1 and ω2 being outside the accessible frequency window. In this situation, it is helpful to study cases where the transitions are both visible. In particular, one would like to find an example where, in contrast to γ -RbAg 4 I 5 , the quadratic frequency dependence, σ (ω) ∝ ω2 , comes into view at low angular frequencies, ω < ω 1 , without being swamped by the translational component. By chance, such an example has been found [98]. It is nominally pure amorphous boron oxide, B 2 O 3 , which contains traces of water. The locally mobile charge carriers are supposed to be hydrogen ions that switch from one neighbouring oxygen ion to another, their number density being about 4 × 1019 cm−3 . Figure 34 is a plot of log σ vs. temperature, with data taken at three fixed frequencies, while Fig. 35 shows the frequency dependence of the 185 K conductivity isotherm as well as its variations with decreasing temperature [98]. From both representations it becomes apparent that this is quite different from the second universality in the sense of Fig. 2. Notably, the iso-frequency

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Fig. 34. Semi-log temperature-dependent conductivity data sets of nominally pure amorphous boron oxide at three different frequencies. The maxima shift to higher temperatures as frequency is increased.

Fig. 35. Log–log plot of the frequency-dependent conductivity of nominally pure amorphous boron oxide at 185 K along with an NCL-type model curve obtained from Eqs. (30) and (6). The data measured at 185 K have been complemented at both high and low frequencies by use of data taken at different temperatures. For this procedure, see Ref. [98]. The arrows indicate changes observed in the spectrum as temperature is decreased.

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Fig. 36. Non-vibrational conductivity spectrum of AgI · AgPO3 glass at 293 K, in a log–log representation, along with a model spectrum. The latter is composed of a MIGRATION-type and an NCL-type component, with ω 1 = 2πν1 larger and less activated than Γ0 [30,97,99].

conductivities now display a pronounced temperature dependence, and this is also reflected in the temperature-dependent features of the conductivity spectrum. A model line for 185 K, as obtained from Eqs. (30) and (6), is included in Fig. 35. At the crossover points, the change of slope is found to be more gradual in the experimental data than in the model curve. However, in view of the structural and dynamic heterogeneities in the amorphous material, this is not unexpected. In Fig. 35, the “two-to-one” crossover point is seen to move rapidly to the lower left as temperature is reduced, the activation energy for ω 1 being (0.36 ± 0.03) eV. At the same time, the linear part of the spectrum moves towards the lower right, thus indicating that the temperature dependence of the highfrequency limiting value of the conductivity is more pronounced than that of ω 2 . A similar temperature-dependent NCL-type contribution to the conductivity has been detected in a glass with a much higher number density of locally mobile ions, where it does, indeed, develop the characteristics of the second universality at low temperatures [30,97,99]. In this silver-ion conducting glass, viz. AgI·AgPO 3 , the broadband non-vibrational conductivity spectrum, measured at 293 K, see Fig. 36, is reminiscent of γ -RbAg 4 I 5 at 113 K. Again, an NCL-type component is seen to be superimposed onto the high-frequency part of the MIGRATION-type spectrum. However, the rates Γ0 and ω1 now appear unrelated, with the larger quantity, ω 1 , having the smaller activation energy. Evidently, the two components reflect ionic movements of quite different kinds. The weight of the NCL-type feature suggests that the rapid localised motion must be ubiquitous in this glass [97].

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Fig. 37. Conductivity spectra of AgI · AgPO3 glass. The spectrum taken at 293 K is the same as in Fig. 36. The 20 K and 30 K spectra show the persistent and dominant non-activated second universality feature, and the dotted line is an indication where its HF plateau is expected to lie. See main text for details.

As in nominally pure amorphous B 2 O 3 , the “constant loss” linear part of the spectrum of Fig. 36 moves to the lower right as temperature is reduced. Eventually, however, this shifting comes to a halt, completing the transition into the second universality, with σ (ν) ∝ ν independent of temperature in the entire impedance regime, see Fig. 37 [30,99]. The reason for this is simple. The processes that “survive” at low temperatures are those that require no (or almost no) activation, while the thermally activated ones have “died out”. In terms of a time-dependent double-well potential, as introduced in the previous section, the remaining non-activated processes are those sketched in the lower panel of Fig. 38. With increasing temperature, these are complemented by slightly activated ones as in the upper panel, where barriers have to be surmounted. The single-particle potentials shown in Fig. 38 are meant to be snapshots taken at times when the ion is at the right or at the left. Their time dependence is thus found to bear resemblance to a see-saw [98]. However, the potential seen by the ion itself, while it moves in the course of time, is flat between neighbouring positions in the lower panel and contains a small barrier in the upper panel of Fig. 38. In summary, it can be stated that, at any given frequency, the temperature independence of the conductivity, which is a characteristic of the second universality, will no longer hold at temperatures where activated localised movements come into play. This effect was already observed in Fig. 29, at the end of Sect. 4.1.

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Fig. 38. “Snapshots” of time-dependent single-particle potentials taken at times when the ion is at its right or left point of return. The lower panel shows processes during which the ion, moving forward and backward in the course of time, does not see a potential barrier. These occur at cryogenic temperatures. At higher temperatures, see the upper panel, they are complemented by processes in which a barrier has to be surmounted. The time-dependence of the single-particle potentials, as visualised in the figure, bears resemblance to a see-saw [98].

In the second-universality low-temperature regime, an interesting consequence results from the fact that the kinetic energy of the moving ions must be supplied by the thermal energy, k B T . Within this regime, all characteristic frequencies and times appear to be independent of temperature, including the times required for the localised ionic displacements themselves. This implies that the kinetic energy must be proportional to the square of the elementary displacement, a2 , and the consequence mentioned above can be formulated as [98] a2 ∝ k B T .

(31)

This proportionality has a role to play, as shown in the next section, when the mean square displacement of the locally mobile ions and the second universality are related to each other.

4.4 Mean square displacement and “two-to-one” crossover ˙ L (t) /WL (t) = ω1 + ω2 WL (t), In the rate equation introduced in Sect. 4.2, −W the validity of the second universality implies that the upper, “one-to-zero” crossover angular frequency, ω 2 , does not show any noticeable temperature dependence. On the other hand, the lower, “two-to-one” crossover angular frequency, ω1 , which is too small to be detected experimentally, may in fact vary widely with temperature. In this section, we use the mean square displacement of the locally mobile ions to show that this view is, indeed, appropriate. In doing so, we consider only cases with ω 2 = const. and ω1  ω2 , but distinguish those where ω1 remains outside the experimental window from those where

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Fig. 39. Normalised mean square displacement as a function of time for an ion in a local void, according to Eq. (34). For details, see main text and Ref. [99].

it moves into it and thus becomes detectable. For clarity, the latter case will in the following not be referred to as second universality, but rather as “genuine” NCL. In this context, “genuine” is meant to signify the absence of any temperature dependence in the linear part of σ(ν).  2 To derive the shape of the time-dependent mean square displacement, r (t) L , we recall that WL (t), see Eq. (30), is its normalised derivative. Therefore, r 2 (t) L is proportional to the integral over WL (t  ), taken from time t  = 0 to time t  = t. This integral is t 0

    

  ω2 1 ω2 ln − ω1 t . WL t dt = + 1 exp (ω1 t) − ω2 ω1 ω1

(32)

In a first step, the shape of the mean square displacement is considered at times that are much shorter than 1/ω 1 . In this regime, the term exp (ω1 t) may be approximated by 1 + ω 1 t, and the expression in the wavy brackets becomes ln (1 + ω2 t). Equation (32) is then the integral of the short-time version of WL (t), i.e., of 1/ (1 + ω2 t), which does not involve ω1 (T ). Being proportional to ln (1 + ω2 t), the mean square displacement is zero in the limit of short times and becomes a linear function of ln (ω2 t) at intermediate times, 1/ω2  t  1/ω1 , see Fig. 39.   In a second step, we consider Fig. 39 and ask for the slope of r 2 (t) L vs. ln (ω2 t) in the intermediate time regime. This slope will have to be proportional to temperature, since the mean square displacement is expected to be proportional to the square of an elementary ionic displacement and hence to the thermal energy, as shown by Eq. (31) in the previous section. In a third step, we ask for the conductivity, σ L (ω, T ), at intermediate angular frequencies, ω1  ω  ω2 . Corresponding to WL (t) ∝ 1/t at 1/ω2  t  1/ω1 , the frequency dependence will be σ L (ω) ∝ ω. Concerning the tempera-

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ture dependence, we recall that, according to linear response theory and with the argument to the Fourier trans in Ref. [33], σ L (ω, T ) is proportional   given  form of d 2 r 2 (t) L /dt 2 , divided by k B T . As r 2 (t) L itself is proportional to NCL behaviour is k B T , this term cancels out and the temperature-independent   reproduced. The linear increase of r 2 (t) L vs. ln (ω2 t) and the second universality or “genuine” NCL are thus equivalent features [99].  From Fig. 39 it is evident that the time regime where r 2 (t) L depends linearly on ln (ω2 t) must be limited at both short and long times. In particular,   there must be a long-time crossover time t = t1 = 1/ω1 , when r 2 (t) L   2 at some merges into its limiting value, r (∞) L , reflecting the fact that only a finite local volume is accessible for each ion. This crossover corresponds to the “twoto-one” transition in plots of log σ L vs. log ω. In Sect. 4.2, ω1 was introduced as a Debye rate of non-interacting ions hopping in double-well potentials. This view, to be revisited at the end of this section, is still valid. In the low-temperature case, however, with ω 2 = const. and ω1  ω2 , we need to consider ω1 and its activation energy, E 1 , in a quite different fashion, and a new meaning is thus seen to emerge for the temperaturedependent crossover angular frequency,  2  ω 1 (T ). From the proportionality, r (t) L ∝ k B T ln (ω2 t), which holds at interme diate times, one obtains r 2 (∞) L ∝ k B T ln (ω2 t1 ) by inserting the crossover time, t1 . This time and its inverse, ω1 , thus depend on the size of the accessible void and on the thermal energy. As suggested by Fig. 39, the temperature dependence of ω1 is now formulated as ω1 = ω2 exp (−E 1 /k B T ), and k B T ln (ω2 t1 ) can thus be rewritten as E 1 . This implies  2   2  r (∞) L r (t) L = k B T ln (ω2 t) for 1/ω2  t  1/ω1 . (33) E1 To include short times, we replace ln (ω2 t) with ln (1 + ω2 t). To include long times as well, this is replaced with the expression in the wavy brackets in Eq. (32). In fact, the isotherms of the normalised mean square displacement presented in Fig. 39 have been calculated from  2       r (t) L ω2 kB T ω2 ln − ω1 t . = + 1 exp (ω1 t) − (34) r 2 (∞) L E1 ω1 ω1 This is a continuous function of time, describing the crossover at t1 = 1/ω1 in a smooth fashion and approaching unity in the limit of long times. The main results obtained are the following. (i) The finite local volume accessible for each ion necessitates the “two-to-one” crossover at angular frequency ω1 , including its temperature dependence. (ii) The rate equation, Eq. (29), is reinterpreted without requiring a double minimum potential that is fixed in space. (iii) However, the sketch of Fig. 38 is still appropriate, with the displacements involved decreasing with decreasing temperature. They become much smaller than the local void, when the second-universality regime is

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attained, and then vary with temperature as T 1/2 . (iv) In this regime, the individual ions do not encounter any potential barriers in the course of their displacive movements. (v) According to Fig. 39, the effect of the Coulomb interactions is such that the location at which each ion performs its non-activated forwardbackward displacive movements is allowed to shift in space. (vi) This shifting is restricted to the finite size of the accessible volume, i.e., of the local void provided for the ion by the network structure. At this point, let us revisit the model treatment as introduced in Sect. 4.2, which is still considered valid at sufficiently high temperatures. Being based on the same function, WL (t), see Eq. (30), the mean square displacement is the same as in Fig. 39, once the crossover angular frequencies are placed at the same positions. However, in contrast to the second universality, both ω 1 and ω2 as well as the limiting HF conductivity now depend on temperature. This is because the mobile ions, when hopping from one potential minimum into the other, have to cross non-zero potential barriers. As discussed earlier, the time dependence of the resulting single-particle potentials produces the characteristic non-Debye NCL-type behaviour. The temperature dependence of the above parameters, however, results in NCL-type conductivity spectra that are temperature-dependent in all of their three parts. Examples have been discussed in Sects. 4.2 and 4.3.

4.5 Low-temperature effect identified as “two-to-one” crossover In this section, we report on the experimental detection of the expected lowfrequency behaviour, σ L (ω) ∝ ω2 , in sodium borate glasses of composition x Na2 O·(1 − x)B2 O3 , with x = 0.05 and x = 0.10. At sufficiently low temperatures, say 10 K, these glasses are found to display the second universality, with both ω 1 and ω2 outside the accessible frequency range. For values of x ranging from 0.01 to 0.10, the second universality type conductivities, measured at 10 K, depend linearly not only on frequency, but also on x. Therefore, the effect can be attributed to the presence of the sodium ions [99]. However, there is one important feature that does not become apparent from the data taken at 10 K [99]. What at this temperature appears to be one single effect is, in fact, a superposition of two. As temperature is increased, one of them is found to remain unchanged, while, at any given frequency, the contribution of the other is found to decrease continuously. As will be shown below, this decrease is caused by a “two-to-one” crossover coming into view. Figure 40 is a plot of log σ vs. temperature, at a fixed frequency of 87.8 kHz [98]. The figure contains data taken from three compositions of x Na2 O·(1 − x) B2 O3 glasses, with x = 0.0, x = 0.05 and x = 0.10, respectively. The maximum observed for x = 0.0 is the one already shown in Fig. 34 and discussed in Sect. 4.3. Clearly, it is not caused by the sodium ions. However, it is still seen to contribute to the conductivities measured on the glasses

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Fig. 40. Semi-log plot of conductivity vs. temperature for three glasses x Na2 O·(1 − x) B2 O3 at 87.8 kHz. Experimental data are represented by open symbols. For x = 0.10, the MIGRATION concept is used to represent the contribution due to hopping, and the second-universality component is indicated by a broken line.

with x = 0.05 and x = 0.10. In these glasses, the first-universality-type hopping motion of the mobile ions causes a rapid increase of σ (T ) at temperatures above some 350 K. This increase is well reproduced by the MIGRATION concept, see the solid line for x = 0.10. For the same composition, a horizontal broken line has been drawn to represent the familiar effect of the second universality [98]. Once the contributions mentioned in the preceding paragraph are removed from the total conductivities, a “low temperature component”, to be abbreviated as LTC, is seen to remain, which decreases with temperature at fixed frequency. In the following, we will show that this component is of the “genuine” NCL type, displaying a “two-to-one” crossover of its conductivity, σ LTC (ν, T ), at fixed temperature. In Fig. 41, log σ LTC is plotted vs. inverse temperature, for three different fixed frequencies [99]. The following properties of the LTC effect become apparent from this figure. (i) At high inverse temperatures, log σ LTC (ν) increases linearly with frequency and becomes independent of 1/T . (ii) At each fixed frequency, there is a crossover which corresponds to a “twoto-one” transition at fixed temperature. (iii) Approximating the data in the two regimes by straight lines, idealised crossover points can be constructed. Evidently, crossover frequency and crossover conductivity are activated with the same activation energy, which is (0.15 ± 0.02) eV.

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Fig. 41. Semi-log plot of the low-temperature conductivity component vs. inverse temperature in 0.10 Na2 O · 0.90 B2 O3 glass measured at three fixed frequencies.

Most notably, the conductivity, σ LTC , does not display any temperature dependence at frequencies that are well above the crossover, where it is exactly proportional to frequency. In this sense, it is a “genuine” NCL component. It is now easy to estimate the relative weights of the two “genuine” NCL components that contribute to the conductivity at low temperatures, e.g., to the 10 K data. At 10 K, the linear frequency regime is surely attained at 87.8 kHz, cf . Fig. 41. Therefore, the relative weights are those with which the LTC component and the second-universality component contribute to the total conductivity at 10 K, see Fig. 40. From the figure, they are estimated to be of about equal weight. In Fig. 42, which is a log–log representation of σ LTC (ν), the slopes of conductivity isotherms, as measured in a restricted frequency window, are seen to change from one to two, as temperature is increased from 5 K to 115 K. As the isotherms shown in the figure will all merge into the same NCL line at high frequencies, broadband conductivities can be constructed from the present data. To this end, the isotherms of Fig. 42 are joined by superimposing those data points that have the same (vertical and horizontal) distance from the NCL line with a slope of one. The result of this procedure is presented in Fig. 43, for a temperature of 55 K. The “two-to-one” crossover is now clearly visible. The inset shows a set of three broadband isotherms obtained for different temperatures. Evidently, the frequency squared low-frequency part shifts in an activated fashion, the activation energy being in agreement with Fig. 41, i.e., (0.15 ± 0.02) eV. As in the case of nominally pure amorphous boron oxide, cf . Fig. 35, the actual “two-to-one” transition is much broader than expected from the model

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Fig. 42. Log–log plot of the frequency-dependent low-temperature component of 0.10 Na2 O· 0.90 B2 O3 glass, as measured at six different temperatures: 5 K, 35 K, 55 K, 85 K, 105 K, 115 K (rightmost).

Fig. 43. Broadband log–log representation of the frequency-dependent low-temperature conductivity component of 0.10 Na2 O · 0.90 B2 O3 glass at 55 K. The inset also includes equivalent plots for 35 K and 75 K, cf. main text.

sketched in the previous section. Again, this reflects  theexistence of local voids of different sizes and hence different values of r 2 (∞) L , resulting in different times, t1 , at which these values are attained. An interesting question is the following. Why are there two different NCL effects in the sodium borate glasses with x = 0.05 and x = 0.10? According to

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the argument given along with Fig. 39, it must be concluded that two classes of voids are available for the localised motion of mobile ions, differing in size considerably. The class of larger voids is then associated with values of ω 1 that are so small that they remain outside the experimental frequency window. Correlated movements of ions within these voids are thus expected to constitute the dynamics of the second universality, while the ions in the smaller voids contribute to the “genuine” NCL effect which shows the “two-to-one” transition. For obtaining more detailed information on possible ionic positions and their different local neighbourhoods, suitable experimental techniques would include EXAFS and advanced solid state NMR.

4.6 More examples, below and above the glass transition Here we first report on another ionic solid with a conductivity that consists of two NCL components. Again, one of them is temperature-independent at a given frequency, while the other is not. This solid is BMIm-BF 4 , now not considered in its more famous liquid state, cf . Sect. 3.5, but below its glass transition temperature, which is Tg = 188 K. Below Tg , similar features have also been noted and discussed in BMIm-PF 6 [108]. In the case of BMIm-BF 4 we analyse our data in the spirit of the two scenarios shown in Fig. 38. Figure 44 is a plot of log σ vs. temperature, comprising data taken on BMIm-BF 4 at eight frequencies from 95 Hz to 410 kHz [109]. At temperatures well above the glass transition, the data collapse in a single curve corresponding to the temperature-dependent DC conductivity. Increasing dispersion is observed as temperature is reduced below 230 K, and at the glass transition itself log σ is found to be continuous at each frequency. Below the transition, the DC conductivity is zero, while the AC conductivity may be described as a superposition of two components. One of them is of the second-universality type, see the horizontal lines, i.e., it is proportional to frequency and becomes temperature-independent at sufficiently low temperatures, while in the vicinity of the glass transition activated localised movements come into view at low frequencies. The shape of the other component is like a hump, with the position of its maximum shifting to higher temperatures with increasing frequency. This is reminiscent of nominally pure amorphous boron oxide, where a similar behaviour was observed in Fig. 34. Careful removal of the second-universality type contribution and subsequent transformation of the remaining data into frequency-dependent conductivities result in Fig. 45, which is a plot of log σ vs. log ν at a fixed temperature of 123 K. Again, there is considerable similarity to amorphous boron oxide, cf . Fig. 35. The frozen structure of BMIm-BF 4 thus appears to permit two different kinds of correlated localised movements of charge. One of them shows the characteristics of the second universality and resembles the respective feature in 0.3 Na2 O · 0.7 B2 O3 , cf . Fig. 29. In the other, both ω1 and ω2 are clearly

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Fig. 44. Semi-log plot of the temperature-dependent conductivity of BMIm-BF4 at eight frequencies. The glass transition temperature is 188 K. Symbols represent the experimental data. Horizontal lines indicate a second-universality component seen in the glassy state.

Fig. 45. Experimental conductivity isotherm of BMIm-BF4 at 123 K (circles), reduced by the second-universality component. At low frequencies, these data are complemented by data taken at higher temperatures (crosses). The solid line is a model spectrum, while the Debye-type dashdotted line is included for comparison.

temperature-dependent, reflecting a mechanism in which ions need to cross a barrier between potential minima in either direction, as outlined in Sect. 4.2 and at the end of Sect. 4.4.

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Fig. 46. Log–log plot of σnon-vib vs. frequency for BMIm-BF4 in the liquid state, at 273 K. Open symbols represent the experimental data (reduced by a minor “vibrational” component). The solid line is a model spectrum obtained by superimposing a MIGRATION-type component (dash-dotted line) and an NCL-type component (dotted line). The frequencies Γ0 /2π and ν1 cannot be distinguished.

Remarkably, an NCL-type effect is still observed when BMIm-BF 4 is heated above the glass transition temperature, into the liquid phase. This proves that, even in the liquid, the structure provides voids for the ions to move in, in a locally restricted fashion. Note that such an effect could not be detected in a much simpler ionic melt, i.e., in calcium potassium nitrate, CKN, see Fig. 14. Figure 46 is a log–log plot of the frequency-dependent conductivity of BMIm-BF 4 at 273 K, i.e., above the glass transition [109]. This spectrum is reminiscent of γ -RbAg 4 I 5 , cf . Fig. 30, because of two features that are found in both systems. (i) The low-frequency part of the spectrum is well reproduced in terms of the MIGRATION concept, cf . Sect. 3.5, while the high-frequency part is swamped by an NCL-type component, the resulting slope of one being clearly visible. (ii) As in γ -RbAg 4 I 5 , the experimental data are perfectly reproduced, if the values of two parameters are chosen to be identical, not only at 273 K, but also at other temperatures. These are the elementary hopping rate, Γ0 (T ), as used in the MIGRATION concept, and the “two-to-one” crossover angular frequency, ω1 (T ), as used to describe the NCL-type effect. Here we recall that the MIGRATION-model parameters, Γ0 (T ) and σ HF (T ), are both well-known from the construction presented in Sect. 3.5.

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Fig. 47. Conductivity spectra of a polymer electrolyte (1 molal NaPF6 in a PEO : PPO random block copolymer) measured at four temperatures above the glass transition (circles). For the MIGRATION concept model curves (dashed lines) the values of σHF (T ) have been taken from the construction of Eq. (20) and Eq. (21), with K = 2.0. The data are found to exhibit a clear linear regime distinct from the MIGRATION-type component. The vibrational component is also shown.

It is thus tempting to envisage the dynamics of the mobile ions in BMIm-BF 4 in a view that is similar to that for γ -RbAg 4 I 5 . The essence of this would be as follows. At short times, when the arrangement of the ions is still close to rigid, there would be two kinds of elementary displacement in the melt, viz., those with and without further connectivity, i.e., with and without a chance of translational motion. At the same time, the barriers to be crossed in an elementary displacement would be the same. An NCL-type effect has also been detected in a salt-in-polymer electrolyte. Here, we consider the dynamics of 1 molal NaPF 6 in a PEO : PPO random block copolymer [21,22]. In this class of polymer electrolytes, the anions are delocalised and the dominant charge carriers, while the cations are coordinated to the host. Conductivity spectra have been measured in the amorphous phase above the glass transition temperature, which is Tg = 231 K. Four such conductivity isotherms are shown in Fig. 47. Again, the low-frequency part conforms to the MIGRATION concept, Summerfield scaling is valid, and the temperature dependence of the DC conductivity is well described by Eq. (20), see Sect. 3.5, E ∗ being close to 0.25 eV. MIGRATION-type conductivity spectra, constructed as in Sect. 3.5, have been included in Fig. 47. From the figure it is evident that an NCL-type linear frequency dependence of σ (ν) is observed at frequencies above 10 GHz. In contrast to liquid BMImBF 4 , however, ω1 (T ) is now found to be larger than and unrelated to Γ0 (T ). This is similar to AgI·AgPO3 glass. In view of the structural features of the polymer electrolyte, the most plausible interpretation for the NCL-type com-

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ponent is in terms of localised movements of dipolar units on the polymer backbone, containing oxygen ions as well as sodium ions, which are attached to them.

5. Conclusion The fascination of the two universalities lies in their ubiquity, i.e., in the occurrence of either or both of them in different kinds of disordered ion-conducting materials. These include crystalline, glassy and polymer electrolytes, molten salts and ionic liquids. Evidently, the existence of the universalities is not primarily a consequence of phase, structure and composition, but rather of some common laws that govern the many-particle dynamics of the mobile ions. On our expeditions towards and beyond the universalities, it was our major goal to arrive at an understanding of the origin and nature of those common laws. To this end, we have taken conductivity spectra over unusually wide ranges of frequency and temperature. We have considered the corresponding real functions of time and found rate equations that reproduce them as well as the spectra themselves. These rate equations may, therefore, be regarded as manifestations of the underlying common laws. At the same time, they also form a sound basis for understanding and visualising the phenomena in terms of simple physical pictures such as Figs. 7 and 32. Expressing basic rules, the rate equations, Eqs. (8) to (10) and Eq. (29), remain unchanged when used to describe the many-particle dynamics in different ionic systems. The meaning of certain “ingredients”, however, is neither addressed nor specified in the equations. Rather, it may change from one system to another. For instance, ions or vacant sites may act as “charge carriers”, and the forces that build up and reduce “mismatch” need not necessarily be Coulombic, but may also be steric in character (e.g., in polymer electrolytes). Moreover, there are “boundary conditions” that may be expressed in a particular temperature dependence of the values of the parameters, cf . Sect. 3.5 for the example of ionic liquids. It should also be noted that the rate equation for strictly localised motion, Eq. (29), has the remarkable property of allowing two complementary interpretations, being valid at high and low temperatures, respectively, see Sect. 4.4. In our present overview we hope to have convinced the reader of the strength and versatility of our model concept. To conclude, two of our major findings shall be mentioned in particular. (i) In fragile ionic systems, the characteristics of their first-universality-type conductivity and fluidity spectra are used to trace the dynamics of the mobile ions from short to long times. A new equation is thus obtained for σ DC (T ) and f DC (T ), which can meaningfully replace the empirical VTF relation. (ii) The typical second universality behaviour, with σ (ν) ∝ ν, cannot continue indefinitely at low frequencies.

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Rather, there must be a crossover into a quadratic frequency dependence which has, indeed, been detected experimentally in a sodium borate glass.

Acknowledgement The results reported in this overview have been obtained under the scientific umbrella of Sonderforschungsbereich 458. We gratefully acknowledge the establishment and constant financial support of SFB 458 by Deutsche Forschungsgemeinschaft, DFG. It is a great pleasure to thank all those who have over the years participated in our common effort to gain a better understanding of the dynamics of the mobile ions in materials with disordered structures. This includes friends and colleagues both inside and outside SFB 458, as well as KF’s past coworkers. To name just a few, in alphabetical order, we would especially like to thank A. Bunde, C. Cramer-Kellers, W. Dieterich, H. Eckert, A. Heuer, M. D. Ingram, H. Jain, F. Krok, J. Maier, H. Mehrer, A. Moguˇs-Milankovi´c, A. S. Nowick, S. Pas, B. Roling, M. Schönhoff, P. Singh, H.-D. Wiemhöfer and D. Wilmer. Besides thanking our friends at SFB 458 for an atmosphere of intense engagement and healthy exchange of views, we also would like to cite a few relevant results of theirs [110–126] which have added new insights while addressing issues complementary to those treated in this overview.

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519

Author Index Abouzari, R., 363 Badr, L. G., 459 Banhatti, R. D., 459 Bastek, J., 275 Bawohl, M., 73 Berkemeier, F., 363 Bhide, A., 123 Biermann, C., 459 Bracht, H., 245, 399 Brinkmann, Chr., 103 Burjanadze, M., 7 Cramer, C., 123 Dinges, T., 43 Echelmeyer, T., 303 Eckert, H., 43, 159, 339 Faske, S., 103 Fellberg, E. M., 459 Fögeling, J., 275 Friedrich, R., 423 Funke, K., 459 Gallasch, T., 363 Gentschev, A.-C., 7 Greiwe, G., 363 Grofmeier, M., 245 Heuer, A., 423 Hiller, M. M., 7 ´ W., 123 Imre, A. Karatas, Y., 7 Kaskhedikar, N., 7 Kloss, S., 7 Koch, B., 103 Kogel, L. M., 7 Köster, T. K.-J., 303 Kunze, M., 339

Lange, S., 73 Laughman, D. M., 459 Lühning, L., 423 Messel, J., 73 Müller, R. A., 7 Mutke, M., 459 Nilges, T., 73 Obeidi, S., 275 Osters, O., 73 Pöttgen, R., 43 Roling, B., 223, 399 Röthel, S., 423 ˇ Santi´ c, A., 459 Schiffmann, G., 303 Schirmeisen, A., 399 Schmitz, G., 363 Schönhoff, M., 123, 339 Schulz, A., 339 Sreeraj, P., 43 Staesche, H., 223 Staskunaite, R., 245 Stockhoff, T., 363 Stolina, R., 7 Stolwijk, N. A., 275 Sunder, K., 245 Taskiran, A., 399 Vettikuzha, P., 7 Vogel, M., 103 Voigt, N., 303 Wiemhöfer, H.-D., 7, 43, 339 Wiencierz, M., 275 Wrobel, W., 459 Wüllen, L. v., 303 Wunde, F., 363

520

Keywords Alkali 245 Alkaline-Earth 245 Battery Electrodes 363 Cation–Anion Pairs 275 Complex Coacervate 123 Computer Simulations 423 Conductivity 245, 339 Crystal Chemistry 43 DC Conductivity 275 Dynamical Heterogeneities 103 Electrochemistry 7 Electrostatic Atomic Force Microscopy 399 Fokker–Planck Equation 423 Glass Ceramics 223 Glasses 223, 245 Glass Membranes 363 Glass Structure 159 Impedance Spectroscopy 123 Ion and Electron Conductors 73 Ion Conduction 459 Ion Dynamics 103, 399 Ion Dynamics in Solids 73 Ionic Conductivity 7, 363, 423 Ionic Mobility 43 Ionic Motion 223 Ionic Transport 303 Ion-Transport Model 275 Layer-By-Layer 123 Lithium 43

Materials Science 459 Microscopical Analysis 363 Model 459 Multi-Time Correlation Functions 103 Network Formers 245 Network Modifiers 245 Non-Linear Effects 423 Nonlinear Ionic Conductivity 223 Nuclear Magnetic Resonance 103, 339 Oxide Nano-Particles 275 Polymer Electrolyte 7, 123, 339 Polymer Synthesis 7 Polyphosphazene 7 Polysiloxane 7 Proton Conductor 123 Pulsed Field Gradient NMR 339 Salt Precipitation 275 Self-Diffusion 245, 459 Silicate 245 Solid-State Electrolytes 103 Solid State NMR Spectroscopy 159 Spin Relaxation 339 Structure-Property Relations 159 Tetrelides 43 Thermoelectrics 73 Thin Films 363 Transference Number 275