Electronic & ionic conduction & correlated dielectric relaxations in molecular solids

Electronic & Ionic Conduction & Correlated Dielectric Relaxations in Molecular Solids MANESH ZACHARIAH Supervisors: Dr. Roberto MACOVEZ Prof. Dr. Josep Llus TAMARIT MUR Barcelona, September 2016 PhD programme in Computational and Applied Physics Departament de Fisica All the contents of this work are licensed under the Creative Commons https://creativecommons.org/licenses/by-nc-nd/4.0/ Abstract The study of crystalline materials has played a prominent role in the traditional approach to solid state physics: the study of the solid state emerged from crystallography, and the basic theories of solid state physics were formulated for the case of crystalline matter. However, many practical applications use materials which are more abundant in nature, and that are weakly or strongly disordered, such as molecular crystals, glasses (amorphous solids), plastic crystals, liquids, liquid crystals, etc. In glasses, for example, the arrangement of the constituent atoms or molecules lacks the slightest vestige of long range order. The advances that have been made in the physics and chemistry of amorphous solids have contributed to the Nobel awards earned by N. F. Mott, P. W. Anderson, and P. J. Flory. Much of the intellectual fascination about disordered solids arises from the fact that scientific insight must be achieved without the help of the well-mastered solid-state concepts associated with periodicity which describe the crystalline solid state. While some old approaches remain useful for disordered solids, significant advances have been made only by developing new approaches such as localization theory and percolation. From an applied perspective, much of the intense research interest in disordered solids is driven by the technological importance of these materials, which includes the use of ultra-transparent optical fibers in telecommunications, the use of amorphous semiconductors in xerography and solar cells, the use of liquid crystals in display technology, and the ubiquitous everyday uses of polymers and organic glasses as structural materials. From a fundamental viewpoint, a deeper understanding of the properties of disordered materials is needed to explore the many fascinating condensed matter issues related with disorder. Disordered materials display low electrical conductivities than their crystalline counterparts, due to localization of valence electrons, so that electron hopping is the main charge transport mechanism. On the other hand, some disordered materials are able to conduct electricity by the diffusion of ions through interstitial sites and their ionic conductivity is normally higher than the crystalline counterparts. The disorder of these materials may be dynamic, rather than static, and the study of the dynamics both in the glass state and at higher temperature helps unveiling the origin of the physical properties of the glass state. The type of disorder present in a material, for example, whether it only involves orientational degrees of freedom (as in a plastic crystal) or only translational (as in a liquid crystal) or both (as in window glass), or whether it is static (as in amorphous solids) or dynamic (as in a liquid) all these factors have an important impact on the conductivity and other physical properties such as viscosity, plasticity or stiffness. This thesis focuses on the experimental study of the conduction properties and molecular dynamics of molecular solids made of fullerene derivatives, and plastic co-crystals based on the succinonitrile molecule. The studied materials display, depending on the case, electronic, protonic and ionic conduction. The experimental technique employed to investigate these materials is broadband dielectric spectroscopy, which allows studying simultaneously molecular dynamics and electrical conductivity in broad range of frequency and temperature (Chapter 3). Fullerenes are relatively simple molecules; pristine fullerenes such as C60 and C70 and several of their derivatives are excellent electron acceptor and n-type semiconductors, in some cases with high electron mobility. Some fullerene solids even display superconductivity at low temperatures, while some fullerene salts with small cations show remarkably high ionic conductivities. We study in particular the intrinsic and water-induced charge transport in a highly symmetric organic fullerene derivative, C60(ONa)24, which is synthesized as a polycrystalline hydrate and which can be obtained as a pure material by heating to sufficiently high temperature. We show that while the pure material is an n-type (electron) semiconductor, exposing it to humid atmosphere leads to a dramatic conductivity enhancement which is due to charge transport through the hydration layers present on the surface of the crystalline grains, likely due to a proton exchange mechanism. We also show that the dc conductivity of the hydrate is strongly temperature dependent across the dehydration process, and that both pure and hydrated materials display a conductivity-related dynamic process associated with accumulation of electrons at grain boundaries (Chapter 4). We argue that presence of water has strong impact on the conduction properties, both in the case of the dc transport and the frequency-dependent chargeaccumulation dynamics. In Chapter 5 we focus on a brominated fullerene derivative, namely C60Br6, which shows n-type electronic conduction below room temperature and a non-trivial phase behavior. Finally, in Chapter 6 we analyze the relaxation dynamics and the ionic conductivity of plastic-crystalline ionic conductors based on the succinonitrile (C4H4N2) molecule, which behave as a solid ion or proton conductor in the presence of ionic impurities or when doped with acids or lithium salts, suggesting a possible application as plastic electrolyte. We observe that the plastic co-crystals of succinonitrile with a similar molecule, glutaronitrile (C5H6N2), represent the first ever known plastic crystals to display a perfect correlation between the ion drift and the on-site reorientational dynamics. These surprising results, never reported before in an ordered solid, are interpreted in terms of a perfect correlation between the time scale of translational diffusion and that of purely reorientational on-site dynamics, which is reminiscent of the similar relaxation timescales of ethanol in its supercooled liquid and plastic-crystalline phases. Doping the co-crystals with lithium salts boosts the conductivity but breaks such perfect correlation, which indicates that the rotation-drift correlation is only valid when charge transport is dominated by self-diffusion of molecular ions intrinsic to succinonitrile or glutaronitrile, while the motion of smaller atomic (Li+) ions is decoupled from the molecular dynamics. Contents 1. Introduction 1.1 Conduction Mechanisms in Condensed Matter 2 1.2 Molecular Dynamics in Condensed Phases 6 1.3 Materials of Choice and Main Experimental Tool 9 1.3.1 Fullerenes and their derivatives 9 1.3.2 Succinonitrile (SN) 11 1.3.3 Method 12 1.4 Outline of the Thesis 12 2. Models of Dielectric Relaxation and Charge Transport in Disordered Systems 2.1 Introduction 19 2.2 Polarization Mechanism 22 2.3 Detailed Frequency-Dependent Response 28 2.4 Dielectric Relaxation Models 32 2.4.1 Debye Model 32 2.4.2 The Havriliak-Negami Function 34 2.4.3 Cole-Cole and the Cole-Davidson Functions 35 2.4.4 The Kohlrausch-Williams-Watts Function 36 2.5 Charge Transport Mechanism and Conductivity-Induced Dielectric Losses 37 2.5.1 Dc Transport 39 2.5.2 Ac Transport 41 2.5.3 Relation between Dc and Ac Conductivities 44 2.5.4 Scaling of Ac Conductivity 46 3. Experimental Techniques and Data Analysis 3.1 Basic Characterization Techniques 55 3.1.1 Thermogravimetric Analysis 55 3.1.2 Differential Scanning Calorimetry 57 3.1.3 Fourier-Transform Infrared Spectroscopy 59 3.1.4 X-ray Powder Diffraction 59 3.2 Broadband Dielectric Spectroscopy 61 3.3 Dielectric Data Analysis 67 3.3.1 Complex Permittivity and Dielectric Relaxations 68 3.3.2 Ac Conductivity and Space-Charge Losses 71 3.3.3 Temperature Dependence of the Relaxation Times and Dc Conductivity 76 4. Water-Triggered Conduction and Polarization Effects in a Hygroscopic Fullerene 4.1 Introduction 85 4.2 Synthesis and Experimental Methods 87 4.3 Preliminary Characterization 89 4.4 Detailed Characterization by BDS 96 4.4.1 Pure C60(ONa)24 97 4.4.2 Effect of the Surface Hydration Water on Pure C60(ONa)24 108 4.4.3 C60(ONa)24 16H2O Hydrate 114 4.5 Conclusions 124 5. Hopping Conduction and Conductivity Cross-Over in Bromofullerene 5.1 Introduction 133 5.2 Synthesis and Experimental Methods 135 5.3 Thermodynamic Characterization 137 5.4 BDS Results and Discussion 138 5.4.1 Variable Range Hopping Conduction in C60Br6 143 6 proton conduction, the dc conductivity is often measured as function of relative humidity (and possibly temperature), although this approach is not conclusive to discriminate between proton and ion conduction as ionic charge transport is also affected by the relative humidity.8,9,10,11 As a graphical summary, Figure 1.2 shows the different conduction mechanism in condensed phases4 (see also section 2.5 of Chapter 2). Figure 1.2: Scheme of the different conduction mechanisms in condensed phases. 1.2 Molecular Dynamics in Condensed Phases In physics, a state of matter is one of the distinct forms that matter takes on. Three states of matter are observable in everyday life: solid, liquid and gas. In gaseous state, particles are well separated and in constant motion with no binding force between them. The particles in a liquid or solid are fairly close together and their mutual interactions cannot be ignored; these two states of matter, and those of similar density such as glasses, polymers, colloids, or liquid crystals, constitute what is called “condensed matter”. By cooling (lowering the temperature) or compressing (increasing the pressure), the physical state of matter changes from gas to liquid, and main charge carrier (type of conduction) Electron Band-like conduction (crystals) electron or hole hopping (disordered & molecular solids) Ion (except H+ ) Diffusion (liquid electrolytes) Hopping between empty sites (solid electrolytes) Proton Vehicle mechanism Grotthus shuttling (systems with extended Hbond networks) 7 further on from liquid to solid. When the temperature is decreased from the gas phase, interactions between molecules become more and more important and correlations between the motions of different constituents emerge and a new dense phase appears as the liquid, in which molecules still move around and do not display any long-range order. Liquids are perhaps the simplest case of disordered condensed phases, because the structural randomness is most pronounced in this case. As the temperature of the liquid is further decreased, the molecules starts to pack together in a more regular way and reach a higher density state. Then the liquid has crystallized, with each molecule occupying a specific location on a lattice: a solid crystal is formed. In many systems, the liquid phase does not crystallize right away, but rather it can be supercooled below the melting point without acquiring any translational or orientational order. In the “supercooled liquid” state, further cooling forces the system to freeze into a disordered state: a glass has formed12,13 (see Figure 1.3). In some molecular crystals the centers of mass of the molecules form a lattice, but the molecular orientations are dynamically or statically disordered. In the case of dynamic orientational disorder, the orientational degrees of freedom of the molecules give rise to a characteristic state, called “plastic phase”, which exists between the liquid phase and the perfectly crystalline phase. The molecular compounds which display a plastic state are called orientationally disordered crystals (ODIC).14 The dynamic orientational disorder can freeze yielding glassy crystals or orientational glass (OG). Another class of disordered phase called liquid crystals in which molecules keep an at least average orientational order but organize in layers with inner translational disorder.15 In addition to the translational and orientational disorder, another type of disorder that can be present in molecular materials is the conformational disorder. This can be displayed by materials whose constituent molecules can exist in more than one possible isomeric form. If different molecular isomers are present in the liquid phase, they are usually preserved in the glass phase and they may even be present in the crystalline or ODIC phases (conformationally disordered materials may also display at the same time also translational and/or orientational disorder). 8 The structural glassy state can be defined as a state with a collective molecular motion (“relaxation”) time above 100 seconds or equivalently a state in which the viscosity takes a value of the order of η = 1013 poise. The temperature at which the glass transition takes place is called glass transition temperature, Tg and it is different for different cooling rates due to the dynamic character of the glass transition, which is a kinetic transition rather than a thermodynamic one, although this is still in a lively discussion.16 A smaller cooling rate allows the sample to stay closer to equilibrium (i.e., the supercooled liquid state) until lower temperatures. Figure 1.3: Schematic representation of the possible transitions from the liquid state of dipolar molecules into a structural glass (SG), an ordered crystal or an orientational glassy phase (OG).14 9 In the study of glass-forming system, the most important dynamic process is the so-called primary (α) relaxation process, which corresponds to the cooperative, collective rearrangement of the constituent molecules and is closely associated with the glass transition.17 The frequency of the α process decreases with decreasing temperature, and displays in fact a continuous, dramatic slow-down by several decades in a short temperature interval upon approaching Tg. Other molecular dynamic processes can occur in addition to the primary relaxation. These molecular motions are usually less cooperative and occur at longer frequency than the α relaxation, and are called secondary () relaxations. These secondary relaxations may be quasi single molecule motions,18 or else correspond to the motion of mobile side groups or parts of the constituent molecule. 1.3 Materials of Choice and Main Experimental Tool 1.3.1 Fullerenes and their derivatives Fullerenes are the fourth known allotrope of carbon after diamond, graphite and amorphous carbon. They exist in the form of hollow molecular cages of quasi-spherical or ellipsoidal shape, as nested cages (carbon onions), and as hollow or nested tubes (carbon nanotubes). An interesting type of fullerene is the bukmintserfullerene C 60 , which is built up from 60 carbon atoms in alternating hexagonal and pentagonal rings to form a shape similar to a football (see Figure 1.4). The structure of a buckminsterfullerene is a truncated icosahedron with 60 vertices and 32 faces (20 hexagons and 12 pentagons where no pentagons share a vertex) with a carbon atom at the vertices of each polygon and a bond along each polygon edge (each carbon atom in the structure is bonded covalently with 3 others). The van der Waals diameter of the C 60 molecule is approximately 1 nm. Fullerenes and several of its derivatives are excellent organic semiconductors, with some rather unique features, such as high electron mobility and low LUMO level. They have therefore been used extensively as the 10 active material in OFETs, and as the electron acceptor material in Organic Photovoltaics (OPV) devices. 19,20 Figure 1.4: Molecular structure of C60. In the room-temperature solid phase of C60, the molecules form a (face-centered) cubic lattice and rotate very rapidly, resulting in an orientationally disordered phase. Below 260 K this free-rotor motion is reduced to a ratcheting motion between two preferred orientations. 21 , 22 , 23 Such order-disorder phase transition displays a temperature hysteresis of 5 K, and takes place at unusually high temperature compared with other systems. The merohedral twinning motion of the C60 molecules finally freezes out at a glassy transition at 90 K, below which the lattice structure is simple cubic. What makes C60 a nice “playground” to study organic solids are its simple chemical formula, its symmetric shape, and its ability to form many different compounds with other elements or organic molecules. When salts of C60 (“fullerides”) are formed with alkali metals, C60 converts from a semiconductor into a relatively good organic electronic conductor, and even to a superconductor at low temperature.24 Some alkali fullerides even display ionic conduction.25 In their pure form, fullerenes and their derivatives form electrically insulating or semiconducting solid-state phases. Since fullerenes and their derivatives are electron acceptors, they typically behave as “n-type semiconductors”, in which charge carriers are electrons in the upper band derived from empty molecular orbitals rather 11 than holes in the lower band derived from filled molecular orbitals. If cations or protons present either as constituents or as impurities, can also contribute to and even dominate the overall charge transport.26 In this thesis we have studied two fullerene derivatives, namely C60(ONa)24, which is a strongly hygroscopic material that even forms a crystalline hydrate, and C60Br6 a dipolar bromofullerene derivative. Both display electronic hopping conduction in suitable temperature ranges. 1.3.2 Succinonitrile (SN) The succinonitrile molecule (NC–CH2–CH2–CN) forms a plastic crystalline phase with significant structural disorder, resulting in greater mechanical plasticity and enhanced self-diffusion compared with most other plastic crystals (including solid C60, which displays no visible mechanical plasticity) (Figure 1.5). The orientationally disordered phase also displays conformational disorder, with most conformers possessing a dipole moment, which leads to a dielectric constant of 55 at room temperature in the solid state.27, 28,29 In contrast to other plastic crystals, it is difficult to supercool and at T < 233 K the material transforms into an orientationally ordered crystal. The plastic-crystalline phase of succinonitrile has (body centered) cubic structure; the disorder in such phase is associated both with isomeric fluctuations involving a rotation about the central C−C bond of the molecules, and to molecular jumps from one diagonal position of the bcc cell to another.30,31,32 Figure 1.5: Molecular structure of Succinonitrile. 12 The plastic-crystal phase of SN is limited to the temperature range of about 233 - 331 K. However, the addition of the related molecular compound glutaronitrile (NC–(CH2)3–CN) strongly extends the plastic-crystalline region, enabling the transition into an orientational glass. In this thesis we study the conduction properties, phase behavior, and molecular dynamics of co-crystals of succinonitrile with two other nitriles (glutaronitrile and acetonitrile), both pure and doped with lithium salts. We find that the succinonitrile-rich co-crystals with glutaronitrile display a perfect correlation between the molecular self-diffusion and the on-site reorientational dynamics. In detail, these co-crystals are found to obey the Walden rule7 which is usually seen in (ideal) liquid electrolytes. 1.3.3 Method In order to probe molecular dynamics and charge transport in disorder system we used broadband dielectric spectroscopy (BDS). Due to its unique ability to probe molecular fluctuations and charge transport over a broad frequency and temperature range, BDS has proven indispensable in the quest to understand the underlying mechanisms of charge transport and dynamic motion in disordered systems. Its advantage is that it allows studying in the same experiment both the conduction properties and the dielectric polarization response of a sample. BDS studies can be performed on a very wide range of disordered organic and inorganic systems ranging from amorphous semiconductors and glasses to metal-cluster compounds to polymers and polymer composites.33,34,35 1.4 Outline of the Thesis This thesis focuses on the molecular dynamics and conduction mechanisms of three different materials. Chapter 2 gives an introduction to the theoretical concepts of the dielectric relaxation processes and charge transport mechanisms in disordered systems, while Chapter 3 provides information about the experimental techniques and the methods used for the acquisition and analysis of experimental data. The presentation, analysis and discussion of the obtained results are presented in the 13 following three chapters (one for each organic solid under scrutiny). Chapters 4 and 5 discuss the conductivity mechanism and related dielectric loss in two fullerene derivatives. In particular, Chapter 4 deals with solid C60(ONa)24, which is studied both in pure form and as crystalline hydrate, and is found to display both electronic and protonic conduction. Chapter 5 deals with the electron transport properties and phase behavior of the C60Br6 derivative. The last chapter on experimental results (chapter 6) deals with the cooperative and non-cooperative relaxations and ionic conduction in pure and lithium-salt doped succinonitrile co-crystals. 14 References: 1 Safa Kasap and Peter Capper, Handbook of Electronic and Photonic Materials, Springer 2006. 2 Brutting, W.; Adachi, Ch. (Ed.s). Physics of Organic Semiconductors, 2nd Ed. Wiley 2012. 3 Stallinga, P. Electronic transport in organic materials: Comparison of band theory with percolation/(variable range) hopping theory, Adv. Mater. 2011, 23, 3356– 3362. 4 Baranovski, S. Charge Transport in Disordered Solids with Applications in Electronics. Wiley 2006. 5 van Staveren, M. P. J.; Brom, H. B.; de Jongh, L. J. Metal-Cluster Compounds and Universal Features of the Hopping Conductivity of Solids. Phys. Rep. 1991, 208, 196. 6 Kwan Chi Kao, Dielectric phenomena in solids with emphasis on physical concepts of electronic processes, Elsevier 2004. 7 Walden, P. Z. Organic solvents and Ionization Media. III. Interior Friction and its Relation to Conductivity. Phys. Chem. 1906, 55, 207-246. 8 Cramer, C.; De, S.; and Schönhoff, M. Time-Humidity-Superposition Principle in Electrical Conductivity Spectra of Ion-Conducting Polymers. Phys. Rev. Lett. 2011, 107, 028301. 9 Gränicher, H.; Jaccard, C.; Scherrer, P.; Steinemann, A. Dielectric Relaxation and the Electrical Conductivity of Ice Crystals. Discuss. Faraday Soc. 1957, 23, 50–62. 10 Vilčiauskas, L.; Tuckerman, M. E.; Bester, G.; Paddison, S. J.; Kreuer, K. D. The Mechanism of Proton Conduction in Phosphoric Acid. Nat. Chem. 2012, 4, 461–466. 15 11 Knight, C and Voth, G. A. The Curious Case of the Hydrated Proton. Acc. Chem. Res. 2012, 45, 101–109. 12 Anderson,P. W. Through the Glass Lightly. Science 1995, 267, 1615-1616. 13 Lunkenheimer, P.; Schneider, U.; Brand, R and Loidl, A. Glassy Dynamics. Contemporany Physics 2000, 41, 15-36. 14 Brand, R.; Lunkenheimer, P. and Loidl, A. Relaxation Dynamics in Plastic Crystals. J Chem. Phys. 2002, 116, 10386-10400. 15 Drozd-Rzoska, A.; Rzoska, S.J.; Pawlus, S.; Martinez-Garcia, J.C.; Tamarit, J.Ll.; Evidence for Critical-Like Behavior in Ultraslowing Glass-Forming Systems. Phys. Rev. E 2010, 82, 031501. 16 Albert, S.; Baue, Th.; Michl, M.; Biroli, G.; Bouchaud, J.-P.; Loidl, A.; Lunkenheimer, P.; Tourbot, R.; Wiertel-Gasquet, C.; Ladieu, F. Fifth-Order Susceptibility Unveils Growth of Thermodynamic Amorphous Order In GlassFormers. Science 2016, 352, 1308-1311. 17 Angell, C. A.; Ngai, K. L.; McKenna, G. B.; McMillan, P. F.; Martin, S. F. Relaxation in Glassforming Liquids and Amorphous Solids. J. Appl. Phys. 2000, 88, 3113-3115. 18 Johari, G.P. and Goldstein, M. Viscous Liquids and the Glass Transition. II. Secondary Relaxations in Glasses of Rigid Molecules. J. Chem. Phys. 1970, 53, 23722388. 19 Lai, Y.-Y.; Cheng, Y.-J.; Hsu, C.-S. Applications of Functional Fullerene Materials in Polymer Solar Cells. Energy Environ. Sci. 2014, 7, 1866–1883. 20 Anthopoulos, T.D.; Singh, B.; Marjanovic, N.; Sariciftci, N.S.; Ramil, A.M.; Sitter, H.; Cölle, M.; de Leeuw, D.M. High Performance N-Channel Organic Field-Effect Transistors and Ring Oscillators Based on C60 Fullerene Films. Appl. Phys. Lett. 2006, 89, 213504. 22 2.2 Polarization Mechanisms The polarization mechanism can be classified based on the type of material and the constituents that undergo polarization8. For very high (i.e., above optical/UV) frequencies even electrons cannot keep up with the changing field, resulting in a relative permittivity εr =1. When placing a non-polar medium into a capacitor, under the influence of the electric field two types of polarization may arise, known as electronic and atomic polarizations, which are dominant at relatively high frequencies (in the UV, visible, and IR ranges). Electronic polarization takes place on a time scale of 10–15 s, which lies in the visible or near UV region of the electromagnetic spectrum, while the atomic polarization takes place at a longer time scale of 10-12 s (IR region). The electronic polarization (Pe) occurs due to the displacement of the negatively charged electron cloud relative to the positive nuclei in the atom and the electron cloud, which can be easily displaced under an applied electric field due to the low mass of the electron. In this way, induced dipole moments arise, producing the electronic polarization. The atomic polarization (Pa) is due to the relative displacement of ions in ionic materials and to molecular vibrations in molecules made up of different atoms. In heteronuclear molecules, the electron shells are distorted and point in the direction of more electronegative atoms. Application of electric field causes the equilibrium position of charges to change, with the inter-atomic separation increasing or decreasing, which leads to a modification of the molecular dipole moment. If a material contains permanent dipoles, another polarization mechanism may be present in the radiofrequency range or for quasi-static fields, known as “orientational” or “dipolar polarization” (Figure 2.2). When the polar molecule is subjected to an electric field of frequency between 1 Hz and 100 MHz, the permanent dipoles will tend to orient parallel to the applied field, 23 causing a net polarization in the field direction. The tendency of orientation of permanent dipoles with the applied field is called orientational polarization9 (Po), and contrary to the previous two cases, it is strongly temperature dependent. As the temperature increases, the thermal motion and thus the characteristic frequency of molecular orientations increases; at the same time, the overall alignment of permanent dipoles decreases since the tendency of the dipoles to align along the applied field is disturbed by their thermal motion. Figure 2.2: Schematic representation of electronic, atomic and dipolar polarization mechanisms. At yet lower frequencies, interfacial or space charge polarization effects and dc conductivity contributions dominate the linear response. Interfacial or space charge polarization (Psc) occurs when there is an accumulation of charge at the boundary between different media, i.e., at the electrode surface or at the interphase in multi-phase materials. It is different from orientational and atomic polarization because instead of arising only from bound charges (i.e. ionic and covalently bonded structures), interfacial polarization also stems from free charges. At the electrode surface, a quasi-static field will 24 cause a charge imbalance (accumulation of free charge from the metal side) because of the dielectric material's insulating properties. However, the mobile charges in the dielectric will also tend to migrate, in order to maintain charge neutrality. Both effects are responsible for interfacial polarization (Figure 2.3). Figure 2.3: Schematic representation of interfacial electrode polarization which shows how the free positive (resp. negative) charges inside the dielectric material migrate towards the negative (resp. positive) charge build-up on the right (resp. left), caused by the external electric field. Finally, the dc conductivity gives rise to ohmic losses that show up in the conductivity response of dielectric materials, which will be further discussed in Section 2.5. The conductivity contribution in a dielectric is especially visible at higher temperatures, where the conductivity strongly increases. In the rest of this Section we will deal with the polarization effects listed earlier. Neglecting the ohmic losses due to the dc conductivity, the total polarization can be written as: (2.5) P= Pe + Pa + Po +Psc where Pe is the electronic polarization, Pa the atomic polarization, Po the orientational polarization and Psc the space charge polarization. Both Pe and Pa are due to induced local dipoles, i.e., they are due to dipole moments that are absent in the absence of the applied field. Such induced dipoles are 25 linearly proportional to the local field Eloc = E+P/30, which is the field present at a particular molecular site, and can thus be expressed as: (2.6) 𝑝𝑖𝑛𝑑𝑢𝑐𝑒𝑑 = 𝛼𝑖𝑑𝑬𝑙𝑜𝑐 =𝛼𝑖𝑑 𝐄+P 3𝜀0 , where αid is the molecular polarizability of each molecule (which is measure of mobility of negative and positive bound charges). In non-dipolar media where permanent dipoles are absent, the relation between the macroscopic electric field and the polarization due to N induced dipoles per unit volume V is given by the theory of Mosotti and Clausius2 as: (2.7) 𝑷𝑖𝑑 = 𝑁 𝑉𝒑𝑖𝑛𝑑𝑢𝑐𝑒𝑑 = 𝑁 𝑉 𝛼𝑖𝑑𝑬𝑙𝑜𝑐 = 𝑁 𝑉𝛼𝑖𝑑 1−𝑁𝛼𝑖𝑑 3𝑉  0 E Eq. (2.8) can be readily derived combining Eq. (2.4) with Eq. (2.7) as well as the definition P = (N/V)pinduced. The Clausius-Mosotti formula Eq.2.7 can also be rewritten as: (2.8) r−1 r+2 = Nαid 3V0 While in non-dipolar media equations (2.7) and (2.8) are valid for the static permittivity, in media with mobile permanent dipoles they are correct only at frequencies above the characteristic frequency of molecular orientations, and below the characteristic frequency of atomic (1012 Hz) and electronic polarization (1015 Hz). In this latter case, the relative permittivity r must be replaced with the constant value  at frequencies intermediate between radiofrequency and IR, and the Clausius-Mosotti equation (2.8) must be rewritten as: 26 (2.9)  −1  +2 = 𝑁 𝑉 𝛼𝑖𝑑 3  0 At the beginning of 20th century Debye generalized equation (2.9) by adding the effect of orientation polarization, i.e. the contribution of permanent dipoles. The orientational polarization can be expressed as the macroscopic volume density of the vectorial sum over all N permanent dipoles contained in the unit volume V in a form reminiscent of Eq. (2.6), as: (2.10) 𝐏0= 𝑁 𝑉𝛼𝑜𝑬𝑙𝑜𝑐 =  2 3𝑘𝐵𝑇𝑁 𝑉 𝑬𝑙𝑜𝑐 where αo the orientational polarizability, which Debye found using an argument based on statistical physics, and 2 is the mean square dipole for non-interacting dipoles. The total polarization due to both the permanent and the induced molecular dipoles is the sum of equations (2.7) and (2.10): (2.11) 𝐏= 𝑁 𝑉(𝛼𝑖𝑑 +𝛼𝑜 )𝑬𝑙𝑜𝑐 From the expression for the local field 𝐄𝑙𝑜𝑐 =𝐄+ 𝐏/3  0 and the linear constitutive relation P = 0 (r-1) E, equations (2.11) can be rewritten as: (2.12)  𝑠−1  𝑠+2 =𝑁 𝑉3  0 (𝛼𝑖𝑑 +𝛼𝑜 ) where s is now the static permittivity that takes into account the polarization due to both permanent and induced dipoles. By means of equation (2.9), equation (2.12) can finally be written as: (2.13)  𝑠−1  𝑠+2 −−1 +2 =𝑁 𝑉2 90kBT . 27 Equation (2.13) is known as Debye formula6. Onsager10 extended the Debye-formula by considering the enhancement of the permanent dipole moment of a molecule  by the polarization of the environment (reaction field) as given below: (2.14) s− = 1 30 s(+2)2 3(2s+ ) 2N kBTV and the quantity  = s –  appearing in Eq. (2.14) is called “dielectric strength” of the dipolar relaxation. The drawback of both Debye and Onsager formulas is that these equations are only valid for systems where dipole-dipole interactions are negligible such as dipolar gases and certain dipolar non-associating liquids, while they are not actually valid for most condensed phases containing permanent dipoles interacting through dipolar interaction or steric inter-dipole interactions. Onsager‟s extension fails in polar associating liquids because it does not consider the static orientational correlations between molecules. Later Kirkwood11 and Frohlich7 introduced a phenomenological correlation factor (symbol gK) to model the local interaction between dipoles and rewrote the Onsager formula as: (2.15)  =  𝑠−  = 1 3  0  𝑠(  +2)2 3(2  𝑠+  )𝑔𝐾  2𝑁 𝑘𝐵𝑇𝑉 If we consider only the nearest neighbours of a dipole, the gK factor can be approximated as: (2.16) 𝑔𝐾 = 1 + 𝑧  𝑐𝑜𝑠  where z is the coordination number and  is the angle between a test dipole and one of its neighbors, and the bracket denote the time average. If there is a correlation 28 between the orientations of the neighbouring dipoles, gK will differ from 1 (since then cos  1). When the value of gK 1 which gives the parallel alignment of dipoles, gK 1 correspond to an anti-parallel alignment and gK =1 for non-interacting dipoles. In practice gK is complicated to calculate theoretically12, but it can be estimated from the experimental value of the dielectric strength  using Eq. (2.15). 2.3 Detailed Frequency-Dependent Response The values of the dielectric permittivity discussed so far are only those in the static and high-frequency (relative to orientational polarization processes) limits. Much dielectric studies are, however, concerned with the detailed knowledge of frequencydependent phenomena, where dielectric dispersion occurs, because from such knowledge dynamic information can be obtained. When a harmonic alternating electric field 𝐸 𝑡 =𝐸0 exp⁡(𝑖  𝑡), of angular frequency  = 2𝜋𝑓, is applied to a sample containing permanent dipoles, all polarization mechanisms described earlier take place. The polarization due to induced dipoles has very fast response times (between 10-17 and 10-14 seconds for the electronic polarization and between 10-13 and 10-12 seconds for the atomic polarization), so that this contribution to the polarization may be considered, in dielectric experiments, to rise instantaneously with the change in electric field. In contrast, orientational polarization has relatively long response time (between 103 and 10-10 seconds, depending on temperature) and therefore lags behind the rise in the electric field, which results in the phase shift between electric field E and the polarization P. This lag is commonly referred to relaxation, which is defined as the delay in the response of a system to changes in the forces to which it is subjected. Similarly, if the applied electric field is suddenly switched off, the polarization does not go to zero instantaneously. The time delay necessary for a dielectric to respond to a change in the applied electric field is called characteristic relaxation time. These phenomena occur because the molecular reorientation motions take much longer time than electronic transitions or molecular vibrations. 29 Figure 2.4: Different polarization mechanism in both real and imaginary part of relative permittivity as a function of frequency. Under such conditions, the dielectric permittivity has to be treated as a complex dielectric function of the form  ∗=  ′  −𝑖  ′′() where ε‟(ω) and ε‟‟(ω) are the real and imaginary parts of the permittivity, respectively. The real part is related to the reversible energy stored in the material, while the imaginary part is proportional to the dissipated energy. Both provide quantitative information about the relaxation processes associated with the reorientation of the dipoles (see Figure 2.4). Dielectric spectroscopy, which is one of the main experimental techniques employed in this thesis, and which is described in detail in Chapter 3, allows simultaneous measure of both the real and imaginary parts of the complex permittivity. The imaginary part of the permittivity contains information about Joule losses due to the dc conductivity, as explained later in Section 2.5. In the imaginary part (see Figure 2.5), a dielectric relaxation appears as a (usually asymmetric) peak, named dielectric loss peak, whose maximum defines the characteristic relaxation frequency max and the corresponding relaxation time 𝜏 = 1/max. In the real part, a step-like decrease is observed with increasing frequency 30 across max, whose height is the dielectric strength  of the relaxation process. At low frequency, the real part of the permittivity reaches its static value. At high frequency, the orientation polarization is unable to follow the time variation of the field; hence this contribution to the overall polarization (Eq. 2.5) drops out and the real part of the dielectric function reaches the value ε∞. For example, for smallmolecule polar liquids of low viscosity, the time required for the dipole or orientation polarization process is about 10-11 to 10-10 seconds, corresponding to frequencies in the microwave region. The orientation polarization contributes at lower frequency, but it does not show up in the IR response, since the typical required time for vibrational processes is about 10-12 to 10-14 s, corresponding to the frequency of infrared light (the electronic polarization is the most rapid process and the time required is about 10-15 s, which corresponds to the frequency of ultraviolet light). Figure 2.5: Scheme of the real ‟() (solid line) and the imaginary ‟‟() (dashed line) part of the complex dielectric function for system displaying a single relaxation process and ohmic (a) or non-ohmic (b) conductivity. In the latter case, an electrode polarization effect is also observed at low frequency. The dynamic processes that contribute to the orientational polarization in the complex permittivity may be interor intra-molecular, i.e. involve the motion of whole molecules or else of a subunit of a larger molecule, and they can be more or less cooperative. In glass-forming system, for example, the most important dynamic 31 process is the so-called primary (α) relaxation process, which corresponds to the cooperative, collective rearrangement of the constituent molecules that is directly associated with the viscosity of the system.13 The frequency of the α process (as well as the viscosity) decreases with decreasing temperature, and displays in fact a continuous, dramatic slow-down by several decades in a short temperature interval upon approaching the glass transition temperature Tg. Figure 2.6: Schematic representation of the frequency dependent dielectric loss of a dynamically disordered material. Two distinct features are shown; (a) primary α-relaxation and (b) secondary relaxation. Other (usually less cooperative) molecular dynamic processes can occur in addition to the primary relaxation. These molecular motions occur at longer frequency than the α relaxation, and are called secondary relaxations as they are less cooperative in character. These secondary relaxations may be in some cases single-molecule motions,14 or else correspond to the reorientational motion of mobile side groups or larger parts of the constituent molecule. The typical spectral lineshape of a material displaying both a primary and a secondary relaxation is shown in Figure 2.6. The temperature dependence of the characteristic relaxation time of the α process deviates from a thermally activated behavior (Arrhenius) and follows instead a 38 (2.30)  ′′  =  𝑜𝑟  + 𝜎0 𝜀0  𝑠 where or describes the orientational polarization, σ0 is a phenomenological parameter, and s is a phenomenological exponent that has a value of one in the case of pure ohmic conduction24 (Figure 2.8(a)). In this latter case no conductivity contribution is present in ε′, σ0 is equal to the dc electrical conductivity σdc, and increases linearly with decreasing frequency (𝜀′′  ~ 𝜎𝑑𝑐 𝜀0  ). In the conductivity representation the real part σ′(ω) is then constant (σdc), and the imaginary part σ′′(ω) increases linearly with frequency. In experiments the slope is often not one because of the not perfectly ohmic character of the conductivity, in which case σ0 is just a fitting parameter. The dielectric properties can also be expressed in the modulus representation M*() = M‟()+iM‟‟(). A conductivity relaxation time9 can be calculated from the imaginary part of modulus by fitting it as: (2.31) 𝑀′′  =𝑀∞  𝑐𝑜𝑛𝑑 1+  𝑐𝑜𝑛𝑑 2 Eq. (2.31) is similar to the imaginary part of M*(ω) for a Debye-like relaxation process, and exhibits a peak for ωMτCond = 1 with τCond = ε0ε∞/σdc. Therefore the dc conductivity can in principle be estimated from the position of modulus maximum loss (ωM) (Figure 2.8). In practice, the dc conductivity contribution to the dielectric response may be masked by non-ohmic conduction, dipolar relaxation peaks, or by electrode polarization or other space-charge effects. 39 Figure 2.8: (a) Theoretical example for a complex dielectric function with a pure ohmic contribution: σ0/ε0 = 1 (dashed line), σ0/ε0 = 104 (solid line), ε‟ = 5. (b) Real part M‟ and imaginary part M‟‟ of the complex electric modulus according to the complex dielectric function given in (a):σ0/ε0 = 1 (dashed line),σ0/ε0 = 104 (solid line) 2.5.1 Dc Transport Theoretical and experimental studies of the steady state or dc conductivity (dc) in hopping electronic systems began in the 1950's25,26 with the discovery of impurity hopping conduction in compensated semiconductors such as germanium and silicon. The theoretical analysis by Miller and Abrahams25 postulated that the hopping probability Wij for the electronic charge carrier to hop from a site i to an unoccupied site j is given by: (2.32) 𝑊𝑖𝑗 =𝑓0exp −2𝛼𝑅𝑖𝑗 −  𝐸𝑖𝑗 𝑘𝐵𝑇 Here f0 represents the number of hop attempts per unit of time at high temperature and small distances, which is of the order of magnitude of the optical phonon frequency; Rij is hopping the distance between site i and site j (see Figure 2.9); Eij is the energy difference between site i and site j; and α is the inverse of the decay length of the (localized) electronic wave function. If Rij and Eij refer only to nearest neighbor hopping sites, Eq. (2.32) leads to a simply-activated form of 40 the nearestneighbor hopping (NNH) dc conductivity (since Rij and Eij are then the same for all hopping processes): (2.33)  𝑑𝑐 =𝐴exp − ∆𝐸 𝑘𝐵𝑇 where A is a constant. The concept of variable range hopping (VRH) was introduced by Mott to cover the situation where the energy between (non-nearestneighbor) hopping sites is a function of their spatial separation. Figure 2.9: Illustration of a disordered material with localized wave functions and energies that differ from site to site. Two sites (i and j) are shown. α-1 is the localization length and Rij is the distance between the sites. Mott 27 , 28 used the expression for the hopping probability Wij to derive an expression for the (low-)temperature dependence of the dc conductivity of a disordered material with a constant density of states (DOS) around the Fermi level. From Mott‟s law the temperature dependence of the dc conductivity can be represented as: (2.34)  𝑑𝑐 𝑇 =𝐴 𝑇𝑏 exp −𝑇0 𝑇 𝑛≈  0exp −𝑇0 𝑇 𝑛 41 The constants A and T0 depend on the overlap and number density of the electronic states involved. The exponents b and n depend on the distribution of states around the Fermi level; the value of n is related to the dimensions of the transport process and usual values of n are ½ and ¼. Mott‟s calculation assumed that the density of localized states near the Fermi level does not depend on energy; under this assumption n = ¼. However, Efros and Shklovskii29 pointed out that in some disordered systems, the DOS is not constant but rather displays a gap at the Fermi level. This occurs because, when an electron hops from one site to another, it leaves behind a hole and the system must have enough energy to overcome the resulting electron-hole Coulomb interaction. The resulting DOS vanishes near the Fermi level (Coulomb gap), and such Coulomb-type correlation leads to a value n = 1/2.29,30,31 Compared to the NNH model, the VRH model accounts for the contribution, at low temperature, of charge carriers hopping not to first neighbor molecules, but to more distant states, energetically more favorable. A key quantity of the hopping mechanism is the critical rate Wc, i.e., the fastest rate at which a macroscopic continuous path across the material can exist. Such a path consists of all the i–j pairs of states with a transition rate Wij  Wc. The rate Wc determines the dc conductivity value so that dc and Wc are proportional and exhibit the same temperature dependence. In the case of ionic conductors, such as salt solutions, ionic liquids and some plastic crystals, the temperature dependence of dc conductivity can be modeled with empirical Vogel-Fulcher-Tammann (VFT) equation (see chapter 3 for more details). 2.5.2 Ac Transport A typical example of ac conductivity spectrum (real part of σ′(ω)) is shown in Figure 2.10. The conductivity spectrum displays three distinct frequency regions with distinct behavior, namely: a characteristic bending at low frequency called electrode polarization effect, a frequency-independent plateau at low or intermediate frequencies, called dc conductivity which bends off at a certain critical frequency ωc marking the onset of a third region in which the conductivity increases with frequency, showing power law dependence for ωc. The critical frequency can be 42 determined by calculating the maximum in the second derivative of σ′ with respect to ω. Electrode polarization9,32 is caused by the (partial) blocking of charge carriers at the electrode/sample interface which lead to the separation of positive and negative charges when a slowly varying electric field is applied. EP occurs mainly for moderately to highly conducting samples and masks the dielectric response of the sample for slowly varying fields. This effect, giving rise to giant values of the dielectric constant and a strong drop of conductivity towards low frequencies, arises when the charge carriers arrive at the metallic electrodes and accumulate in thin layers immediately beneath the sample surface forming a so-called space-charge region. Figure 2.10: Logarithmic plot of dielectric loss with corresponding ac conductivity for a specific temperature in solid C60(ONa)24. Apart from the electrode polarization effect, which will not be analyzed in detail in this thesis, the variation of the real part of conductivity with frequency is empirically expressed by Jonscher‟s universal dielectric response (UDR)33 as: (2.35)  ′  =  𝑑𝑐 + 𝐴  𝑠 Here σdc is the dc conductivity, A is constant for a particular temperature, and s is the dimensionless frequency exponent. In this model the ac response in the high -1 0 1 2 3 4 5 6 1 2 3 4 -1 0 1 2 3 4 5 6 -7 Log(f /[Hz]) Log(f /[Hz]) Log('') -7 Log(M'') Log('/ [S.cm-1]) -6 -5 -4 -3 -2 43 frequency limit is considered as the sum of individual (not correlated) responses of pairs of sites randomly distributed through the material, as described by the pair approximation model introduced by Pollak and Geballe34. Eq. (2.35) is a common feature for amorphous semiconductors and some other disordered systems.33 The values of s in an ideal Debye-dielectric and ideal ionic-type crystals are one and zero respectively. Typically, it is understood that a value of s closer to zero indicates that charge carriers are free to move (delocalized) through the material, while an s value between 0.5 and 0.8 is observed in materials with more localized charge carriers. In some disordered solids, ac conductivity measurement at sufficiently high frequency results in a crossover from power law to a linear increase with frequency, called „second universality.‟35,36 The movement or hopping of the charge carriers is influenced by their neighborhood, and the exponent s is a measure of the degree of interaction. The behavior of the frequency exponent as a function of temperature can be used to determine the origin of ac conduction mechanism. Besides the VRH theory, various other models have been proposed in the scientific literature37,38 to account for the ac conductivity line shape, such as quantum mechanical tunneling (QMT) model, the correlated barrier hopping (CBH) model, the overlapping large-polaron tunneling (OLPT) model and the non-overlapping small-polaron tunneling (NSPT) model, to name a few. In the QMT model, s depends upon frequency but is independent of temperature. If the exponent s depends on both frequency and temperature, and it, increases with increase in temperature then it is in agreement with the predictions of the NSPT model, whereas if s decreases at first, reaching a minimum and increases thereafter with increase temperature, such behavior can be rationalized within the OLPT model. In the CBH model, the exponent s always decreases with increasing temperature. The frequency dependence of ac conductivity does not seem to follow the simple Jonscher‟s power law in glassy materials. Instead, they sometimes follow the socalled jump relaxation model (JRM), introduced by Funke39,40,41 to account for the ionic conduction in solids, in which the full conductivity spectrum can be described by the double power law42: 44 (2.36)  ′  =  𝑑𝑐+ 𝐴  𝑠1+ 𝐵  𝑠2 Here A and B are constants, the power-law term with s1 < 1 corresponds to the grain-boundary conductivity arising from the translational hopping motion, while the last term with exponent s2 corresponds to well localized relaxation/reorientational motions. 2.5.3 Relation between Dc and Ac Conductivities The UDR line shape is observed in samples that do not exhibit, at least close to ωc, any relaxations due to permanent molecular dipoles. In such case, when the dielectric data are plotted as loss spectra rather than as ac conductivity, a bump-like feature is observed at ωc, as in the spectra of Figure 2.10. This loss feature originates from localized hopping motions of charge carriers, but it can be described in a quite similar way as for dipolar relaxations, albeit with parameters having a different meaning and temperature dependence. In particular, the shape parameters, as well as the relaxation strength (Δε) and the dielectric loss peak (max) actually depend on the dc conductivity value. An empirical relation introduced by Barton, Nakajima and Namikawa and thereby known as BNN relation 43 correlates the electrical conductivity to the dielectric strength ε of the corresponding dielectric loss: (2.37)  𝑑𝑐 = 𝑝  0  𝑚𝑎𝑥 In the BNN relation, p is a loosely defined parameter, expected to be of order 1, and ε is only weakly dependent on temperature, so that the frequency position of the loss peak (ωmax) is linearly correlated with dc conductivity (dc). 44 The conductivity-induced relaxation, which we will refer to also as BNN relaxation is, in general, quite broad and asymmetric, and thus far from a Debye-like relaxation. The corresponding dielectric strength Δε may arise entirely from mobile charge effects and not involve bulk dielectric effects at all. 45 Figure 2.11: Logarithmic plot of modulus M‟‟ with corresponding ac conductivity, for a specific temperature in solid C60(ONa)24. Figure 2.11 shows the comparison between an ac conductivity spectrum and the corresponding imaginary part of the modulus. The latter displays a peak; often called “conductivity relaxation”, 45 which in the case of Figure 2.11 it has actually two components. The low-frequency component of the M‟‟ feature lies in the range of frequencies in which charge carriers can perform successful hopping from one site to the neighboring sites (leading to effective charge transport). The highfrequency component in M‟‟ lies in the range of frequencies in which the charge carriers are spatially confined to their potential wells and the charge can make only localized motions within the well. The spectral position of the modulus feature marks roughly the transition from long-range to short-range mobility with increasing frequency. When equation (2.37) is satisfied, it entails that both ac and dc conductivity are closely related to each other, and therefore that the charge transport is based on the same mechanism, regardless of the frequency at which it is studied. Also ionic charge conduction (due e.g. to cations diffusing through a disordered matrix) occurs by hopping, only not between molecules but rather between empty interstitial sites. In order to effectively model the conductivity spectrum in -1 0 1 2 3 4 5 6 1 2 3 4 -1 0 1 2 3 4 5 6 -7 Log(f /[Hz]) Log(f /[Hz]) Log('') -7 Log(M'') Log('/ [S.cm-1]) -6 -5 -4 -3 -2 46 disordered ionic materials such as liquids or glasses, Dyre 9,46,47,48 has developed an analytical model known as the symmetric hopping model or random free energy barrier model that ascribes both ac and dc conductivity to the same underlying physical process, in which charge carriers in a disordered matrix experience random, spatially varying potential energy barriers which prevent or allow their hopping to new sites in the matrix. In this model, jump rate (jump probability per unit time) is assumed symmetric i.e., the same for the charge carrier to jumps forwards and backwards across the barrier. According to this model, the complex conductivity can be expressed as: (2.38) 𝜎∗(  ) = 𝜎0 𝑖  𝜏𝑒 𝑙𝑛 1+ 𝑖  𝜏𝑒 , where 𝜏e is the attempt frequency to overcome the largest barrier, determining the dc conductivity. The frequency which characterizes the onset of dc conductivity σ0, is related to the BNN relation and if σ0 = 1/e which implies that they are due to same charge transport process. 2.5.4 Scaling of the Ac Conductivity Many dynamical processes in disordered materials exhibit what is often referred to as „thermorheological simplicity‟ (TRS), also known as „time-temperature superposition‟.48,49 This means that, while the characteristic frequency or time scale governing the process varies with temperature, the inherent spectral features of the relaxation remain in the same proportion. Consequently, the frequency-dependent line shapes of the relaxation spectra are undistorted by changing temperature and are only shifted in frequency by virtue of the temperature dependence of the characteristic frequency. The scaling of the ac conductivity spectra requires both division by the dc conductivity (dc) and shifting by some characteristic frequency f0‟ (2.39) σ(f) σdc = G f f0 ′ . 47 Here the characteristic frequency f0‟ refers to a temperature-dependent reference frequency which may be chosen for example as the onset frequency fc = ωc/2π of the dispersive part of the conductivity spectra, or in some cases as f0‟ =2dc, as shown by Kahnt.50 In this scaled representation, the data collapse onto a single, common curve, a so-called “master curve”, whose line shape is given by the function G(x) in equation (2.39). The validity of such scaling arises from the fact that the horizontal and vertical scales are generally related by the Nernst-Einstein relation,51 so that the macroscopic quantity dcT is proportional to a microscopic quantity such as the characteristic frequency of hopping. When a scaling of the vertical and horizontal scales is obtained by setting dcT= fc, the resulting scaling is referred to as Summerfield scaling.52,53 This type of scaling is most often observed in temperaturedependent measurements of disordered samples with fixed (temperatureindependent) charge-carrier concentration, provided the disordered matrix remains isostructural. An example is provided in Chapter 4 of this thesis. 54 55 Chapter 3 Experimental Techniques and Data Analysis 3.1 Basic Characterization Techniques 3.1.1 Thermogravimetric Analysis Thermogravimetric analysis (TGA) is a method of thermal analysis in which changes in physico-chemical properties of materials are measured as a function of increasing temperature with constant heating rate, or as a function of time with constant temperature and/or constant mass loss. In this thesis TGA is employed to measure the mass loss of a sample as a function of temperature. Such mass loss may occur as result of decomposition, evaporation of volatile impurities attached to sample such as water vapor, carbon dioxide, or by desorption of adsorbed molecules, among others. TGA can provide information about physical phenomena such as phase transitions, vaporization/sublimation, absorption, and adsorption and desorption, and decomposition. 56 The TGA experiments presented in thesis were performed with a Q50 thermobalance from TA-instruments. The instrument is equipped with a titanium sampleholder supported by a precision balance and residing in a furnace for heating or cooling the sample. In a typical experiment, a sample of initial mass between 0.5 to 10 mg was placed in an aluminum pan (TA-instruments) and the mass was monitored while heating the sample under nitrogen flow. Typically, the temperature was varied between room temperature (300 K) and 600 K, at a rate of 2 to 10 K per minute. The use of nitrogen flow ensures that no oxidation takes place during heating. The plot of mass value versus temperature is referred to as the thermogravimetric curve (TG curve). The mass is usually normalized to the initial mass, and the TG curve is shown as percentage, as shown in the example of Figure 3.1. The derivative thermogravimetric (DTG) curve is shown as dotted line. Such curve represents the rate of mass change as a function of temperature when substance is heated at uniform rate.1 Figure 3.1: A percent TG curve, referenced to the sample’s initial mass, exhibiting a lowtemperature plateau of constant mass (A), the mass loss region (B), and another plateau of constant mass (C). The dotted line shows the first derivative of the percent mass loss with respect temperature (DTG Curve). 57 3.1.2 Differential Scanning Calorimetry Differential scanning calorimetry (DSC) is a thermoanalytical technique used for measuring the characteristic phase-change properties of a sample such as melting and crystallization as well as the glass transition. The enthalpy of transition is determined by measuring the difference in the amount of heat required to increase the temperature of a sample and that of a known reference as a function of temperature. The reference used for DSC has a heat capacity that varies slowly over the range of temperatures measured; indium is normally used as a standard for calibration of temperature and enthalpy changes due to its well-defined melting point and heat of fusion. Both sample and reference are maintained at nearly the same temperature throughout the experiment. When the sample undergoes a physical transformation such as a phase transition, it requires either a larger or lower heat flow, compared to the reference, to remain at the same temperature as the reference. Whether less or more heat must flow into the sample depends on whether the transformation process is exothermic or endothermic, respectively. For example, as a solid sample melts to a liquid it will require an extra heat flow corresponding to the latent heat of transformation, because the phase transition from solid to liquid is an endothermic process. On the other hand, if the sample undergoes an exothermic process such as crystallization it will release energy in the form of heat. By analyzing the difference in heat flow between the sample and reference, DSC is able to measure the amount of heat absorbed or released during such transitions.2,3 The DSC experiments were performed with a Q100 analyser from TA instruments equipped with a refrigerated cooling system, yielding a wide operating range of temperatures from 190 to 600 K, with cooling/heating rates between 2 and 10 K per minute. In order to carry out DSC measurements at temperatures below 190 K, we employed a different calorimeter equipped with a liquid nitrogen dewar, namely a TA-2920-MDSC analyser (also from TA-instruments). The sample atmosphere during DSC experiments is controlled by connecting purge gases (nitrogen or helium) to the setup and the flow rate of the gas is monitored by a mass 58 flow controller. A sample of size between 1 and 30 mg is placed in an aluminum pan (TA instruments) or in a high-pressure stainless steel pan with gold plated seal (Perkin-Elmer). The last one is used for samples that can react readily with the aluminium sample holder or in the case of materials with high vapour pressure. Figure 3.2 shows an example of DSC thermogram, acquired upon heating a molecular glass. The glass transition temperature (Tg) is signaled by a small bump in the baseline which marks the corresponding change in the heat capacity of the sample (more visible in the inset). Above the glass transition temperature, the material is in a metastable supercooled liquid state. As the temperature increases further, heat is released in the crystallization process, as the entropy of the sample decreases, leading to an exothermic (downward) feature. Upon further heating this crystal phase, an intense endothermic (upward) peak appears which signals the melting temperature (Tm). Figure 3.2: A DSC thermogram of a glass forming material. The main peak corresponds to the melting of the solid phase (Tm), which is formed by crystallization of the supercooled liquid obtained by heating the glass above the glass transition temperature (Tg). 59 3.1.3 Fourier-Transform Infrared Spectroscopy Fourier-transform infrared spectroscopy (FTIR) is an optical absorption spectroscopy technique that allows detecting the dipolar inter-atomic vibrations of individual molecules of a sample. When infrared radiation is passed through a sample, some of the radiation is absorbed by the sample. The absorbed IR radiation is resonant with a specific vibration of the inter-atomic bonds about their equilibrium distance. Only vibrations resulting in the change in the local dipole moment can be excited in IR spectroscopy. Because each molecule is a unique combination of atoms, the IR spectrum is like a molecular fingerprint, as no two molecular structures produce the same infrared spectrum. Infrared spectroscopy can thus be used to identify the chemical composition of unknown material; the intensity of the peaks in the spectrum is a direct indication of the amount of each molecule present. It can also be employed to investigate the presence of impurities or of specific intermolecular bonds and supermolecular structures.4 For the measurements presented in this thesis we used a Thermo Scientific NicoletTM 6700 FT-IR spectrometer equipped with a He/Ne laser source, a CsI beamsplitter and DTGS-CsI detector with a spectral range between 4000 and 400 cm-1 and wave number resolution of 1 cm-1. The measurements are carried out in transmission mode, using free-standing pellets made by mixing a sufficient amount of powder sample with KBr powder, which is transparent in the probed wavenumber range. 3.1.4 X-ray Powder Diffraction X-ray powder diffraction (XRPD) is an analytical technique used for structural characterization and phase identification of polycrystalline materials. X-ray diffractometers consist of three basic elements: an x-ray tube, a sample holder and an X-ray detector. X-ray diffraction is based on the constructive interference of monochromatic X-rays diffracted by a crystalline sample. The X-rays are generated by a cathode ray tube, filtered to obtain monochromatic radiation and collimated to 60 achieve a beam with a well-defined direction, which impinges on the sample. The scattering of the incident rays from the sample yields constructive interference (and thus a diffraction peak) when Bragg's Law (nλ = 2dsinθ) is satisfied. Bragg’s law provides the correlation between the diffraction angle θ and the lattice spacing d along a particular crystallographic direction in the sample, for a given wavelength λ of the impinging electromagnetic radiation. The diffracted X-rays are then detected at different scattering angles. While a single crystal sample provides diffraction spots in well-defined 3D directions, in a polycrystalline powder sample the crystalline domains have random orientations, which results in a series of diffraction ‘cones’, each one at a different scattering angle. In typical powder diffraction experiment, the X-ray tube (and thus the direction of the incoming X-rays) is maintained fixed, and the detector is moved (Bragg-Brentano geometry). By varying the scattering angle, i.e., by moving the detector along an arc of circumference covering a sufficiently large angle, all possible diffraction directions of the lattice should be obtained due to the random orientation and large number of crystalline grains. Consider the scattered intensity from crystallographic planes oriented at an angle θ with respect to the incoming X-rays. The Bragg-scattered beam will emerge at an angle equal to θ with respect to the same planes, that is, at an angle 2θ with respect to the incident beam. Therefore, when a scattering maximum is detected by the detector, the angle between the X-ray tube and the detector at that moment is equal to 2θ. From the conversion of the angular position θ of the Bragg-planes to dspacings by means of Bragg’s law, it is possible to identify the material's lattice symmetry as well as the lattice parameters. The high-resolution XRPD profiles used in this thesis were recorded by means of a vertically mounted INEL cylindrical position sensitive detector (CPS120). The detector was used in the Debye-Scherrer geometry (transmission mode), enabling simultaneous recording of the diffraction profile over a 2θrange between 4˚ and 120˚ (angular step ca. 0.029˚ (2θ)). Monochromatic Cu Kα1 radiation (λ= 1.54059 A) was selected by means of an asymmetrically focusing curved quartz monochromator. Temperature control was achieved with a liquid nitrogen 700 series Cryostream Cooler from Oxford Cryosystems operating from 500K to 90 K with a 61 temperature accuracy of 0.1 K. The generator power was commonly set to 35 kV and 35 mA. External calibration with cubic phase Na2Ca2Al2F4 was performed for converting measured 4096 channels into 2θ-degrees, using cubic spline fittings.5 For the measurements, the samples are introduced into typically 0.5-mm-diameter Lindemann glass capillaries in the liquid or in the solid state at room temperature and are continuously rotated perpendicularly to the X-ray beam during data collection to improve averaging of the crystallites. The peak positions are determined by pseudo-Voigt fits performed with DIFFRACTINEL software. 3.2 Broadband Dielectric Spectroscopy The main experimental technique used in this thesis to investigate the dielectric and charge transport properties of the studied materials is broadband dielectric spectroscopy (BDS), also known as impedance spectroscopy. It is called “broadband” as it allows probing the interaction of matter with electromagnetic waves in the frequency regime between 10–6 and 1012 Hz. In this extraordinarily extended dynamic range, molecular and collective dipolar fluctuations, charge transport and polarization effects at inner and outer boundaries take place, and determine the dielectric properties of the material under investigation. BDS enables studying the dynamics of permanent dipoles and induced mesoscopic dipoles, and the conduction properties of materials, in a single experiment.6 62 Figure 3.3: Survey of measurements techniques used in the frequency range between 10-6 Hz and 1015 Hz.6 BDS experiments can be performed either in time or frequency domain. In this work, dielectric spectroscopy is performed in the frequency domain; the measurements are carried out by varying directly the frequency of the applied AC field. No single BDS setup allows measuring in the whole range between 10–6 and 1012 Hz; rather, depending on the electromagnetic spectral range, different setups based on different measurement principles have to be used (Figure 3.3). In this thesis, the low-frequency range between 10-2 Hz and 107 Hz was probed using a 63 Novocontrol Alpha analyzer and the range from 106 Hz to 1.8 ×109 Hz by using the Agilent HP4291 impedance analyzer. In both setups the sample is placed in a capacitor cell of empty capacitance equal to C0, and from the experimentally measured complex capacitance C* of the capacitor filled with the material under study, the material’s complex dielectric function is obtained as: (3.1) Here is the angular frequency with =2πf =2πT-1 (T is the period, or time for a complete oscillation of the field), and ε′(ω) and ε′′(ω) describe the real and imaginary part of the complex dielectric function. To measure C*, a sinusoidal electric field E*(ω) = E0 exp(iωt) at angular frequency ω is applied inside the capacitor by applying an ac voltage V*(ω) = V0 exp(iωt), chosen so that the field strength E0 = V0/d (where d is the sample thickness, i.e. the distance between plates) is within the linear response range (the upper limit for most materials is taken to be E0 ≤ 106 V cm–1). The dielectric function can be obtained by measuring the amplitude of the current through the capacitor and its relative phase with respect to the voltage, thereby measuring the complex impedance Z*(ω) = V*(ω)/I*(ω) of the sample. Since for a capacitor of capacitance C the complex impedance is Z* = 1/iωC, the frequency-dependent complex dielectric function is then given, using Eq. (3.1), as: (3.2) The value for C0 can be obtained from a measurement of the empty cell or directly from the knowledge of the geometry of the cell. At high (microwave) frequency the geometrical dimensions of the sample capacitor become important and different cells have to be used in the two different experimental setups. With the Alpha analyzer, a parallel plate capacitor configuration is used. In order to measure low dielectric loss (ε′′), small plate distances and large areas of the 70 Figure 3.7: Schematic representation of the real and imaginary dielectric permittivity parts of equation (3.7) with only one Havriliak-Negami function. For ohmic contacts and no space-charge or Maxwell-Wagner-Sillarspolarization effects (see subsection 3.3.2), the exponent N is equal to 1, but in most cases 0.5 < N < 1 is obtained. For each relaxation process, the dielectric strength gives the step-like difference in ′ at frequencies lower and higher than the frequency of loss maximum, being also proportional to the area below the corresponding ′′ relaxation peak. The value ′ at infinite frequency is determined by . For common values of the Havriliak Negami shape parameters α and , the maximum of the relaxation peak in ′′ is approximately situated at fHN = 1/(2π HN). The width parameter α specifies the slope of the low frequency side whereas −α gives the slope of the high frequency side of the relaxation in ′′. The HN function contains as special cases the Cole-Cole (β = 1) and Cole-Davidson (α = 1) functions, and if α = β = 1 it reproduces the Debye behavior. The Debye behavior is particularly simple as it corresponds to the dielectric response of a sample with a unique relaxation time; compared to it, all other functions are said to represent a material displaying a continuous distribution of relaxation times. 71 The dielectric loss peaks of dipolar origin, in particular at low frequencies may obscured by the (ionic or electronic) dc conductivity, so that the ohmic conduction term should be eliminated to elucidate low-frequency relaxation processes. In order to remove frequency independent conduction from measured loss spectra an approximate logarithmic derivative approach is used10: (3.8) Equation 3.8 approximately equals the ohmic-conduction-free dielectric loss for rather broad peaks, like those of primary or secondary relaxations in molecular glass formers. In this thesis we have fitted only the imaginary part of the dielectric permittivity (loss spectra), or equivalently the ac conductivity spectra, without fitting simultaneously the real permittivity spectra. To this aim, we employed the standard software package WinFIT, which is especially designed for dielectric and impedance fits. It gives a fast routine for the optimization of the imaginary part of equation (3.7), allowing an analytical evaluation of dielectric spectra. The main feature of WinFIT is nonlinear curve fitting of the measured data in the frequency and time domains. The measured data can be imported in several binary and flexible ASCII formats, displaying the data and fit function in an online window. 3.3.2 Ac Conductivity and Space-Charge Losses Figure 3.8 shows the typical ac conductivity spectrum ’( ) of a disordered material without any relaxation due to permanent dipoles (if such a relaxation were present, it would be visible as a bump-like feature in the ac conductivity, just as in the ′′( ) spectra, only on top of a non-horizontal background). The ac conductivity in such disordered materials is mediated by hopping (rather than band-like) processes, and it displays a flat response at low frequency corresponding to the dc limit, and a temperature-dependent power-law-like increase at high frequency. This behavior is characteristics of the charge transport in disordered materials and is 72 known after Jonscher as “universal dynamic response” (UDR).11,12 The frequency behavior of the dispersive part of the ac conductivity can be described roughly by: (3.9) , where dc is the (frequency-independent) dc conductivity and s is the power-law exponent, in the range 0  s ≤ 1. In several cases, s is not constant but actually depends on frequency (and temperature). Figure 3.8: Schematic representation of real part of ac conductivity and dielectric constant in a disordered material. In the UDR, the dc conductivity corresponds to long-range hopping processes, while the ac conductivity stems from local processes that have limited extent or restricted range (for example, electronic hopping between two next neighbor molecules with low-lying energy levels from where the electronic charges have nowhere to hop to). The change from dc to ac regimes is marked by a so-called onset frequency or crossover frequency ( c), which represents the typical hop frequency of restricted-range hopping of charge carriers between sites. If the hopping mechanism underlying the dc conductivity is the same as that of the ac 73 conductivity (except for the larger possible hopping range), the temperature dependence of the cross-over frequency c is virtually identical to that of the dc conductivity dc. In some cases, at the onset frequency of the ac conduction regime a dielectric loss peak is visible, associated with accumulation of free charge carriers at spatial inhomogeneities of the sample such as grain boundaries. Such loss peak is referred to as conductivity-induced (or space-charge) relaxation and is usually described by the Barton–Nakajima–Namikawa (BNN) relation13: (3.10)  where p is a constant of order 1, ε is the dielectric strength of the space-charge relaxation, and ε0 is the permittivity of free space. Since ε has only weak temperature dependence, while σdc and ωc are thermally activated, eq. (3.10) implies that the temperature behaviors of the two quantities are strongly correlated, as stated above (see Subsection 3.3.3 for a discussion of the typical temperature dependence). The ac conductivity spectrum can also display a characteristic “bending” at low frequency (Figure 3.8) with a large increase in dielectric constant, which is due to the electrode polarization effect, i.e., the accumulation of charge at the sample/electrode interface. The dc value of the conductivity is usually assigned in such case to the plateau value observable at intermediate frequency. Experimentally, since the plateau is not perfectly horizontal, in the absence of an electrode polarization effect the value of σdc can be taken to be the low-frequency value of the σ′ spectrum. In spectra exhibiting a more pronounced electrode polarization effect, σdc was taken to be the value of σ’ in the middle of the plateau or at the point of inflection, that is, at the frequency for which dσ′/df was minimum. It is worth noticing that the ac conductivity obeys a time-temperature superposition principle, which means that the shape of ’( ) in a log-log plot is temperature independent. This makes it possible to construct a master curve. The shape of the master curve is roughly the same for all disordered solids (universality).12 74 In heterogeneous media made up of distinct phases, these display usually distinct static permittivity (and possibly different dc conductivity). When an ac electric field is applied, this heterogeneity results in the build-up of space charges (both free and bound ones) near the interfaces between the various phases. Such an interfacial polarization may result in a characteristic dipolar loss, referred to as MaxwellWagner-Sillars polarization14,15 which usually occurs at frequencies lower than the time scales typical of molecular orientational polarization contributions. The accumulation of charges at internal phase boundaries causes a strong increase in ε′ with decreasing frequency, just as in the electrode polarization effect. The simplest models to characterize the dielectric properties of a inhomogeneous media is a layered structure of two materials with frequency independent dielectric constants ε1 and ε2, in which only one medium is electrically conductive or else a spherical intrusion of one phase in an otherwise uniform background of the other phase. In these models, each region is characterized by a different relative permittivity and dc conductivity. Experimentally, it is found that if the phase boundaries are well defined, the Maxwell-Wagner-Sillars polarization has a single, well-defined relaxation time and can be fitted accordingly by the Debye model function (special case of HN function, Eq. 3.5, with α = β = 1). The more complicated case in which both media are electrically conductive and have dc conductivities σ1 and σ2, respectively, as sketched in Figure 3.9, the complex dielectric function can be written as: (3.11)  Equation (3.11) is similar to Debye formula but the parameters have a completely different meaning and physical origin. The relaxation time MW of the interfacial polarization can be computed as: (3.12)  75 Figure 3.9: Two dielectric layers in series. i and ri are the corresponding dielectric permittivities and dc conductivities. 6 We conclude this subsection with a short discussion of the modulus formulism. In the complex modulus representation, an increase in ε′′ or a decrease in σ′ with decreasing frequency is transformed into a step-like change in the real part of the modulus and as a peak in the imaginary part of modulus. Hence a modulus peak is often observed which is associated with the conduction process.16,17 In materials exhibiting a relaxation due to the motion of permanent dipoles, such dipolar relaxation is also visible as a (separate, higher-frequency) peak in the imaginary part of M. The advantage of representing dielectric data in modulus formalism is that the electrode polarization effects are suppressed in this representation. This is primarily because of the insensitiveness of the frequency dependence of the imaginary part of the modulus to the polarization processes, provided these are characterized by capacitances that are much larger than the bulk capacitance.18 An HN function can be defined in the modulus formalism, and used to fit relaxation or conductivity losses visible in the modulus spectrum. The phenomenological HN function used to analyze the frequency dependence of the complex modulus is given by 6: (3.13)  76 As in the permittivity HN function, τHN-M is a characteristic relaxation time, and α and β are shape parameters describing the distribution of relaxation times.19 3.3.3 Temperature Dependence of the Relaxation Times and Dc Conductivity Once the relaxation times and dc conductivity have been determined with the procedures described in the previous sections, the temperature dependence of the relaxation times and dc conductivity can be analyzed. In practice two expressions are commonly used to express the temperature dependence of the dipolar relaxation and ionic motion. The first one is the Arrhenius equation, originally introduced to describe chemical reactions. The second one is the Vogel–Fulcher–Tamman (VFT) equation20, introduced to describe the non-Arrhenius dependence in many glassforming systems.21 Arrhenius and Vogel-Fulcher-Tammann Temperature Dependence The mathematical expression characterizing the temperature dependence of the rate R of a simply activated process is the Arrhenius law: (3.14) Here A is an independent factor, Ea is the activation energy and kB is the Boltzmann constant. Linearization of this equation shows that an Arrhenius process shows up as a straight line when the rate R (relaxation time or dc conductivity) is plotted versus the inverse temperature (the so-called Arrhenius plot), and the slope of this line is proportional to the activation energy. Thus, Arrhenius plots are commonly used to display the temperature dependence of the dynamics. If the temperature dependence of a process is only approximately described by the Arrhenius Eq. (3.14), it is convenient to define effective or apparent, activation energy, as: 77 (3.15) In the case of perfect Arrhenius dependence, Ea as calculated from Eq. (3.15) is exactly constant, while for subor super-Arrhenius dependence Ea increases or decreases with increasing temperature. The variation with temperature of the primary relaxation time of glass formers and so-called rotator phases (phases in which the average molecular centers of mass occupy lattice points, as in an atomic crystal, but where the molecules display rotational motions) is generally non-Arrhenius (Figure 3.10). In particular, upon cooling the relaxation time almost always increases faster than predicted by the Arrhenius equation (3.14). One of most commonly accepted models to mimic such behavior is the Vogel-Fulcher-Tammann (VFT) equation20, which was introduced as a fitting function for the relaxation time of glass-forming liquids. The VFT equation is usually given in the form: (3.16) , where 0 is the high temperature limit of the relaxation time, D is strength coefficient which related to the fragility of the material and TVF is the Vogel temperature, associated by some authors with a would-be “ideal” glass transition temperature. 78 Figure 3.10: Schematic representation of VFT behavior and the Arrhenius dependence of relaxation process. Sub-Arrhrenius and Super-Arrhenius Dependence of the dc Conductivity In the case of Arrhenius plot of dc conductivity, several systems display nonArrhenius dependence. Interestingly, depending on the sign of the variation of effective activation energy with a change in temperature, different conduction mechanism can be distinguished (Figure 3.11). In the case of electronic or polaronic charge carriers moving by hopping, the dc conductivity may either follow or deviate from the Arrhenius behavior. For example, the so-called nearest-neighbor hopping (NNH) model, in which an electronic charge carrier may only jump from an initial molecule to a first neighbor molecule, the predicted temperature behavior is a perfect Arrhenius law: (3.17) σdc = σ0 exp(−T0/T). Many semiconducting materials, however, follow the prediction of the so-called variable-range hopping (VRH) theory,22,23 first introduced by Mott, according to which the temperature dependence of the dc conductivity is given by: (3.18) σdc = σ0 exp[−(T0/T)n], 79 entailing that the Arrhenius plot of σdc follows a power-law dependence of the type Log(σdc) = A – B/Tn. The result of the fit for different samples gives values of n close to ½ or ¼.24,25 Figure: 3.11 Schematic representation of the Arrhenius plot for the (a) dc conductivity and (b) its activation energy. While in the NNH hopping conduction always occurs via neighboring sites, and hence the hopping range (typical hopping distance) are independent of temperature, according to the theory proposed by Mott, the hopping range may vary as temperature decreases. The basic idea is the following: while in the NNH model it is assumed that all sites are equivalent so that the activation energy (typical energy barrier) between all possible pairs of molecules is always the same, in real materials hops between different sites and energy levels will also have different activation energy. While at high enough temperature the thermal energy is sufficiently high that next-neighbor hops will always be the most likely to occur, at low temperature the charge carriers might find it more efficient to hop (tunnel) through larger distances if this allows finding a new site characterized by a smaller energy barrier (the hopping rate decays exponentially the longer the hopping distance, but also the higher the barrier). As a result, the typical hopping range will increase as the temperature is decreased, whence the name variable-hopping range. While Mott 86 semiconductors in which the main conduction mechanism is electron (or hole) hopping.6 Localization and polarization effects are especially important:7 charge carriers are localized on single molecules and are surrounded by a polarization cloud, hence hopping of a charge carrier to a neighboring site is hindered by interelectron Coulomb repulsion,8 and it leads to charge transport only if the polarization cloud follows, since otherwise the charge carrier jumps back attracted by the excess bound polarization charge.9 While this general framework is widely accepted, the mechanism of charge transport in organic molecular materials is subject of debate, and in particular the observation of metallic-like behaviour and superconductivity in some alkali fullerene salts (fullerides), is surprising, since most other organic materials behave as semiconductors.10 Many organic materials containing metals or oxygen groups are hygroscopic. The presence of water has in general a strong impact on the conduction properties both in the case of the ionic and electronic transport, introducing in particular new mechanisms for charge generation and conduction and thus boosting the overall dc conductivity.11,12,13,14,15 This well-known effect is exploited in many kinds of humidity sensors, based both on inorganic materials such as ceramics and on organic materials such as polymers.16,17 The presence of water can also have an important effect on the dielectric response of materials, leading to an increase of the dielectric constant due to the high orientational polarizability of the dipolar H2O molecules. Determining the impact of water adsorption or intake on charge conduction properties is not only interesting from a fundamental point of view but it is also crucial for the implementation of organic materials in electronic or sensing devices.18 Since the value of the conductivity of organic semiconductors in their pure form is usually quite low, the introduction of new charge carriers or conduction paths through adsorption of water leads generally to an increase of the material’s conductivity even by several orders of magnitude. The exact origin of such effect, both in organic and inorganic materials, has been the subject of debate. To shed some light on these issues, we have investigated the intrinsic and waterinduced charge transport in a relatively simple organic system, namely the ordered solid phases of a highly symmetric organic fullerene derivative, C60(ONa)24, which 87 is synthesized as a polycrystalline hydrate and which can be obtained as a pure material by heating to sufficiently high temperature. The pure form is hygroscopic, which allows studying the effect on conductivity of water molecules captured from the atmosphere surrounding the sample. The existence of two well-defined phases (pure material and hydrate) allows investigating the impact on the electric conduction of two distinct types of water molecules exhibiting different interactions with the organic matrix: namely, structural water – which is an integral part of the hydrate’s crystal structure – and surface hydration water – which is present at the surface of crystalline domains. 4.2 Synthesis and Experimental Methods Sodium oxofulleride (C60(ONa)24) was synthesized in the last step of a synthetic route starting from Buckminster fullerene C60 (Aldrich, 99.8 %). The route involves first the synthesis of fullerol (chemical formula C60(OH)24), as detailed in refs 19, 20 and 21. C60(ONa)24 was then obtained by neutralizing C60(OH)24 (80 mg) with an aqueous NaOH 1 M solution (1.7 ml). The salt was precipitated with acetone, centrifuged, washed with acetone and dried at room temperature. The product was a hygroscopic polycrystalline brown powder, soluble in water, which was stored in air prior to measurements (Figure 4.1). The material was characterized by thermogravimetric analysis (TGA), differential scanning calorimetry (DSC), Fourier-transform infrared (FTIR) spectroscopy, X-ray powder diffraction (XRPD), and broadband dielectric spectroscopy (BDS). For both infrared and dielectric spectroscopy the powders were mechanically pressed into pellets of submillimeter thickness. 88 Figure 4.1: Molecular structure of the Sodium oxofulleride C60(ONa)24 For the FTIR measurements it was necessary to add KBr powder to the organic material to achieve free standing pellets, and the spectrum of a pure KBr pellet was used as baseline. FTIR spectra were measured in the mid-infrared range (4000 – 400 cm–1) using a Nicolet 6700 spectrophotometer equipped with a He/Ne laser source and DTGS-CsI detector. Each spectrum was the average of 32 scans collected with 1 cm−1 resolution. TGA scans were acquired while heating the sample under N2 flow between room temperature (300 K) and 600 K at a rate of 2 K min–1, by means of a Q50 thermobalance from TA-Instruments. DSC measurements were carried out in an open vessel between 300 K and 600 K, at a rate of 2 K min–1, using of a Q100 calorimeter from TA-Instruments. High-resolution X-ray powder diffraction (XRPD) profiles were recorded with a vertically mounted INEL cylindrical positionsensitive detector (CPS120). The peak positions were determined by fits with pseudo-Voigt functions using the DIFFRACTINEL software (see chapter 3 for more details). Dielectric measurements were carried out in the frequency (f) range from 10–2 to 106 Hz with a Novocontrol Alpha analyzer, using stainless steel electrodes in a parallel-plate capacitor configuration. Isothermal frequency scans were acquired in the temperature range between 200 and 550 K (with a temperature stability of 0.3 K) in a N2 flow Quatro cryostat. The complex impedance data are displayed in four 89 representations: real and imaginary part of the permittivity (ε’ and ε”), real part of the conductivity (σ’), and imaginary part of the modulus (M”, where M is defined as the inverse of the complex permittivity, M = 1/ε). The frequency-dependent complex conductivity and the complex relative permittivity are related by the equation σ = 2f ε0 ε (see chapter 3 section 3.3). All these quantities are displayed as function of the frequency f of the applied electric voltage. 4.3 Preliminary Characterization As mentioned in the introduction, sodium oxofullerene (C60(ONa)24) is a hygroscopic material that is synthesized as a polycrystalline hydrate. Figure 4.2(a) displays the room temperature FTIR spectra of the as-stored hydrate powder and of the powder heated to 423 K, which no longer contains the structural water. The spectra are normalized to the height of the most intense band at 1458 cm–1, which corresponds to the bending mode of the covalent C–O bonds of the C60(ONa)24 molecules. Such normalization is equivalent to rescaling the spectra to the relative fullerene content. The presence of water in the as-stored powder is confirmed by the observation of intense bands (indicated by arrows in Figure 4.2(a)) at 3465 cm–1 (stretching vibration of the O–H bonds of water) and 1690 cm–1 (bending mode), whose intensity is significantly reduced after heating to 423 K. 90 Figure 4.2: Room-temperature FTIR spectra (a) and XRPD patterns (b) of the as-stored hydrate and of the pure material (after heating to 423 K). The XRPD patterns are normalized to acquisition time and displayed with an offset for clarity. Inset to panel b: close-up of the XRPD pattern in the 2θ range between 37 and 41 degrees. The crystalline nature of both the as-stored and pure materials is revealed by the XRPD patterns of Figure 4.2(b). The diffraction pattern of the pure material, obtained by heating the hydrate, exhibits a much higher scattering background and significantly broader peaks, suggesting only partial order and smaller grain size in the pure material than in the hydrate. The average grain size was estimated in both phases from the angular width of non-overlapping diffraction peaks using the Scherrer equation,22,23 and found to be 32 ± 4 nm for the pure material and 50 ± 10 nm for the hydrate. A smaller linear dimension of the pure grains may be expected considering both the loss of water volume and the possible formation of defects upon the structural change. The structure obtained after heating to 500 K exhibits no clear peaks below 2 = 20° and displays main peaks at much higher scattering angles than the hydrate. This indicates that the pure material is characterized by a smaller first-neighbor distance, as expected due to the loss of structural water. 4000 3000 2000 1000 816 24 32 40 48 56 64 300 350 400 450 500 75 80 85 90 95 100 350 400 450 500 0 1 2 3 4 Pure Wavenumber [cm-1] Absorbance (a.u.) Hydrate (a) Hydrate Pure Intensity (a.u.) 2[degrees] P3 P2 (a) % mass P1 T [K] T [K] (b) Heat Flow [W.g-1] (b) 38 39 40 91 Figure 4.3: TGA (a) and DSC (b) curves measured on the as-stored hydrate powder. Three different processes associated with the loss of water may be identified: P1 - desorption of surface water; P2 - dehydration of structural water accompanied by phase change; P3 - desorption of migrated water. Figure 4.3(a) displays the TGA scan acquired while heating the hydrate. The curve displays an initially slow decrease of mass (marked in the Figure as P1) starting at room temperature up to approximately 350 K, the temperature that marks the onset of the main water loss (marked as P2). The clear step in the TGA curve indicates a sharp transition from a hydrated to a dehydrated phase upon heating to 370 K which is 15.6% mass loss corresponding to roughly 16 water molecules per fullerene unit, whose stoichiometry can be represented as C60(ONa)24 ∙16 H2O . The temperature of maximum mass loss (obtained by taking the first derivative of the TGA curve) is approximately 370 K, i.e. roughly the boiling point of pure water, which moreover coincides with the crystallographic transition between the hydrate and pure material (see below). We assign the initial mass loss between room temperature and 350 K to desorption of H2O molecules adsorbed onto the outer surface of the crystalline grains, which are less tightly bound than the structural (interstitial) ones. The main loss corresponds instead to the decomposition of the hydrate. Both assignments will be further corroborated in Section 4.4. 4000 3000 2000 1000 816 24 32 40 48 56 64 300 350 400 450 500 75 80 85 90 95 100 350 400 450 500 0 1 2 3 4 Pure Wavenumber [cm-1] Absorbance (a.u.) Hydrate (a) Hydrate Pure Intensity (a.u.) 2[degrees] P3 P2 (a) % mass P1 T [K] T [K] (b) Heat Flow [W.g-1] (b) 38 39 40 92 It may be observed that the TGA graph (Figure 4.3(a)) does not exhibit the same slope throughout the main water loss. Similarly, the DSC curve (Figure 4.3(b)) exhibits a structured peak with weak shoulders on both sides. All observed DSC features correspond to endothermic processes. Such multiple-component spectra are reminiscent of those of other C60-derived systems,24 and result from the different processes accompanying the loss of water, namely the breaking of hydrogen bonds between water and fullerene units, the re-crystallization into a new lattice structure, the migration of water to the outer surface and its final desorption, which is the last process that takes place (we label it as P3 in Figure 4.3(a)). The temperature range of each process and the corresponding mass loss are highlighted with dashed lines (see caption of Figure 4.3(a)). The powder diffraction spectra measured near the transition temperature of 370 K (not shown) reveal a single structural change between a crystalline hydrate and a partially ordered pure phase which is obtained irreversibly by heating to high temperature. The structural change therefore occurs simultaneously with the main water loss. The XRPD pattern of the hydrated phase could be indexed by patternmatching as a monoclinic P2/m phase (Figure 4.4). The indexation was carried out using the DICVOL91 program implemented in the FullProf Suite.25 The monoclinic P2/m was the only solution obtained in the range of values of unit cell volume between 1000 and 2000 Å3, which is typical of other fullerene derivatives such as C60Br24(Br2)2, C60F36 or C60F48.26,27 Taking as a starting point this solution, the lattice parameters were refined using the Pattern Matching option of the FullProf program. After the last step the following parameters were obtained: a = 19.554(1), b = 9.287(1), c = 6.208(4), β = 92.27(5), corresponding to a unit cell volume of 1126.5(1) Å3. The reliability factors were Rp = 3.57, Rwp = 4.73, Rexp = 2.75 and 2 = 2.96. It was not possible to obtain a reliable indexing for the pure material, perhaps due to the low intensity and relatively large width of the low-angle diffraction features in the corresponding pattern (upper curve in Figure 4.2(b)). As it occurs for C60 powders obtained from solutions and C60 solvates in which solvent molecules remain trapped into the re-built lattice,28 the pure material resulting from the dehydration process may contain structural defects (such as the well-known solvent- 93 induced stacking faults) which produce broadening, shift or asymmetry of diffraction peaks that vary for different Miller indexes.24 Figure 4.4: Experimental (red circles) and calculated (black line) XRPD patterns along with the difference profile (blue line) and Bragg reflections (vertical sticks) for the pattern matching refinement of the P2/m monoclinic phase of the hydrate at room temperature (same data as those of Figure 4.2(b)). Figure 4.5 shows the frequency-dependent dielectric spectra acquired on the asstored material (hydrate) during heating from room temperature to 433 K. The data are shown both in the dielectric loss (left-hand panels) and ac conductivity (righthand panels) representations, and displayed in separate temperature ranges to highlight the observed changes in line shape. The high-temperature spectra shown in Figure 4.5(d) (and partially in Figure 4.5(c)) represents the pure C60(ONa)24 salt, which will be discussed later in this chapter. 94 Figure 4.5: Dielectric loss (left panels, 1) and ac conductivity (right panels, 2) spectra acquired on heating the as-stored hydrate, in separate temperature ranges: (a) 298 to 333 K; (b) 338 to 363 K; (c) 368 to 393 K; (d) 398 to 433 K. 0 1 2 3 -11 -10 -9 -8 -7 0 2 4 6 -9 -8 -7 -6 0 2 4 6 -8 -7 -6 -1 0 1 2 3 4 5 6 0 2 4 6 -1 0 1 2 3 4 5 6 -7 -6 333K 328K 323K 318K 313K 308K 303K 298K (a2) Log('/ [S.cm-1]) (d1) (c1) (b1) (a1) Log('') 363K 358K 353K 348K 343K 338K (b2) 393K 388K 383K 378K 373K 368K 363K (c2) 433K 428K 423K 418K 413K 408K 403K 398K (d2) Log(f /[Hz]) 95 The dielectric loss (ε”) is characterized, both in the hydrate and in the pure material, and at almost all investigated temperatures, by a conductivity background proportional to reciprocal frequency at low frequency9 and a single loss feature at radiofrequency. As is visible in Figure 4.5(a1), in the as-stored hydrate the loss feature is particularly intense and broad, indicating a wide distribution of relaxation times. The relaxation feature in the pure material (Figure 4.5(d1)) is characterized by a much lower intensity and is observed in a different frequency range. The origin of this spectral feature is ascribed to charge-carrier-related loss associated with the accumulation of charge at crystalline grain boundaries which will be discussed in the section 4.4.1 in detail. Given the high water content of the as-stored material, it is tempting to ascribe the loss feature in Figure 4.5(a1) to the dielectric relaxation of water molecules present in the sample, as was proposed for other organic systems.29 We will show in the next Section that the origin of the dielectric loss in the hydrate is likely charge accumulation at grain boundaries, although the presence of water molecules affects its spectral intensity. The conductivity spectra exhibit an almost constant value at low frequency, corresponding to σdc, and a temperature-dependent increase at high temperature (the so-called dispersive region). As seen in panels c and d of Figure 4.5, at higher temperatures the conductivity displays a characteristic low-frequency “bending” (also visible, although less pronounced, in the corresponding dielectric loss spectra) due to the electrode polarization effect. The dielectric response of the hydrate exhibits a complex dependence on temperature. Below approximately 315 K (Figure 4.5(a)), both the frequency of the relaxation maximum and the dc conductivity are temperature-activated, as seen from the quasi-rigid shifts (in logarithmic frequency scale) of both the permittivity and conductivity spectra. Between 315 and 333 K, the shift of the relaxation frequency slows down, until it appears to stop at 333 K (Figure 4.5(a1)). Between 338 and 363 K (Figure 4.5(b)), both the relaxation frequency and σdc are observed to shift much faster with temperature than they do at lower temperatures. Finally, in the range between 368 and 393 K, that is, in the temperature interval of the main mass loss and structural change, a dramatic change of spectral profile is observed in the 102 It is important to note that, compared to the nearest-neighbor hopping model which predicts a simple Arrhenius behavior (see Chapter 3 section 3.3.3), the VRH model can more satisfactorily describe the experimental results for σdc in the whole probed range of temperatures. Figure 4.8(b) shows the typical hopping distance R in the whole temperature range. The typical hopping distance is given by35 (4.3) R = ξ (T0/192T)0.25 Here ξ is the decay length of the localized electronic wavefunction, which is assumed to be equal to 1 nm, i.e. equal to the van der Waals radius of the C60 molecule., while T0 = 8.72±1.26 · 105 K is the parameter entering in Eq. 4.2(a), which was determined from the fit of the σdc values. The value of R varies from 1.96 to 1.75 nm as the temperature increases from 313 K to 473 K. The apparent activation energy shown in the same plot was calculated both using Eq. (4.1) and according to the formula: (4.4) Ea = 0.5kBT00.5T0.5 which is valid for the case of VRH with n = 0.5. As visible from Figure 4.8(b), Eq. (4.4) gives values more or less consistent with those obtained directly from the raw data using Eq. (4.1). These results therefore indicate that the charge conduction mechanism in pure material is due to variable-range hopping of localized electronic states. We suggest that the charge carriers are electrons, rather than holes, based on the known electron affinity and n-channel behavior of pure C60 and of many of its derivatives.5,2,3,42 Figure 4.9(a) shows the Arrhenius plot of the relaxation frequency fmax of the loss peak of the pure material. Just as the dc conductivity, also the relaxation frequency does not obey a simply-activated Arrhenius behavior. The apparent activation energy for fmax is defined, in line with Eq. 4.1, as: 103 (4.5) As visible in panel Figure 4.9(b), despite its noisier profile the apparent activation energy of fmax displays the same mild temperature dependence as σdc, increasing with increasing temperature. The Arrhenius plot of fmax (panel a) could be fitted with a power law of the same form as for the dc conductivity, with a slightly higher exponent equal to 0.62 ± 0.08. Figure 4.9: Dielectric relaxation data of pure C60(ONa)24. (a) Arrhenius plot of the frequency fmax of the loss maximum and corresponding power-law fit (line). (b) Apparent activation energy of fmax as function of the inverse temperature. Bars represent typical uncertainties. (c) Arrhenius plot of the frequency fM”max of the conductivity relaxation peak of the modulus spectra and of fσ (see text). The Arrhenius plot of fmax is also shown for comparison. (d) Arrhenius plot of the dielectric strength ε, as extracted from the fits to the loss spectra. 2.2 2.4 2.6 2.8 3.0 3.2 3.4 30 40 50 0 1 2 3 4 5 2.2 2.4 2.6 2.8 3.0 3.2 3.4 0.6 0.8 1.0 0 1 2 3 4 5 (d)  (a) (b) Log(fmax/[Hz]) Ea[eV] 1000/T [K-1]1000/T [K-1] Log(f/ [Hz]) (c) Log(fM''max) Log(f ) Log (f max) 104 A loss feature with these characteristics cannot correspond to the dipolar relaxation of polar impurities (e.g. water). In fact, the same feature was observed even after annealing at 575 K (the highest temperature reached in our experiments) in nitrogen atmosphere, where no volatile dipolar impurity such as water can be present in the sample. Moreover, the positive curvature of the Arrhenius plot of fmax is incompatible with a dipolar relaxation peak. The strong similarity between the temperature variation of the loss frequency and the dc conductivity indicates instead that the loss feature is associated with the hopping of charge carriers. As may be gathered by visual inspection of Figure 4.9(c), the Arrhenius plot of the loss feature fmax has a temperature dependence that is very similar to that of σdc. This confirms that the dielectric loss is associated with the electrical conduction (namely, with hopping processes of electronic charge carriers), as already suggested in the second part of Section 4.3. The interpretation of the loss peak as being a conductivityinduced effect is further corroborated by the scaling behavior of the spectra (see below). As mentioned in Section 4.3, the simplest explanation of the conductivity-induced loss feature is the accumulation of (free) charges due to the spatial variation (inhomogeneity) of the conductivity, the so-called space-charge effect. The frequency of the loss feature marks the onset of the dispersive conductivity regime dominated by back-and-forth hops of charges between first-neighbor sites, whereas the dc value is determined by long-range correlated hopping processes.43,44 Given that the grain size is quite small (roughly 30 nm, see the discussion of Figure 4.2(b)), it is natural to ascribe the observation of a space-charge effect to dissipative processes accompanying electron accumulation at grain boundaries. It is convenient for our analysis of the Arrhenius behavior to introduce another characteristic frequency, the so-called space-charge relaxation frequency fσ, which represents the characteristic relaxation frequency of spatial charge fluctuations in a conducting medium,45,46 and which is defined as fσ = σdc/(2ε0ε). Figure 4.9(c) displays the Arrhenius plot of fmax, fσ and of the frequency fM”max of the modulus peak. The values of Log (fM”max) overlap almost perfectly with those of Log (fσ). Since usually the dielectric strength ε is at most a slowly varying function of 105 temperature, Log (fσ) is basically proportional to Log(σdc). On the other hand, the frequency of the conductivity peak in the modulus spectra basically represents the frequency f* which marks the onset of the dispersive part of the conductivity,47 defined as the frequency at which σ’(f*) = 2σdc.48 The fact that Log (fM”max) and Log (fσ) coincide therefore implies that the activated behavior of σdc, a quantity that is related to the long-range charge transport, is identical to that of the onset frequency f* above which the conductivity is dominated by hopping processes between nearest-neighbor molecules. This entails that the mechanism behind the long-range charge transport is the same as for the dispersive regime, namely hopping processes between neighboring molecules, as may be expected. Panel d of Figure 4.9 shows the temperature dependence of the strength ε of the dielectric loss as obtained from fits to the ε” spectra. The slow decrease of ε may be expected in the case of a conductivity-related dielectric loss in a molecular material, since the thermal motions limit the extent and dynamics of the polarization clouds surrounding the charge carriers. It should be pointed out however that several distinct microscopic models predict an inverse proportionality between ε and T (see e.g. Ref. 49). Figure 4.10 shows the results of the scaling analysis of the dielectric spectra acquired upon cooling from 473 to 313 K. As visible in panel a, the ac conductivity (σ’) spectra display the same line shape at all temperatures and may be superposed to the same master curve. The spectra are found to obey a time-temperature superposition principle following in particular the prescription of the so-called Summerfield scaling relationship σ’(f)/σdc = G(f/σdcT) (see e.g. Ref 50). In this expression, G is the shape function (master curve) of the normalized conductivity spectra, common to all temperatures, and f is the frequency in Hertz. 106 Figure 4.10: Summerfield scaling analysis of the normalized ac conductivity σ’/σdc (a) and of the permittivity ε’ (b) between 473 and 313 K. Inset to panel a: logarithmic plot of the onset frequency f* (defined in the text) versus σdcT. As shown in the inset to Figure 4.10(a), the onset frequency f* of the dispersive part of the conductivity spectra, defined as the frequency f* at which σ’(f*) = 2σdc,48 is directly proportional to the product σdcT. This is further proof that the same mechanism involved in the long range charge transport (dc conductivity) is responsible for the transition to the dispersive part of σ’. The conductivity is the product of the charge carrier density N, the mobility , and the charge e of the carriers. The mobility has activated behavior scaling with the same activation energy as f*, while the charge carrier is constant or only weakly dependent on temperature.45,51The real part of the permittivity (Figure 4.10(b)) also obeys the Summerfield scaling. The height of the step-like bump in the ε’ spectrum, which as mentioned is equal to the strength ε of the loss feature in ε”, is observed to decrease slowly with increasing temperature, in agreement with Figure 4.9(d). 4 6 8 10 12 14 0 1 2 3 4 6 8 10 12 14 0 1 2 3 4 5 6 78910 11 12 13 0.4 0.6 0.8 1.0 1.2 01x10-6 2x10-6 3x10-6 0.0 2.0x10-7 4.0x10-7 6.0x10-7 8.0x10-7 1.0x10-6 1.2x10-6 -8 -7 -6 -5 1 2 3 4 Log{(f/dcT)/ [Hzcm.K-1]} Log('/dc) (a) 473K 458K 453K 448K 443K 438K 433K 428K 423K 418K 413K 408K 403K 398K 393K 388K 383K 378K 373K 368K 363K 358K 353K 348K 343K 338K 333K 328K 323K 318K 313K Log(') (b) Log(f*/Hz) Log[dcT/(S.K.cm-1)] 107 Figure 4.11: (a) Plot of σdc versus 2ε0ε fmax to verify the validity of the Barton-NakajimaNamikawa relationship (see text). (b) Summerfield scaling analysis of the exponents s’ and s” characterizing the complex ac conductivity (see text). Error bars are relative to the data acquired at 373 K but represent the typical uncertainty of all spectra. Figure 4.11(a) displays the linear relationship between the dc value of the conductivity and the quantity 2ε0ε fmax. The slope of the linear fit is of the order of unity (the actual value is 0.35), which shows that σdc  2ε0ε fmax, or fmax  fσ. This relationship, known as Barton-Nakajima-Namikawa (BNN) condition,52 is further confirmation that the frequency position fmax of the loss peak is determined by the dc conductivity. The validity of the time-temperature superposition principle and the fulfillment of the BNN relationship together entail.34 that the mechanism driving conduction is basically the same at all temperatures and that the high-temperature relaxation feature is associated with the motion of the charge carriers that contribute the dc conductivity, a situation encountered in many inhomogeneous or disordered materials and even in organic and inorganic ionic conductors.34, 53 If the sample exhibited two equally important contributions to the conductivity coming respectively from the electronic and ionic mobility, one would expect that the temperature-dependence of both would differ; hence the observation of Summerfield scaling implies that the electronic contribution is the dominant mechanism, and that ionic contribution, if present, plays only a minor role, despite 7 8 9 10 11 12 13 0.4 0.6 0.8 1.0 1.2 010 20 30 0 2 4 6 8 10 12 (b) s'' Log{(f/dcT)/[Hzcm.K-1]} 313K 343K 373K 403K 433K s' dc  105[S.cm-1] (a) max0 [F.Hz.cm-1] 108 the high density of sodium moieties in the material. An electrode polarization effect is observed only at the highest temperatures probed (see Figure 4.6), confirming that the ionic displacements are negligible at low temperature. These observations imply that the Na+ ions are relatively tightly bound to the oxygen atoms of the fullerene cages. The real and imaginary parts of the complex conductivity in the dispersive regime, that is, above the onset frequency f*, are usually analyzed in terms of a power-law-like dependence of the type σ’  f s’ and σ”  f s”.34 Figure 4.11(b) shows the frequency-dependent exponents s’ and s” characterizing the conductivity spectra of the material at the indicated temperatures. As may be expected from the scaling analysis carried out so far, the Summerfield scaling is found to apply also to both exponents, except for s’ at low frequency. The discrepancy is due to the different extent of the electrode polarization at different temperatures. Such polarization effects also lead to the observed decrease of the s” exponent as the frequency is lowered (left-hand portion of panel b of Figure 4.11). At intermediate frequencies, the s’ exponent displays a small bump indicative of the presence of the dielectric loss, which as discussed is centered near f*, in correspondence with which s” exhibits a broad local minimum. The values of both exponents coincide near f*, where s’ ≈ s” ≈ 0.6. At higher frequencies (right-hand portion of Figure 4.11(b)) the frequency-dependence of both exponents matches the behavior observed in other systems,9 with s” becoming larger than s’ above f*. It is interesting to compare the origin and character of induced polarization in the oxofullerene sodium derivative with those of Buckminster fullerene. Compared to the pristine C60 molecule, the formation of covalent C–O bonds in C60(ONa)24 reduces the aromaticity and conjugation of the carbon-carbon bonds of the fullerene cages. Since the locations of the remaining fewer double bonds in C60(ONa)24 are fixed with respect to the positions of the oxygen moieties, a much lower molecular orbital polarizability results, compared with that of the molecular orbitals of C60 and its anions. In other words, contrary to solid C60 and to solid phases containing (C60)n– anions, in which the molecular polarizability is due to the  electrons in the 109 molecular orbitals of the carbon cage,7,54,55,56 the molecular polarizability of C60(ONa)24 is likely to be due mainly to the distortion of the polar C–O∙∙∙Na bonds. 4.4.2 Effect of the Surface Hydration Water on Pure C60(ONa)24 As mentioned at the beginning of subsection 4.4.1, to measure the pure material we preheated the hydrate powder to dehydrate it, getting rid of the structural water. However, between this preheating and the measurements, i.e., during preparation of the pellet, the sample was exposed to ambient air. Due to the hygroscopic nature of C60(ONa)24, this resulted in formation of a surface hydration-water layer on the powder grains, as we describe in this subsection. The spectra acquired on one such quasi-pure pellet obtained by exposure of the pure material to air are displayed in panels a and b of Figure 4.12. As visible from comparison of Figure 4.12 with Figure 4.5, it is seen that the only partially hydrated powder does not exhibit the phase change observed in the hydrate upon heating, but rather a single, non-reversible spectral modification as the temperature increases from room temperature to 360 K. The dc conductivity of the partially hydrated powder (Figure 4.12(c)) was initially much higher than that of the fully dehydrated material, and only dropped to a value comparable with the latter upon heating to 325 K. Above this temperature, the temperature-dependence of the conductivity was the same as in pristine (water-less) C60(ONa)24, and the spectra obtained in subsequent heating-cooling cycles under constant nitrogen flow overlapped with the first cooldown data, confirming that all surface water leaves the sample in the first heating to 325 K. A single, broad permittivity feature is observed in all dielectric spectra of Figure 4.12(a). The Arrhenius plot of the characteristic frequency fmax (respectively, dielectric strength ε) of such feature is shown in the main panel of Figure 4.12(d) (respectively, in the inset). In the same temperature range of the conductivity change, the loss feature undergoes a non-monotonous frequency shift and a significant decrease in strength (by a factor of three). 110 Figure 4.12: Dielectric loss (a) and ac conductivity (b) spectra acquired on a pellet made with powder preheated to 430 K and then exposed to ambient air, in the temperature range between 293 and 373 K. The arrows highlight the spectral variation with increasing temperature. (c) Arrhenius plot of σdc for the same data. For comparison, the σdc values extracted from measurements performed cooling down from 500 K is also shown (same data as Figure 4.6(b)). (d) Arrhenius plot of the relaxation frequency, compared with that of σdc. Inset: dielectric strength ε as a function of the reciprocal temperature. The observed decrease of σdc upon desorption of water is reminiscent of the behavior of many insulating and semiconducting porous inorganic materials11-13 in which a conductivity enhancement is observed after exposure to humidity. It may be gathered from Figure 4.12(c) that the conductivity variation is approximately of four orders of magnitude, a dramatic effect which is in line with reported conductivity enhancements in inorganic systems.11-13 The recovery of the conductivity value of 1 2 3 4 5 6 -1 0 1 2 3 4 1 2 3 4 5 6 -8 -7 -6 2.0 2.4 2.8 3.2 -10 -9 -8 -7 -6 -5 2.0 2.4 2.8 3.2 3 4 5 6 Log('') 303K 313K 323K 333K 343K 353K 363K Log (f /[Hz]) (b) Log (f /[Hz]) Log(' /[S.cm-1]) Increasing T Heating Cooling Log(dc/ [S.cm-1]) 1000/T [K-1] Increasing T (a) (c) (d) 1000/T [K-1] 1000/T [K-1] Log(fmax /[Hz])  Log(dc/[S.cm-1]) -8 -7 -6 -5 2.0 2.4 2.8 3.2 0 10 20 30 111 the pristine material at relatively low temperatures and the absence of the other changes visible in the hydrate (Figure 4.5) together indicate that, in the short exposure to air prior to measurement, the material did not have time to form a stable hydrate phase containing structural water. We thus conclude that air exposure led to condensation of (loosely bound) hydration water onto the surface of pure C60(ONa)24 grains. In what follows, we will refer to this water contribution as “surface hydration water” to distinguish it from the structural water. Such surface hydration water is responsible for the observed conductivity enhancement. As we discussed in the section 4.4.1, pure C60(ONa)24 is a polycrystalline powder with rather low conductivity (10–11 S/cm at room temperature) and relatively large surface area, considering the size of the crystalline grains (30 nm). The large surface area rationalizes the observed dramatic (surface) conductivity increase upon air exposure. The conductivity enhancement is responsible also for the more prominent electrode polarization effect visible in the room-temperature spectra (Figure 4.12(b)). Water-induced conductivity enhancements are known to occur in inorganic metals and oxides, and it is generally accepted that they arise from small-ion transport through chemisorbed and physisorbed water layers.11,14,33 In fact, the conductivity enhancement in these materials is purely a surface effect, as water cannot penetrate inside their tight lattice structure of inorganic materials. The exact nature of the underlying conduction mechanism is debated,30 but the most likely candidates are ion diffusion and proton exchange, the latter being a “shuttle” mechanism by which O–H bonds are interchanged between adjacent water molecules. The proton exchange mechanism is responsible for the electrical conductivity of a wide range of systems such as pure water, ice, phosphoric acid, pharmaceutical phosphate salts, as well as some hydrated organic compounds.57,58,59,60,61 This mechanism, known in water as “Grotthuss shuttling,” is sometimes accompanied by rearrangements of the molecular orientations and of the solvation shells.57 Contrary to what has been proposed for some inorganic materials, where the protons responsible for the surface conductivity enhancement have been suggested to stem from the chemisorption of water onto activation sites,59 the surface effect reported here is unlikely to arise from a chemisorption process, as the extra surface-conductivity contribution vanishes 118 Figure 4.14: Dielectric loss (left panels, 1) and ac conductivity (right panels, 2) spectra acquired on heating the C60(ONa)24 16H2O hydrate preheated under nitrogen atmosphere to 323 K, in different temperature intervals: (a) 293 to 323 K; (b) 328 to 363 K; (c) 368 to 383 K; (d) 388 to 423 K. It may be gathered from Figure 4.14(a) that, prior to the loss of structural water, the permittivity spectra are actually characterized by two loss features, one in the same frequency range of the conduction-related feature of the pure material, and a second one at higher frequency, visible only below room temperature in our -2 0 2 -11 -10 -9 -8 0 2 4 -10 -9 -8 -7 0 2 4 -8 -7 -1 0 1 2 3 4 5 6 0 2 4 -1 0 1 2 3 4 5 6 -8 -7 383K 323K 318K 313K 308K 303K 298K 293K (a1)(a2) (b1) (b2) 363K 358K 353K 348K 343K 338K 333K 328K Increasing T (c1) (c2) 378K 373K 368K Increasing T (d1) Log('') (d2) Log( '/ [S cm-1]) Log(f /[Hz]) 423K 418K 413K 408K 403K 398K 393K 388K 119 experimental frequency range. Both relaxation features exhibit simply activated behavior. While the lower-frequency feature appears to be associated with charge accumulation at grain boundaries, as mentioned above, the low-temperature feature visible at high frequency, which displays activation energy of 0.9 eV (not shown), might have a dipolar origin. As this faster feature is observed only in the hydrate and has no analog in the pure material, we associate it with the dipolar relaxation of structural H2O molecules, which, given the crystalline nature of the hydrate, may only exhibit reorientational motions (e.g., as in a plastic crystal). The activation energy of such fast relaxation is significantly higher (by almost a factor of two) than that observed in supercooled or confined water,61-64 which may result from the much stronger interactions and orientational correlations (steric hindrance) between the structural H2O molecules in the crystal lattice of the hydrate. Figure 4.15 shows the Arrhenius plots of the main (slower) relaxation (panel a) and of the conductivity (panel b) in the whole temperature range of the spectra of Figure 4.14. For comparison purposes, in Figure 4.15(b) we also show σdc for the pure material (same data as those of Figure 4.7(a)). In Figure 4.15(a), together with the maximum loss frequency fmax we show also the space-charge relaxation frequency defined as fσ = σdc/(2ε0ε) (see Section 4.4.1), where ε is the dielectric strength of the slower relaxation in the hydrate. The correspondence between the line shapes of all three Arrhenius plots (σdc, fmax and fσ) upon heating is remarkable. Both fmax and σdc display a crossover to steeper temperature dependence above 323 K. This more pronounced variation with temperature slows down at 350 K, until its trend is reversed at around 370 K (the temperature of the structural transition), with both σdc and fmax decreasing with increasing temperature. The non-monotonic behavior of the dc conductivity is simultaneous with that of the loss feature, and both exhibit a maximum at 365 K. The Arrhenius plot of fσ displays a similar behavior. All these similarities and the continuous evolution of the main loss feature of the hydrate into that of the pure material (which arise from a space-charge accumulation effect as shown in Subsection 4.4.1) are all strong indications that the main loss feature in the bulk hydrate has a space-charge origin. 120 Figure 4.15: Change in the dielectric properties of the C60(ONa)24 16H2O hydrate preheated to 323 K upon the structural dehydration: (a) frequency fmax of the main loss feature, and corresponding space-charge relaxation frequency fσ ; (b) Dc conductivity upon heating (red markers) and cooling (blue markers). Inset to panel a: dielectric strength ε of the data acquired upon heating, as a function of reciprocal temperature. The data upon heating correspond to the spectra of Figure 4.14. As visible in the inset to Figure 4.15(a), the dielectric strength of the loss feature decreases with increasing temperature, in line with the results presented in Figure 4.5 in Section 4.3. As it can be inferred from a visual comparison between the permittivity spectra of panels a1 and d1 of Figure 4.5 or 4.14, and as clearly visible in the insets to Figure 4.12(d) and Figure 4.15(a), the strength ε of the loss feature is always higher in the (even partially) hydrated samples than in the pure material. In particular, upon heating the structural hydrate the dielectric strength shows two abrupt changes to lower values, the first around 323 K, in coincidence with the crossover to steeper Arrhenius dependence, and the second one at 370 K, in correspondence with the structural transition. There is an overall decrease of 2.0 2.4 2.8 3.2 0 1 2 3 4 5 2.4 2.8 3.2 -11 -10 -9 -8 -7  Log(f max/[Hz]) Log (f max) Log(f )(b) (a) Log(dc/[S cm-1]) 1000/T [K-1] Heating Cooling 2.4 2.8 3.2 0 5 10 15 121 strength by a factor of 7 between the hydrate and pure material. The higher value of ε in the presence of water and the fact that at low temperature the Arrhenius plots for fmax and of fσ do not overlap (Figure 4.15(a)), as well as the only approximate validity of the BNN condition and the slight difference between the activation energies of the dc conductivity and of the permittivity feature (Figure 4.13), are all clear indications that, while the loss feature and the conductivity are correlated, the origin of the loss cannot be a pure space-charge effect as in the pure material. In particular, it appears evident that the structural/interfacial water molecules contribute directly to the strength of the permittivity feature. To rationalize our findings, we suggest that the fundamental origin of the main loss feature in the hydrate is accumulation of charge at the sample’s inhomogeneities, as in the pure material; however, such dielectric feature also contains in the surface-hydrated material and in the structural hydrate a partial dipolar contribution associated with the reorientational motions of the structural H2O dipoles which accompany the oscillation of the interfacial dipole associated with the accumulation of charge carriers. Notice in fact that the strength of the space-charge feature is always higher in the presence of water (both in the case of structural water and surface hydration water). We finally analyze the effect of the structural dehydration on the dc conduction properties. The observation of a local maximum of conductivity close to the maximum water loss (Figure 4.15(b)) and the fact that the inter-fullerene spacing is higher in the hydrate than in the pure material, suggest that the hydrate’s σdc is dominated in this temperature range by a non-electronic charge transport mechanism associated with water. The charge carriers associated with the hydrogen-bonded network of structural water molecules are probably protons moving by hydrogenbond shuttling, which as mentioned in Subsection 4.4.2, is the main conductivity mechanism in several hydrogen-bonded and hydrated systems.57-64 Indeed, the abrupt changes in the value of the dielectric strength and activation energies are observed around 323 K, where the temperature dependence becomes steeper; this temperature is well below the onset of the loss of structural water, which takes place at 350 K (Figure 4.2(a)): hence, the conductivity enhancement cannot be ascribed to 122 the formation of internal voids in the hydrate, which makes it unlikely that it can be explained by means of a vehicle mechanism (i.e., diffusion of H3O+ or OH– ions). The observation of a conductivity maximum entails that the density of charge carriers is not constant. In fact, in a series of spectra taken at the fixed temperature of 350 K under constant N2 gas flow (in a different experimental run than that of Figure 4.16) the dc conductivity was observed to decrease with time. These data, displayed in Figure 4.16, clearly show that, in the temperature window where water starts leaving the sample, the measured σdc value is not an equilibrium value, so that no true activation energy can be extracted from the heat-up data of Figure 4.15(b). The fact that the conductivity drops in time further proves that the conductivity enhancement cannot be due to hydronium or hydroxyl ions that start diffusing through the voids left in the lattice by the departing water, for in such case one would expect the conductivity to increase (or, at least, to remain constant) as more water leaves the sample. Having thus discarded an electronic or a vehicle charge transport mechanisms in the hydrate, and given that the surface conductivity enhancement is due to proton shuttling (section 4.4.2), we propose that the protonexchange scenario applies also to the σdc enhancement preceding the dehydration process. 123 Figure 4.16: Series of conductivity spectra acquired at 348 K after heating for the first time a hydrate pellet to this temperature. To summarize, the C60(ONa)24 16H2O hydrate exhibits two dielectric relaxations. The fastest one (at higher frequency) may stem from reorientational motions of the structural H2O molecules. The slower and more prominent loss (larger dielectric strength) is observed to evolve, upon dehydration, into the conductivity-induced relaxation feature of the pure C60(ONa)24 salt. While this suggests a common origin associated with charge-carrier accumulation at the sample’s heterogeneities, the dielectric strength and temperature-dependence of this main relaxation indicate that it involves also the reorientational motion of water dipoles at the same heterogeneities. Several conductivity anomalies are reported below and at the temperature of structural dehydration. A decrease of conductivity is observed at the structural transition, obviously related to the phase change, which indicates a direct involvement of water molecules in charge transport through the hydrate. At temperatures below the structural dehydration we observe a cross-over of the conductivity to a more pronounced temperature dependence, which we ascribe to the onset of a water-induced conduction mechanism likely involving hydrogen-bond -1 0 1 2 3 4 5 6 -8.0 -7.5 -7.0 -6.5 Log(f /[Hz]) Log(' /[S.cm-1]) 348K t = 0min t = 5min t = 10min t = 15min t = 20min t = 25min time 124 exchange. The effects of water on conduction and dielectric properties appear to be closely inter-correlated: on one hand, water reorientations accompany the spacecharge relaxation of carriers at heterogeneities; on the other, the onset of proton exchange is likely accompanied by the onset of reorientational motions of the water molecules. 4.5 Conclusions We have analyzed the conduction and dielectric properties of the recently synthesized polycrystalline C60(ONa)24 powder and of its hydrate, of chemical formula C60(ONa)24 16 H2O. In the pure material charge conduction is electronic and well described by Mott’s variable-range polaron hopping model. The effective hopping activation energy Ea and most probable hopping range vary between 0.72 eV and 0.88 eV and 1.95 to 1.76 nm, respectively, as the temperature increases from 313 K to 473 K. The imaginary permittivity spectra display a single loss feature which is associated with polarization effects accompanying the hopping of polaronic charge carriers, as demonstrated by the validity of the Barton-Nakajima-Namikawa condition and of the time-temperature superposition principle based on the Summerfield scaling. While the fully dehydrated, pure material exhibits low conductivity, exposing it to humid atmosphere leads to a four-decade conductivity enhancement below 325 K, due to charge transport through the hydration layers present on the surface of the crystalline grains. Such charge transport is due to proton exchange through second or higher molecular hydration layers. In the hydrate, the dc conductivity is strongly temperature-dependent, and it is higher than that of the pure material by almost two orders of magnitude around 350 K. A cross-over of the conductivity to a more pronounced temperature dependence is observed in the hydrate at a temperature substantially lower than the structural dehydration temperature. We argue that both the higher conductivity in the hydrate, and the cross-over in temperature dependence arise most likely from a proton exchange majority contribution to the long-range charge transport. 125 A space-charge loss associated with charge accumulation at grain boundaries is observed in all spectra. Its dielectric strength is strongly affected by the presence of water at the grains’ surface: it drops by a factor of approximately three upon desorption of the surface hydration water, and by a slightly lower factor upon loss of the structural water. Our results indicate that interfacial water has a strong impact not only on the long-range charge transport but also on energy dissipation processes that accompany the accumulation of charge at crystalline grain boundaries. Our study sheds light on two well-known but not fully understood phenomena, namely, the surface conductivity enhancement in porous materials upon adsorption of water vapour, and the change in conductivity across the structural dehydration of crystalline hydrates. In particular, we have shown that the extra surface conduction takes place through secondary hydration layers and involves proton hopping between whole water molecules; and that the conductivity change actually occurs at lower temperature than that of structural decomposition. Our results also imply that the anomalous non-monotonic behavior of relaxations observed in porous watercontaining systems may arise simply as a result of the change of conductivity induced by the desorption of water. Finally, we have shown that it is the presence of water around the fullerene derivatives which results in proton drift, while the waterfree material does not exhibit protonic conduction, a result which rationalizes earlier findings.11-14 126 References 1 Lai, Y.-Y.; Cheng, Y.-J.; Hsu, C.-S. Applications of Functional Fullerene Materials in Polymer Solar Cells. Energy Environ. Sci. 2014, 7, 1866–1883. 2 Anthopoulos, T.D.; Singh, B.; Marjanovic, N.; Sariciftci, N.S.; Ramil, A.M.; Sitter, H.; Cölle, M.; de Leeuw, D.M. High performance N-Channel Organic Field-Effect Transistors and Ring Oscillators Based on C60 Fullerene Films. Appl. Phys. Lett. 2006, 89, 213504. 3 Riccò, M.; Belli, M.; Mazzani, M.; Pontiroli, D.; Quintavalle, D.; Janossy, A.; Csanyi, G. Superionic Conductivity in the Li4C60 Fulleride Polymer, Phys. Rev. Lett. 2009, 102, 145901. 4 Pontiroli, D.; Aramini, M.; Gaboardi, M.; Mazzani, M.; Gorreri, A.; Riccò, M.; Belli, M. Ionic Conductivity in the Mg Intercalated Fullerene Polymer Mg2 C60 Carbon 2013, 51, 143-147. 5 Quintavalle, D.; Márkus, B. G.; Jánossy, A.; Simon, F.; Klupp, G.; Győri, M.; AKamarás, K.; Magnani, G.; Pontiroli, D. and Riccò, M. Electronic and Ionic Conductivities in Superionic Li4C60. Phys. Rev. B 2016, 93, 205103. 6 Brutting, W.; Adachi, Ch. (Ed.s). Physics of Organic Semiconductors, 2nd Ed. Wiley 2012. 7 Macovez, R.; Hunt, M. R. C.; Goldoni, A.; Pedio, M.; Rudolf, P. Surface Hubbard U of Alkali Fullerides. J. Electron Spectr. Relat. Phenom. 2011, 183, 94–100. 8 Fazekas, P. Lecture Notes on Electron Correlation and Magnetism. World Scientific Publishing: Singapore 2003. 9 Kremer, F.; Schönhals, A. Broad Band Dielectric Spectroscopy. Springer: Berlin 2003. 127 10 Gunnarsson, O. Superconductivity in Fullerides. Rev. Mod. Phys. 1997, 69, 575606. 11 Haspel, H.; Bugris, V.; Kukovecz, Á. Water Sorption Induced Dielectric Changes in Titanate Nanowires. J. Phys. Chem. C. 2013, 117, 16686–16697. 12 Aragoneses, A.; Tamayo, I.; Lebrato, A.; Cañadas, J.C.; Diego, J.A. Arencón, D.; Belana, J. Effect of Humidity in Charge Formation and Transport in LDPE. J. Electrostatics. 2013, 71, 611-617. 13 Cramer, C.; De, S.; Schönhoff, M. Time-Humidity-Superposition Principle in Electrical Conductivity Spectra of Ion-Conducting Polymers. Phys. Rev. Lett. 2011, 107, 028301. 14 Ahmad, M. M.; Makhlouf, S. A.; Khalil, K. M. S. Dielectric Behavior and Ac Conductivity Study of NiO/Al2O3 Nanocomposites in Humid Atmosphere. J. Appl. Phys. 2006, 100, 094323. 15 Faia, P. M.; Furtado, C. S.; Ferreira, A. J. AC Impedance Spectroscopy: A New Equivalent Circuit for Titania Thick Film Humidity Sensors. Sensors and Actuators B.2005, 107, 353–359. 16 Bernard. M, Kulwicki. Humidity Sensors. J. Am. Ceram. Soc. 1991, 74, 697–708 17 Yamazoe, N. and Shimizu, Y. Humidity Sensors: Principles and Applications. Sensors and Actuators. 1986, 10, 379–398. 18 Coropceanu, V.; Cornil, J.; da Silva Filho, D. A.; Olivier, Y.; Silbey, R.; Bredas, J.-L. Charge Transport in Organic Semiconductors. Chem. Rev. 2007, 107, 926-952. 19 Djordjevic, A.; Vojinovic-Miloradov, M.; Petranovic, N.; Devecerski, A.; Lazar, D.; Ribar, B. Catalytic Preparation and Characterization of C60Br24. Fullerene Sci. Technol. 1998, 6, 689-694. 134 mechanism is by polaronic (electron or hole) hopping.8 The hopping-mediated conductivity is a consequence of strong local interactions that tend to localize charge carriers, such as inter-electron Coulomb repulsion and polarization screening,9 coupling to molecular vibrations,10 or trapping at defects. Due to their globular shape, C60 and some of its derivatives display rotational motions and orientational phase transformations in the solid state. For example, solid C60 displays a face-centered cubic (fcc) rotator phase of freely spinning molecules at room temperature, while below 260 K the free-rotor motion is reduced to a ratcheting motion between two preferred orientations.11,12 This merohedral reorientational motion finally freezes out at a glass transition taking place 90 K, below which the lattice structure is simple cubic with four non-equivalent orientations.13,14 Similar transitions are observed at very similar temperatures in simple derivatives such as C60O and the annulene isomer of C61H2,15,16 albeit in the room-temperature fcc phase of the latter compounds the molecules do not behave as free rotors due to the fact that the adducts occupy interstitial sites of the fullerene lattice. The cyclopropane isomer of C61H2 displays instead a different phase diagram than its annulene isomer, exhibiting in particular an orientational melting occurring through a two-step transition around 198–213 K.17 In pristine C60, orientational ordering strongly favors electronic charge transport: across the transition at 260 K, the dc conductivity of C60 raises by more than one order of magnitude,18 and a similar behavior was observed in C60 salts across the fcc to simple cubic transition.1 This would indicate that molecular reorientations in crystalline fullerenes hinder electron hopping, possibly because they favor coupling to intermolecular vibrations and because the disorder induced by molecular reorientational motions leads to even stronger electron localization on single fullerene molecules and yet smaller electronic bandwidth.9 Less is known on the effect of molecular motions in fullerene derivatives or polycrystalline and disordered fullerene systems. In this chapter we employ dielectric spectroscopy BDS to investigate electrical conduction and dipolar molecular dynamics in the solid phase of the halofullerene C60Br6 derivative. 135 5.2 Synthesis and Experimental Methods For the preparation of C60Br6, a procedure based on Troshin et al.19 was followed. In a typical synthesis, 79 mg of pristine fullerene (C60, Aldrich, 99.8% pure) were dissolved in a mixture of 2 mL of elementary bromine and 10 mL of carbon disulfide. The resulting mixture was allowed to stand without stirring and heating for 10 days. The precipitate was filtered off, washed with hexane and dried at room temperature. Figure 5.1 shows the molecular structure of C60Br6. Figure 5.1: Molecular structure of C60Br6 The as-synthesized powder was characterized by thermogravimetry analysis (TGA) and scanning calorimetry (DSC). TGA curves were acquired while heating the sample under N2 flow between room temperature (300 K) and 450 K by means of a Q50 thermobalance from TA-Instruments. DSC measurements were carried out between 200 K and 450 K using a Q100 calorimeter from TA-Instruments. In both experiments the heating rate was set to 10 K min–1, and the powder was placed in an open vessel to allow evaporation of volatile species. For broadband dielectric spectroscopy measurements, the powder was mechanically pressed into pellets of submillimeter thickness between 7 mm- 136 diameter stainless steel electrode disks in parallel-plate capacitor geometry. Dielectric spectra were acquired in the frequency range from 10–2 to 107 Hz using a Novocontrol Alpha analyzer. Isothermal frequency scans were acquired between 125 and 360 K (with a temperature stability of ±0.3 K) in a N2 flow Quatro cryostat (the sample was in a nitrogen flux during all measurements). Dielectric spectroscopy yields the complex conductivity and permittivity of a sample as a function of frequency (f). As mentioned in Chapter 3, the imaginary part of the permittivity ε”(f), or loss spectrum, is related to the real part of the ac conductivity σ’(f) as ε”(f) = σ’(f)/(2fε0). The σ’(f) spectra (real part of the ac conductivity) displayed a plateau value at low frequency, corresponding to the dc value of the conductivity, σdc. In order to study the temperature dependence of σdc, an effective activation energy of the conductivity is computed as (5.1) 𝐸𝑎 𝜎𝑑𝑐 =−𝑑 𝐿𝑛 𝜎𝑑𝑐 𝑑 1𝑘𝐵𝑇 . Each ac conductivity spectrum and corresponding loss spectrum were fitted as the sum of one or more relaxation processes, modeled with the Havriliak–Negami function,20,21 on top of a background representing the dc conductivity contribution (see chapter 3 section 3.3). 5.3 Thermodynamic Characterization Figure 5.2 shows a typical TGA curve of the as-synthesized powder, both as mass percent and as its first derivative. The main mass loss observed approximately up to 425-450 K corresponds to slightly above 35% in weight, which is in agreement with the expected mass decrease upon loss of all bromine atoms in C60Br6 (the theoretical value is 479.4/1200.1 = 40%). The final product after heating to 450 K is polycrystalline C60, as determined from powder X-ray diffraction experiments (not shown). At lower temperature, a smaller mass loss is observed in the range between room temperature and 340 K due to volatile impurities, likely Br2 or carbon disulfide, both of which are employed in the synthesis (see Experimental Section). 137 Figure 5.2: TGA curve measured on the as-synthesized C60Br6 powder upon heating. Here the thin line is the first derivative of the percent mass loss (thick line). Molecules of bromine and/or of organic solvents are known to remain trapped in the solid matrix during the synthesis process of brominated fullerenes and C60Br6 in particular, sometimes even leading to the formation of solvates.19,22,23 As visible in Figure 5.3, the DSC thermogram exhibits two endothermic features, each occurring in correspondence with a mass loss. 300 350 400 450 500 60 65 70 75 80 85 90 95 100 250 300 350 400 450 500 0.0 0.2 0.4 0.6 0.8 d(% mass)/d(T) [K-1] (a.u.) % mass T [K] 210 220 230 138 Figure 5.3: DSC curves measured on the as-synthesized C60Br6 powder upon heating. In light of these results, for the characterization of C60Br6 by dielectric spectroscopy, the powder was pre-heated to 323 K prior to acquisition of the spectra in order to remove all volatile impurities present in the powder24 (heating to 343 K did not further modify the spectral response). 5.4 BDS Results and Discussion Figure 5.4 shows the frequency-dependent ac conductivity spectra (a) and loss spectra (b) of a typical (pre-heated) C60Br6 pellet, as measured upon heating from 125 K. Similar data were obtained by measuring while subsequently cooling the sample from room temperature. 300 350 400 450 500 60 65 70 75 80 85 90 95 100 250 300 350 400 450 500 0.0 0.2 0.4 0.6 0.8 Heat Flow [W.g-1] T [K] 210 220 230 139 Figure 5.4: Logarithmic plot of the ac conductivity σ’ (a) and dielectric loss ε” (b) spectra of C60Br6, shown for selected temperatures (between 308 and 128 K, every 20 K). Continuous lines are fits with Havriliak-Negami components on top of a background representing the dc conductivity limit. The fit components are shown as dashed lines for the spectrum acquired a 128 K. Inset to (b): logarithmic derivative spectrum 𝜀𝑝𝑜𝑙 ,, shown for two selected temperatures (128 and 148 K; see Eq. (5.6) in the text for the definition). The arrows indicate the position of the two components present in the spectrum. The lineshape of the ac conductivity spectra ’(f) is similar to that of a very wide range of disordered systems ranging from amorphous semiconductors and glasses to -1 0 1 2 3 4 5 6 0 2 4 6 -9 -8 -7 -6 -5 0 1 2 3 4 5 6 -7 -6 -5 -4 -3 -2 -1 Log(f/[Hz]) Log('') 308K 288K 268K 248K 228K 208K 188K 168K 148K 128K (a) Log(f/[Hz]) (b) Log('/[S.cm]) 0 1 2 3 4 5 0 1  Log(der'')  140 metal-cluster compounds to polymers and polymer composites. In all these materials the conductivity is frequency-independent at low frequencies, reaching a plateau value corresponding to the dc value (σdc), and then, above a temperature-dependent onset frequency, exhibits a power-law-like dependence on f, described approximately by: (5.2) σ’(f) = σdc + σ0f s . Eq. (5.2) is known as the universal dielectric response.25,26,27,28,29 In the case of C60Br6, the universal ac conduction law gives an only rough fit to the experimental spectra, which in logarithmic scale exhibit a curved rather than linear lineshape for frequencies above the onset of the ac response. A fit of the spectra with Eq. (5.2) gives values of the exponent s in the range of 0.6-0.7, which is consistent with hopping of localized (polaronic) states.30 The only approximate validity of the universal Eq. (5.2) is due to the presence of a dielectric loss, which is clearly visible in Figure 5.4(b) as a broad bump-like feature above the low-temperature linear dcconductivity background. Many disordered conductors have similar ac conductivity spectra, in which the separation between the dc and ac regimes of the conductivity is signaled by a dielectric loss in the radiofrequency range.25 In most cases such loss is associated with charge accumulation due to spatial inhomogeneities of the dc conductivity.31 The origin of the loss feature in the case of C60Br6 is discussed later. 141 Figure 5.5: Semilogarithmic plot of the dc conductivity σdc of C60Br6 vs 1000/T (Arrhenius plot) between 125 and 330 K. Inset: plot of Log(σdc) vs 100/ 𝑇 below 220 K, for the same data as in the main panel. Figure 5.5 depicts the Arrhenius plot of the dc conductivity (σdc), extracted directly as the plateau value in the σ’ spectra of Figure 5.4 (a). It may be observed that σdc does not exhibit a simply activated (Arrhenius) behavior, but rather displays a negative curvature in the whole probed temperature range (125-330 K). 3 4 5 6 7 8 -9 -8 -7 -6 2.1 2.2 2.3 2.4 2.5 -0.9 -0.8 -0.7 -0.6 Log(dc/[S.cm-1]) Log(dc/[S.cm-1]) 1000/T [K-1] 8.5 8.0 7.5 7.0 -7.0 -7.5 -8.0 -8.5 -9.0 100/T0.5 [K0.5] Log(T/[K]) Log(Ea/[eV]) (b) 140 160 180 200 220 0.12 0.13 0.14 0.15 0.16 T(K) Ea(eV) 1.10 1.15 1.20 1.25 R(nm) 142 Figure 5.6: Logarithmic plot of the activation energy Ea (in eV) of the dc conductivity, versus temperature. The continuous lines are linear fits of the data below and above 215 K, respectively. The dashed vertical line indicates the cross-over temperature between the two regimes, as determined by the intersection of both linear fits. Inset: DSC scan acquired on the same sample upon heating from 200 K. To study the temperature dependence of the conductivity in more detail, we have calculated the effective (temperature-dependent) activation energy Ea of the dc conductivity using Eq. (5.1), and represented this quantity as a function of temperature in the logarithmic plot shown in Figure 5.6. Although the effective activation energy is a monotonic function of temperature (always increasing with increasing temperature), it may be observed that it exhibits markedly different temperature dependences above and below approximately 215 K. The logarithmic plot of Figure 5.6 clearly highlights the existence of two conduction regimes, one characterized by a slope of 0.49±0.02 and the other with a slope of 1.36±0.02 in the logarithmic representation. It is interesting to notice that a small endothermic feature is detected at the same temperature in the DSC thermogram, as shown in the inset to 2.1 2.2 2.3 2.4 2.5 -0.9 -0.8 -0.7 -0.6 Heat Flow[W.g-1] Log(T/[K]) Log(Ea/[eV]) 140 160 180 200 220 0.12 0.13 0.14 0.15 0.16 1.10 1.15 1.20 1.25 220 240 260 280 215K T [K] 143 Figure 5.6. The onset of the new conduction mechanism therefore accompanies a fundamental change in the material. 5.4.1 Variable Range Hopping Conduction in C60Br6 The data below 215 K are consistent with an electronic charge transport described by Mott’s variable range hopping (VRH) model.32 According to this model, the logarithm of the dc conductivity should follow a fractional power-law dependence, given by: (5.3) σdc = σ0 exp[−(T0/T)n]. Here σ0 and T0 are constants and the exponent n is usually equal to 1/2 or 1/4. Applying the definition of activation energy (Eq. (5.1)) to Eq. (5.3), one finds that the slope of the logarithmic plot of Ea vs T plot should be equal to 1 – n.30 The slope of 0.49±0.02 obtained below 215 K gives a value of n of 0.51±0.02, which is virtually identical to the theoretical value of 1/2 of the VRH model, observed in many systems characterized by hopping electronic conduction, ranging from metalcluster compounds to granular and ceramic metals, to doped or amorphous organic and inorganic semiconductors.26,27,29, 33 , 34 The value n = 1/2 is determined by electron-electron correlation27 leading to a weak Coulomb gap near the Fermi level 29 and to a square dependence of the localized state density on energy.35,36 The inset to Figure 5.5 shows the logarithmic plot of σdc vs 1/T0.5 below 220 K to highlight the consistency with the VRH model.