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Physics of Solid State Ionics, 2006: 193-246 ISBN: 81-308-0070-5 Editors: Takashi Sakuma and Haruyuki Takahashi

7 Ionic transport in glass and polymer : Hierarchical structure and dynamics

Junichi Kawamura, Ryo Asayama, Naoaki Kuwata and Osamu Kamishima Institute of Multidisciplinary Research for Advanced Materials Tohoku University, Katahira 2-1-1, Aobaku, Sendai, 980-8577, Japan

Abstract Fundamental aspects of the ionic transport in inorganic glass and organic polymer electrolytes are reviewed. The ion dynamics in the random structures of them can be viewed as a hierarchical dynamic structure in different space-time scales, which is created by the fluctuation freezing during the glass transition. Following a brief histry and recent application of the ionic conductor glasses and polymers, the hierarchical structures of them areexplained. Next, some theoretical aspects on the glass transition (strong-fragile, coupling-decoupling, free-volume), ion dynamics in short time scale (generalized

Correspondence/Reprint request: Dr. Junichi Kawamura, Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Katahira 2-1-1, Aobaku, Sendai, 980-8577, Japan E-mail: [email protected]

Junichi Kawamura et al. 194

Langevin equation, mode-coupling), intermediate time scale (jump diffusion, master equation, percolation) are discussed comparing with some classical models and details of glass structures. Finally, some remaining problems and prospects of future are given. 1. Introduction Glass and polymer electrolytes, both characterized as amorphous structures are now widely used for solid state ionics devices such as lithium ion batteries [1] and fuel cells [2,3], electrochromic displays [4] etc. This is mainly owing to their dense, homogeneous and flexible natures in comparison with crystals or ceramics. On the other hand, their amorphous structure and nonequiliburium nature often hampered the fundamental understanding of their ionic transport mechanism.

This short review is devoted to introduce a present understandings on the ionic transport mechanism in polymer and glassy materials. After a brief survey on the history and applications of the ionic transport in glass and polymers in §1, an introduction to the structure of glasses and polymers are explained in §2, particularly focusing on the hierarchical structure concept. In §3, thermodynamic definition of glassy state is explained and in §4, some theoretical background of the ionic transport in liquid and glasses are given. In §5 is devoted to the unique behavior of supercooled liquid and glass transition phenomena. Solid like models are explained in §6 and the percolation theory is separately discussed in §7. Finally, some brief summary and further expects are given in §8.

Brief history of ionic transport in glass and polymer Alkali ion conduction in inorganic oxide glasses was already known in 19th century by Warburg [5]. Since then, the ionic diffusion in glass has been considered as a disturbance of the electrical insulator used for vacuum tube and semiconductor industry. Especially, the sodium diffusion in the silicon dioxide in MOS semiconductor was an infamous criminal of the leakage trouble in 1970th. On the other hand, in mid 1970th. fast silver ion conduction was discovered in some oxyhalide glasses [6-10], which has been called as superionic conductor glass, fast ion conductor glass, vitreous solid electrolyte, etc. Through 1970 to 1980 th. a lot of silver ion conductor glasses were developed in France, Italy and Japan, especially by T.Minami's group in Osaka [11-13]. Also, in 1980th. lithium ion conductor glasses were investigated in detail [14-16] as a solid electrolyte for all solid state lithium ion batteries. At beginning, conventional inorganic oxide glasses were studied, and later chalcogenide glasses especially sulfide or oxysulfide glasses [17-19] have been recognized better ionic conductivity than oxides and investigated actively until

Ionic transport in glass and polymer : Hierarchical structure and dynamics 195

now [20]. After 1990th some all-solid-state lithium ion batteries using the oxysulfide glasses were announced [21,22]. On the other hand, the oxide based glasses have been evolved for thin-film battery applications [23,24]. In mid 60'th, chalcogenide amorphous semiconductors such as As2Se3, GeS2 have been investigated independently by Russian groups [25]. They reported fast dissolving of silver to the chalcogenide glass thin films by ultraviolet irradiation and named it as "photo-doping phenomena". This phenomenon however drew only a few specialists attentions. Now, this is also recognized as a typical result of fast ion diffusion in chalcogenide glasses combined with optically generated electron (hole) migration and probably the photo induced structural change of the glass [25,26] and is compared with the lithium sulfide glasses. Since the development of optical fiber glasses in 1980 th, halide and chalcogenide glasses have been also studied seeking for the lower optical loss in infrared region. During these studies, unexpectedly high fluoride ion diffusivity was discovered and studied extensively [27,28]. Up to now, there have been developed a large number of ionic conductor glasses, in which ionic species such as silver [12,29], cupper [30-32], lithium [18,19,33-35], sodium [36] fluorine [27,28] and proton [37] are major carriers of the electric current. Some typical examples are shown in table 1. Table 1. Ionic conductivity and glass transition temperatures of typical ionic conductor glasses.


Solid polymer electrolytes In 1970th, alkaline ion conduction was found in solid polymer electrolytes as polyethylene oxide (PEO) dissolving lithium perchlorate (LiClO4) and was also proposed to be used for "lithium polymer batteries" [44,45]. Since then, much work has been devoted to the improvement of the lithium ion conductivity of the polymers, which leads to 10-4 S/cm at room temperature in graft polymers [46] of PEO units and dendritic polymer [47]. The structure of the solid polymer electrolyte is a mixture of crystalline and amorphous phases, however the high ionic conductivity is mainly covered by the amorphous phase [44,45]. Some of the polymers have been used now as the electrolyte for lithium ion secondary batteries, however the relatively low conductivity in the polymer electrolyte at low temperature restricts its applicability. This is due to the strong temperature dependence of conductivity due to the crystallization or and glass transition as shown in fig.1, which will be discussed in §5. For the present, so called "gel electrolytes" containing organic liquid electrolyte in polymer matrix have been used as a compromise [48]. Also, "room temperature ionic liquids" (RTIL) are blended with polymers instead of flammable organic solvents [49-51]. The ionic liquid containing organic ions may form a glassy solid by cooling and become an insulator, however some of them exhibit high ionic conductivity even in the glassy state [53], which is very similar to the inorganic superionic conductor glasses and is included in table 1. Since 1960th, a plenty of effort has been devoted to apply perfluorinated polymers as Nafion for an electrolyte and a separator of fuel cells [2,3]. Recently, proton conductors applicable above 100 °C have been strongly demanded for the fuel cells. For this purpose, along with the improvement of perfluorinated polymer membranes, new proton conductor polymers [54] have also been developed. As can be seen from the above examples, the development of good ionic conductors in amorphous or glassy state is a crucial task for contemporary energy devices. For this purpose, the material with high ionic conductivity and high stability is required, which seems a contradictory demand since the weak bond energy is preferred for high ionic conductivity but is not favored for stability. Key strategy to overcome this difficulty is depicted by strong-fragile and coupling-decoupling concepts by C.A.Angell as will be explained in §5. A micro to mesoscopic heterogeneity is recognized to satisfy this demand. How these concepts can be understood from fundamental physics scheme will be introduced in this review. From fundamental point of view, the ionic transport in glass and polymer includes many challenging and prospective theme in condensed matter physics. Typical important problems are as follows [55];


Figure 1. Temperature dependence of the ionic conductivity of typical lithium ion conductors including crystal(c), glass(g), polymers (p) and gels.

Table 2. Ionic conductivity and glass transition temperature of typical ionic conductor polymers and gels.


(1) Glass transition or Ergodic to Non-Ergodic transition phenomenon which is a frontier in statistical mechanics on non-equilibrium system. (2) Non-exponential and non-linear relaxation phenomena near the glass transition or ion localization region, origin of which is relating to the following problems. (3) Diffusion and localization of ions in random potentials, which is an application theme of the random walk and percolation theories. (4) Correlation between the mobile ions as well as the surrounding ions, this is a largest and most challenging theme, which is more important in classical particles with large exclusion volume than in quantum particle as electrons. (5) Quantum mechanical or chemical bond analysis is a new paradigm to give a fresh prospect on the ionic motion in liquid and glasses, which may give us the final answer to the question why silver and copper compounds are the best ionic conductors. 2. Structure of glass and polymers [92,93] 2.1 Structure in randomness

It is widely believed, "Glass has no structure, it is a random arrangement of constituent atoms". It's not true. Although no glass has long-range periodic structure like crystals, it has a local structure; ex. one silicon atom is coordinated by four oxygen atoms in SiO2 glass. Thus, in order to explain the structure of glass, it is necessary to introduce the special words (1) short range order (2) intermediate range order (3) long range order. One can say, “There is no long range order in glass, however there is short range order”. Ionic transport in glass and polymer is strongly affected by those different levels of structures. In order to achieve the high ionic conductivity as well as stable structure in application, most of the "good" ionic conductor glass and polymers have a sort of "dual structure" of a network frame and a soft part for ion conduction. Short range order

The structure of glass means primarily the short-range order, which can be determined by combining many different spectroscopic means as, X-ray and neutron scattering. XANES, EXAFS, infrared, Raman scattering, NMR etc. In particular, recent advancement of solid MAS-NMR enables us to separate the structure units of the glass and to evaluate the ratio of them. From these experimental data, one can discuss the relation between the ionic transport and local structure of the glass.

The ionic transport is most strongly affected by the short range order especially around the mobile ions, which is the consequence of the chemical bonds between the mobile ions and coordinating other atoms. In conventional alkali


silicate glasses, the alkali cation is coordinated by non-bridging oxygens in the silicate network. The chemical bond strength between the lithium and the NBO is the key factor of ionic transport. A more detailed discussion will be given in §2-3.

Intermediate range order

The word of "intermediate range order" is relating to the 2nd or higher order coordination shell structure, ring structure, and a sort of "clusters", which is in the range from 1nm to 10nm scale and can be detected by small-angle X-ray or neutron scattering, as well as a first-sharp diffraction peak (FSDP) in low angle diffraction spectra. Some ring structures of borate or phosphate glasses are well known examples.

Intermediate range order is also very important for ionic transport in fast ion conductor glasses;. ex. silver oxyhalide glasses are composed of oxide network and silver iodide "clusters" dispersed among them. Silver ion conduction has been believed to happen in the silver iodide clusters [56] or interface region [57-59]. The size of the silver iodide cluster is expected to be about 10 Å, which is typical intermediate order. An evidence of the importance of such intermediate range order for ionic conductivity is demonstrated by the observation of a clear percolation transition in AgI-alkylammoniuimiodide glasses [60] as will be shown in §7.

Long range order

It is mentioned above that no long-range order is seen in glass, however there might exist some special "long-range" structure more than 10 nm scale. One example is a segregation of nano-sized crystals such as α-AgI dispersed glasses [61], which can be seen by FE-SEM measurements to be ca. 30 nm in diameters. Another example is a microscopic phase separation due to the immiscibility in supercooled liquid state such as in the Ag-Ge-S glasses, where unusual conductivity jump is observed as the concentration of silver is varied. This conductivity jump is accompanied by a nano-scale phase separation during the glass formation [39].

Structure of polymer electrolytes

Structures of solid polymer electrolytes are more complicated than those of inorganic glasses, since the organic polymer networks can be designed and synthesized artificially. However, most polymer electrolytes consist of long organic molecular chains and dissolved salts such as lithium salts. The salt concentration is usually less than 10%, since the increase in salt concentration will make the links among the polymer chains to result in the decrease in ionic conductivity. Although, a large amount of salts are tried to dissolve the polymers to make polymer-in-salt type electrolytes [62], which is rather similar to the inorganic ionic conductor glasses. When a charged polymer


chain is cut into small peaces, it becomes an ionic liquid [63], and by being frozen into glassy states it becomes an insulator glass [64,65] or a superionic conductor glass [40,53,60]. A schematic structure models comparing those of inorganic glass and organic polymer solid electrolytes are shown in Figure. 2-1.

Figure 2.1. Structure models of inorganic ionic conductor glass and organic polymer solid electrolytes. (a) Structure model of oxyhalide glasses, triangles are tetrahedrally coordinated metal oxides such as PO4, large circles are halides, small circles are mobile cations. (b) Structure model of solid polymer electrolytes, long curves are polymer chains, O are ether oxygens, large circles are anions, small circles are mobile cations.

2.2 Structure hierarchy and ion dynamics

If one can vary the magnification of the microscope continuously from mm scale to pm scale, amorphous or glassy material manifest themselves in various hierarchical structures. Even a glass looks homogeneous in mm scale, it may have inhomogeneity in sub-µm scale and will have some ordered structure in Å scale. Ionic transport is a phenomenon that a mobile ion starts from a locally ordered atomic scale and diffuses to a macroscopic scale. During this process, the mobile ion experiences various hierarchical structures.

Corresponding to the hierarchical structure of glass, one can observe some different frequency or time domain response corresponding to the fluctuations characteristic to the hierarchy; see fig. 2-2. Atomic vibration in local structure is corresponding to ~10-13 s, which can be observed by infra-red (IR) absorption, reflection, Raman scattering and inelastic neutron scattering (INS); (local vibrational dynamics). The vibrational motion of mobile ions, typically the attempt frequency mode is usually observed rather low frequency region below 1000 cm-1 due to the shallow potential well of weakly binding the mobile ions; ex. Li 425 cm-1, Na 192 cm-1, K 170 cm-1 in oxide glass [66],


~100 cm-1 for silver and copper in halides [32,67,68] and 2900 and 3600 cm-1 for protons in oxide glasses [37].

Longer time scale (10-12 s ~ 10-9 s : GHz ~THz) can be detected by quasielastic neutron scattering (QENS), quasielastic light scattering (QELS) and microwave or miliwave responses. In this time region, one may encounter the low energy excitation characteristic to amorphous matter [55,69] and quasi-local vibration and quasi-local translational mode [32,70,72]. The response in this frequency region corresponds to the ionic motion in nm (cluster?) domains (cluster scale dynamics), which is different from the macroscopic diffusion.

Much longer time scale of radio frequency region (10-9 s to 10-3 s : kHz~GHz) is controversial region, where universal frequency responses of amorphous ionic (and also electronic) conductotos are often observed, which is expressed by a constant loss at higher frequency [73] and a power-law or stretched exponential response at lower frequency region [74,75]. The origin of this universal response has been actively discussed for these 20 years [55,76] relating to the possible correlational motion of mobile ions [77-81] and/or dynamical effect of random structure or heterogeneity [82-90] such as fractal network of the ion channels [60,91]. Authors tentatively designate this region as percolation scale dynamics in fig. 2-2.

At very large scale where the glass looks homogeneous, the ionic motion is regarded as a simple diffusion process in continuous medium described by a conventional diffusion equation, which corresponds to the d.c. conductivity.

A schematic picture of this concept is shown in fig.2-2, where broken line at left side is a light corn representing the dispersion relation of photon (x = ct). It should be noted that the dispersion relation of normal diffusion of an ion is x ~ Dt 1/2, along which the structure models are arranged. 2.3 Random network model of the inorganic glasses

Structure model of inorganic glass comes from the Zachariasen's random network model, Dense randomly close packing (DRCP) model by Bernal, and microcrystalline model by Phillips etc [92,93].

Here, we will start from the random network model for oxide glasses. Zachariasen analyzed the glass forming condition of oxide glasses based on crystallographic point of view and concluded that the glass has a random network structure formed by the corner sharing polyhedrons made of a metal atom at center and oxygens at corners. Typical silicate glass is actually composed of the SiO4 tetrahedrons shearing corner oxygens to form three dimensional network. The oxygen at the corner is bridging two silicon atoms and is called bridging oxygen (BO) ; see figure 2-3. Similarly, in borate glass, BO3 planer triangles or boroxisole rings (Fig. 2-5), and PO4 tetrahedrons in phosphate glasses (Fig. 2-4) are the structure unit to form networks. These


Figure 2.2. Space-Time Hierarchy Structure of Ionic Conductor Glass Experimental techniques covering the space and time scales are shown outside of the map. SAX: Small-Angle-X-ray scattering, SANS: Small-Angle Neutron Scattering, SEM: Scanning Electron Microscope, TEM : Transmission Electron Microscope, EXAFS : Extended X-ray Absorption Fine Structure, FIR: Far Infra Red, IR: Infra Red, QENS: Quasi Elastic Neutron Scattering, IENS : Inelastic Neutron Scattering. oxides are called network forming oxides or network formers, which can be regarded as inorganic polymers.

When alkali oxide as Na2O is added to the SiO2 glass, the introduced oxide ion partly destroys the glass network to form non-bridging oxygen (NBO) connecting to only one Si atom (Fig. 2-3). Simultaneously, the introduced Na+ ion is rather weakly bounded the non-bridging oxygen, which contributes to the ionic conductivity at high temperature. These dopant materials which partly modify the glass network are called network modifiers. Some oxides as Al2O3, which themselves never form glass but are incorporated in the network when added to the glass formers are called intermediate compounds. Addition of the modifiers change the glass structure especially the


concentration of the NBO, which strongly affect the mechanical, thermodynamic and ionic transport properties of oxide glasses due to the break down of the framework structure and trapping the cation around the NBO [94]. The mixing of different network formers and modifiers often results in the ionic conductivity enhancement and is called mixed anion effect or mixed former effect, [95-97]; It is mainly due to the change in the binding energy between the oxide and mobile cations caused by the network structure modifications [98-100].

S iO

O

O

O

O

S iS i

O

O

O

O

OO

Na+

Na+--S i

O

O

OO

O

S iS i

O

O

O

O

O

Na2O

Figure 2.3. Non-Bridging oxygen formation in silicate glasses.

Phosphate glasses [101,102] Similar rule is applicable to the phosphate glasses. In methaphosphate

composition (NaPO3), the PO4 tetrahedoron is connected by sharing two corner oxygens to form one dimensional network and rings. Addition of Na2O destroys the network to form P2O7

4- dimers at pyrophosphate composition (Na4P2O7), and monomer PO4

3- at orthophosphate (Na3PO4) composition. The structure units of the phosphate glass are shown in figure 2, which can be detected by P-31 MAS-NMR.

middle unit

O

P OO

O-

O

PO OO*

branching unit

-O

P OO-

O

end unit

O

P OO

O- -

-

monomeric unit

Figure 2.4. Structure units of phosphate glass.

Borate glasses The structure of borate glass is rather complicated, which has been

analyzed mainly by B-11 MAS-NMR by Bray et al. [103] (Fig. 2-5). They concluded the following rules:


(1) Small addition of Na2O changes the borate coordination from 3 to 4 and the BO4 unit will increase. (2) Above th 30mol% addition of Na2O, the BO4 unit will decrease and on the other hand the non-bridging oxygen (NBO) will increase. More detailed analysis in recent years roughly confirmed this rule [92].

OB

B

BB

B OO

OOO

O O

OO

O

B

BB

B O

OO

O

O

O OO

ﾍﾟﾝﾀﾎﾞﾚﾄ群

B

BB

O

O

O O O

O

ﾎﾞﾛｸｿｰﾙ環

Na O2

Na+

--

-

Na+

Na+

Figure 2.5. Structure units of borate glass and formation of tetrahedral boron.

Oxyhalide glass

Oxyhalide glasses show high ionic conductivity. For instance, the ionic conductivity of AgI doped AgPO3 glass is about 10-3 S/cm at room temperature, which is 3 orders of magnitude larger than the value of AgPO3 glass. When alkali halides are added to the oxide glasses, they are not incorporated in the glass network but remained in the open space between the oxide network (fluoride and some chloride are involved in the network). They can be regarded as a weak network modifier in Zackariasen's point of view, although they modifies the network very weakly. Thus, those halides are called as "dopant", "interstitial" or "plasticizer" as is used in organic polymer field [101,102].

As the first approximation the structure of oxyhalide glass is viewed as a phase separated mixture of oxides and halide domains [104,105], although the size and degree of the heterogeneity is still controversial [106-110]. This comes from the difference in chemical bondings between the constituent atoms in the glass, especially the degree of covalency and iconicity. The strongly connected atoms with covalent bonds form the network backborn of the glass framework. On the other hand, rather weak and ionic bonded ions will construct modifier or plasticizer region. This is true to the alkali halide doped oxide glasses, however it is reverse in AgI doped oxide glasses where the AgI bonding is more covalent than those in oxides. Anyway, a subtle balance of the different types of chemical bonds is of significance in appearance of high ionic conductivity in glass, which will be discussed in §8.

Chalcogenide glasses

The structures of sulfide and other chalcogenide glasses are similar to but a little different from those of the oxide glasses discussed above. Most significant difference is the possibility of edge shearing bonds of metal sulfide


chains [35,111]. The edge sharing connection helps to construct disordered layer structure in chalcogenide glasses, which is an origin of the pronounced FSDP of chalcogenide glasses and is favored for fast ion transport and photosensitive diffusion of silver between the layers [112,113]. Larger ionic radius and polarizability of sulfide than oxide anion is favored for cation migration in glasses and the Li2S-SiS2, Li2S-GeS2 etc are studied as the solid electrolyte for solid state lithium batteries [22,114-116], and also Ag-As-S, Ag-Ge-S(Se) etc. are studied extensively in relating to the optical and photo-induced ion migration properties [112,113]. It is interesting to note that the addition of small amount (~5mol %) of oxide to the sulfide glass enhances the stability of the glass without decrease in conductivity, since the added oxygen works as the bridging oxygen between two silicon atoms and non-bridging positions are occupied by sulfurs coordinating to the mobile lithium ions [116,117]. In case of lithium thiohalide glasses as LiI-Li2S-B2S3 [118], which have very high lithium ion conductivity also have a heterogeneous tendency like silver oxyhalide glasses.

Transition metal oxides

Tungsten, Molybdenum, Vanadium and some other transition metal oxides are not the glass forming oxides, since they usually form octahedrons instead of tetrahedrons favored for glass forming in Zachariasen criteria. So, they are used only as a chlomophore in glass technology due to the d orbital splitting by the surrounding coordination field [119]. However, when they are combined with silver halide they tend to form silver ion conductor glasses rather easily [10,120-123]. Considerable covalent character of the silver halide will modify the chemical bonds of the transition elements to form quasi bridging network with the oxides and silver halides. Such a complex behavior is seen in the difference of Mo and W coordination in AgI-Ag2MoO4 [124] and AgI-Ag2WO4 [122], which is a tetrahedral MoO4

2- unit in former and an octahedral tetramer W4O16

8- unit in the latter. In case of CuI-Cu2O-MoO4 glasses, the coordination of Mo is the octahedral and tetrahedral mixtures [32]. 2.4 Network structures of polymer electrolytes

Polymer electrolytes are usually composed of two main components, one is the polymerized network of organic monomer units as shown in Table 3, and another is an inorganic salt such as lithium perchrolate (LiClO4), LiBF4, LiPF6, etc. The counter anions are distributed in the open space between the polymer chains and also contribute to the ionic conductivity, which causes the electrode polarization in lithium battery applications. We can think of the polymer chain as the network former and the doped salts as the modifier or dopant salts in the inorganic glasses as in Fig. 2-1.


Lithium ion is mobile in the polymer electrolyte but is coordinated by ether oxygens in polymer chains. Thus, the diffusion of lithium ion is accompanied by the micro Brownian motion of the polymer chains, which becomes slower at low temperature and frozen below the glass transition temperature Tg. Consequently, the polymers used for electrolytes have the Tg values much lower than the room temperature; see Table 3. This is the significant difference in comparison with inorganic glasses, where the ionic motion is almost independent from the motion of the surrounding ions (decoupling system) and is used below Tg; this problem will be discussed again in §5. In order to increase the mobility of the lithium ions, hyper branched dendritic polymers were synthesized to show very high ionic conductivity is attained as 10-4 S/cm at room temperature [47]. Recently, some organic-inorganic hybrid polymers have been developed containing Si, B and Al in the polymer network [125-127] as shown in Fig. 2-6.

For battery applications, the polymer electrolytes shown above have not enough conductivity especially at low temperature. So, some organic solvents are absorbed in the polymers, which is called "wet" polymer electrolytes or "gel electrolytes" [48]. In this case, a finite sized domains of the solvent are distributed to swell the polymers. The ionic transport mechanism in the gel electrolyte is much similar to those of liquid states. Moreover, some additives of inorganic fillers are known to increase the ionic conductivity and mobility of lithium ions [128]. Probably, the interface region is responsible for these enhancements but it is out of scope of this article.

Table 3. Typical polymer electrolytes and polymer hosts for gel electrolytes [48].

3. Thermodynamics of glass 3.1 Free energy description of liquid and glass

In order to understand the ionic transport in polymer and glassy electrolytes, we should summarize the thermodynamic characterization of liquid, supercooled liquid and glassy states [92,129,130]. Thermodynamical stability of the different phases is expressed by a Gibbs free energy diagram.


Figure 2.6. Some advanced polymers designed for lithium polymer batteries in comparison with a classical polyether type segment(1). The Gibbs free energy G is the sum of enthalpy H and the multiple of the entropy S and the absolute temperature T as,

G = H − ST (3.1)

Then enthalpy H is expressed by the internal energy U plus the multiple of volume V and the pressure p as, H = U + pV (3.2)

Usually, the effect of the factor pV is small in solid and liquid states in ambient condition, the enthalpy H is almost equal to the internal energy U of the material. From statistical point of view, the internal energy U of the material is determined by the structure and interaction potentials among constituent atoms, which can be evaluated by thermodynamic calculations such as Born-Harber cycles [131].

The internal energy Ucry of the crystalline solid is usually lower than that of liquid, since the more densely packed structure of the crystal makes their


interaction potential lower than in liquid1. Thus, at low temperature where the factor TS is small, the crystal is more stable than liquid.

The entropy S is a measure of the randomness of the material, which can be calculated from the temperature dependence of specific heat. It is proportional to the logarithm of the accessible surface area in the Γ phase space under the constant energy condition. The large atomic motion and undistinguishability of the same species ("communal entropy") in liquid makes the entropy Sliq to be larger than the Scry of the crystalline solid at a given temperature. Because of this the decrease in the Gibbs free energy G by heating is larger in liquid sate than in crystalline state. Thus the two curves of Gcry and Gliq cross at a point Tmp which is a melting temperature of the crystal or the freezing point of the liquid. Now, the crystalline state is found more stable than the liquid below Tmp, which means the liquid should make a phase transition to the corresponding crystal at Tmp in decreasing temperature. This is completely true in thermodynamic equilibrium theory. Actually, however the liquid state often survives below Tmp without freezing, which is called "supercooling". The liquid below Tmp is called "supercooled liquid", which is "quasi-stable" in very quiet condition without stirring, mechanical shock or dust for crystal seed. The quasi-stable" state is defined as the state stable against a small perturbation but unstable to a large perturbation, which is depicted as a local minimum in free energy potential surface as in fig. 3-2. This quasi-stable supercooled liquid state may be destroyed when a large perturbation such as the drop of a crystal seed, and will transfer to the crystalline solid; this corresponds to a crystallization or a vitrification. If the temperature is lowered much below Tmp, the atomic motion becomes slower and slower due to the decrease in kinetic energy of the constituting atoms; resultantly the diffusion coefficients of constituent atoms decrease and the viscosity increases up to ca. 1012 poise, which is apparently a solid without flowing. This is a transition from a supercooled liquid to a glassy state (glass transition). 3.2 Definition of "glass" and "glass transition" [92,130]

The term "glass" has long been known as "an inorganic product of fusion which has cooled to a rigid condition without crystallization." However, because of the recent discoveries of some new "glassy" materials, a more general definition of the term "glass" and "glass transition" has been proposed by a committee of the U.S. National Research Council as follows: "Glass is an X-ray amorphous material which exhibits the glass transition, this being

1This is not applicable in case of water, SiO2 and Bi [92]


Figure 3.1. Temperature dependence of the thermodynamic quantities of liquid (liq), crystal (c) and glass (g). Tm and Tg is the melting temperature and the glass transition temperature respectively. T0 and TK are the singular points where the volume or entropy of liquid is equal to those of the crystal.


liq

Tm

scl

Tg

g

crystal Qi

G

Figure 3.2. Schematic free energy diagram of liquid (liq) supercooled liquid (scl), crystal and glass (g). defined as that phenomenon in which a solid amorphous phase exhibits with changing temperature a more or less sudden change in the derivative thermodynamic properties, such as heat capacity and expansion coefficient, from crystal-like to liquid-like values." [132]. The latter definition of the term "glass" seems too phenomenological to be accepted without knowing its background. It will be more meaningful in the present purpose to accept the following definition by Seki [133]: "glass is a solid which has only short-range order, and shows long-time relaxation phenomena.". This definition is based on the "non-equilibrium nature" of glasses. Thus, the glass transition is regarded as a relaxation phenomenon from the non-equilibrium glassy state to the quasi-equilibrium supercooled liquid state, or vise versa.2 From this point of view, the glass is similar to "a mountain which is stable for human eyes but is flowing for the immortals". This concept is expressed by the "Deborah number (DN)" [92] which is defined as the ratio of relaxation time τs and observation time tobs,

DN =τ s

tobs

(3.3)

2The word 'equilibrium state' is used in the meaning of "the stable state to the small perturbation", and applied for the quasi stable state (supercooled liquid) as well as for the most stable state(crystalline phase).


When DN <1, the system is considered to be in the liquid state, and DN >1 it can be regarded to be in the glassy state. The glass transition will occur at DN ~ 1.

Although the argument based on the Deborah number is essentially adequate to understand the glass transition as a relaxation phenomenon, the concept of the "observation time" is rather vague; moreover, it fails to explain the well known experimental evidence that the glass transition temperature depends on the cooling and/or heating rate.

From the experimental point of view, Tg can be determined more easily in the heating process by means of thermal analyses, volume measurements, kinetic properties, etc. The value of Tg determined in this way usually depends not only on the heating rate of the measurement but also the history of the glass, such as annealing process, cooling rate of glass formation etc. Such values of Tg should be called a "recovery temperature" from a nonequilibrium state to the equilibrium state, which is the accumulation of all the history (memory) of the glass since it was made.

On the other hand, from the theoretical point of view, it will be more reasonable to define the glass transition temperature as the temperature in the vicinity of which the equilibrium supercooled liquid branched into the non-equilibrium glassy state (Ergodic to non-Ergodic transition). The bifurcation temperature can be simply estimated using Lillie number,:

|q | dT

dτSeq

T = Tb

= const. ~ 1

(3.4)

Here, τseq is the equilibrium value of the structural relaxation time and q is

the cooling rate q=-dT/dt. The equation 3-4 can be explained as follows: the temperature decrease ∆T=q∆t in the supercooled liquid state creates a fluctuation δ out of the equilibrium state and is followed by an increase in τs, given by

∆τS = dT dτS ∆T =

dT dτS

q ∆t

(3.5);

the condition ∆τs > ∆t would lead to "fluctuation freezing" or bifurcation to the glassy state. When ∆τs<∆t, the structural relaxation is fast enough for the liquid to relax into (quasi)stable state. But, when the condition ∆τs > ∆t, is


reached, the system can not relax to the (quasi) equilibrium point within the ∆t. This means the fluctuation generated by the sudden temperature change of ∆T becomes accumulating and finally frozen in below this temperature corresponding to Tg. 3.3 Non-equilibrium thermodynamics and order parameter theory[134-136]

The equilibrium state of any material is characterized by its thermodynamic independent variables such as T,P,V,S etc.. For example, in the usual laboratory condition, where the independent variables T and P are maintained to be constant, the equilibrium values of S and V of a one-component material are determined as dependent variables minimizing the Gibbs free energy G(T,P). If the system consists of n components, the variables S,V,x1,x2,...xn are all determined by the condition G(T,P,µ1,µ2,...µn) being minimum, where xi is the mole fraction and µi is the chemical potential of the species i. However, even if the variables T,P,µ1,,µ2,...,µn were known, other variables G,S,V,x1,x2,...xn would no longer be determined uniquely in the non-equilibrium glassy state. A possible extension of the equilibrium thermodynamics to the non-equilibrium state is based on the order-parameter theory [137] which introduces one or more quasi-thermodynamic variables characterizing the non-equilibrium state. For example, the Gibbs free energy is not only a function of T,P,µ1,,µ2,...,µn, but also depends on one or more order parameters Zi. which vanish in the equilibrium state as G(T,P,µ1,,µ2,...,µn;Z1,Z2,...Zm) ----> G(T,P,µ1,,µ2,...,µn;0,0,...0)eq (3.6)

The simplest way of this approach is the introduction of excess quantities such as excess free-volume υfex [138] or excess configurational entropy Sex [139] which vanishes in the equilibrium state. They correspond to density and entropy fluctuations. It should be noted that the composition xi may also deviate from equilibrium to generate a concentration fluctuations to manifest themselves as a phase separation or locally ordered cluster formation in supercooled liquid and glasses.

Since the glassy state is no longer the equilibrium state, the physical quantities of a glass depend on time due to the structural relaxation, even if the external variables T,P,µ1,,µ2,...,µn are all fixed. Their time dependence is characterized by the relaxation of order parameters Z1,Z2,...Zm, which is assumed to obey the usual relaxation equation given by


dZi

dt= −

Zi

τ i

(3.7),

where τi is a relaxation time on the variable Zi. It is to be noted that τi is not only the function of T and P but may depend on Zi itself as well as other parameters Zj; as the consequence, the equation becomes intrinsically nonlinear. In most cases the τi is roughly proportional to the viscosity of the supercooled liquid. The structural relaxation of supecooled liquid and the appearance of glass transition can be picturized by a schematic potential diagram as shown in fig. 3-2, where a thermodynamically coarse grained Gibbs free energy is plotted as a function of a hypothetical configurational coordinate Qi. Here we assume there are two minima along this coordinate corresponding to the crystalline state (left) and the the liquid, supercooled liquid states (right). At high temperature the lowest energy point of G locates at right minimum corresponding to equilibrium liquid state. At melting point Tm, the two minima have the same G values and T<Tm the lowest point changes to the left crystal state. Even in the T<Tm, the liquid state can be remained in a quasi-stable equilibrium state, which is a local minimum in the potential diagram as in the case of scl in the picture. Now, we should consider the structural relaxation in detail. If the temperature is suddenly dropped from a temperature to another such as from Tm to Tscl in figure 2, the position of the local minimum in Qi space will usually shift owing to the thermal expansion or specific heat difference. Then the representative point (filled circle) at T = Tm moves to the open circle in the potential at T = Tscl which is not at the minimum point and a fluctuation is generated; this is similar to the Franck-Condon principle in optical transition. Then the temperature is kept within the time interval of ∆t at Tscl. If the open circle relaxes toward the minimum point within the time of ∆t, the system can be regarded as in a quasi-equilibrium state (supercooled liquid state). Thermodynamic restoring force is proportional to the minus gradient of G in Qi space. When the temperature is lowered further the potential gradient around the minimum becomes smaller and smaller, which results in the slower relaxation of the system to the minimum. Finally, below Tg, the deviation from the equilibrium can not relax within the time of ∆t; and the deviation remains to be frozen in the glassy state. This deviation from quasi-equilibrium state Qi-Qieq is corresponding to the order parameter Zi in the non-equilibrium thermodynamic picture. Relaxation of the Zi is experimentally observed as the time dependence of the volume V, enthalpy H by means of thermodynamic or optical measurements [140]. Also, it is detected indirectly by the time dependence of


some physical properties such as refractive index, magnetic susceptibility and ionic conductivity. Most widely used order parameter for describing the nonlinear relaxation near the glass transition point is the "fictive temperature" Tf introduced by Tool and modified by Narayanaswamy [136]." Another one is to use excess free-volume by Cohen and Turnbul [141-143] and Kovacs et al. [137,138]. Some numerical ways to deal with the nonlinear behavior of the structural relaxations have been developed and used for the analysis of volume change, enthalpy change and also for the variation of ionic conductivity of silver iodomolybdate glasses [121]. It should be mentioned here that the thermodynamic potential argument on the glassy state described above is relating to the recent "energy landscape" concept on the supercooled liquid and glasses [144], which is very useful to understand the more detailed aspects of supercooled liquid and glass transition phenomena, such as strong-fragile, coupling-decoupling, dynamic heterogeneity, and percoaltion problems etc. described latter. In the energy landscape picture, the potential energy surface in 3N+1 dimensional space is mapped to a low-dimensional (usually one or two dimensional) surface. Although it is similar to the Gibbs free energy map in figure 3-2, it considers only the potential energy and the level of coarse graining is much precise than those in thermodynamics picture. A thermodynamic state (even in nonequilibrium) is expressed by a point in Gibbs free energy map, but is a domain spreading in equal total energy in energy landscape picture. 4. Ionic transport theory for liquid 4.1 Diffusion coefficient and ionic conductivity Ionic transport in liquid and solid is a consequence of the random migration of ions stimulated by thermal energy as long as the applied electric field is small enough. The migration of a tugged ion is dealt with a random walk theory at sufficiently long time. From the irreversible thermodynamic theory [135,145], the flux jα of an ionic species α, is proportional to the conjugate force Xα which is expressed as the gradient of the electrochemical potential µ

~

α divided by temperature T as,

j LT

αα α

µ−∇⎛ ⎞= ⎜ ⎟⎝ ⎠

(4.1)

where Lα is called phenomenological coefficient. If one consider the concentration and electrical potential gradient,


µ~

α = µα0 + kT lncα (x) + zαeφ(x) (4.2)

then, jα = −Dα∇cα + cαuαzαe∇φ(x) (4.3),

where, Dα = kLα /cα is the diffusion coefficient and uα =Lα /T is the mobility of species α. In case of no electric field, the ion flux is generated by concentration gradient and by using the continuity equation one can derive the following Fick's 2nd law or diffusion equation; ∂cα

∂t= Dα∇cα

(4.4).

By applying the Ohm's law to eq.(4-3), one can easily derive the following equation for ionic conductivity σ, σ = (zαe)2cαuα

α∑ (4.5),

and, the substitution of the Boltzmann distribution cα(φ)=cα0exp(-zαeφ/kT) to eq.(4-3) gives the following Einstein relation between the mobility uα and the diffusion coefficient Dα;

uα =Dα

kT (4.6)3

From eqs.(4-5) and (4-6) one can derive the so called Nernst-Einstein relation as, σ =

(zαe)2cαDα

kTα∑ (4.7)

3Note that the mobility uα is defined in eq.(4-6) in physics, but is defined in electro chemistry field as,

z eDukT

α αα =

(4.6’)

σ = zαecαuαα∑

(4.5’)


This equation is generally applicable to the ionic diffusion in both liquid and solid including glassy state. Diffusion coefficient Dα in equation (4-4) is related to the random walk of ions. If we assume a tagged ion i of the species α is at rαi =0 at time t =0 and starts to walk randomly to a position rαi(t) at time t, then the difference rαi(t)- rαi(0) is expressed at long t limit as,

1N

rαi(t) − rαi(0) 2

i=1

N

∑ ≅ 6Dα t (4.8)

and

D =t →∞lim

16t

1N

rαi(t) − rαi(0) 2

i=1

N

∑ ≈t →∞lim

<| ∆rαi(t) |2>6t

(4.9)

Instead of expressing the D by position r as in eq. (4-9) one can derive the

diffusion coefficient in terms of velocity of the ion υi(t). A convenient way is to use the velocity auto correlation function <υi(0)υi(t)>, where the bracket means a system average. For a single tagged ion, frequency dependent diffusion coefficient D[ω] and whose long time limit of a self-diffusion coefficient D can be calculated from a fluctuation-dissipation theorem as,

Dα[ω] =13

vαi(t)vαi(0) eiωt0

∞∫ dt ω= 0⎯ → ⎯ Dα =13

vαi(t)vαi(0)0

∞∫ dt

(4.10)

On the other hand, the electrical current due to the motion of the ion is,

J(t) =1V

(zαe)υαi(t)α ,i∑ = (zαe)cαυα (t)

α∑

(4.11),

where υα is the average drift velocity of species α. Then, the frequency dependent conductivity σ[ω] and the dc conductivity σDC can be also calculated by the following Kubo formula [146,147]; σ[ω] =

1kT

J(t)J(0) eiωt0

∞∫ dt ω= 0⎯ → ⎯ σDC =1

kTJ(0)J(t)

0

∞∫ dt (4.12)

When substituting eq. 4.11 to 4-12 then,


σ DC =1

kT(zie)2

i∑ υ i(0)υ i(t)0

∞∫ dt +1

kTziz je

2

i≠ j∑ υ i(0)υ j (t)0

∞∫ dt (4.13).

One can see the first term of (4-13) is equal to the Nernst-Einstein formula

(4-7), however the second term containing cross-correlation between ion i and j is neglected. So, it should be noted that the N-E formula is an approximation (neglect of cross correlation) and if the motion of an ion j at time t is correlated with the other ion j at time t = 0, the N-E equation is not always fulfilled.

It should be noted that the shear viscosity j, which dominates the structural relaxation time in supercooled liquid state and responsible for glass transition is also defined in the fluctuation-dissipation theorem as,

η =k →0lim

1kTV

ˆ σ yx (k,t) ˆ σ yx0

∞∫ ( 0) dt− ⟩k, (4.14)

where, σyx is a microscopic stress tensor [148,147].

Equations (4-12) and (4-14) are quite general and rigorous formula from the statistical dynamics for irreversible processes, at least for small fluctuations about equilibrium. Thus, one can calculate the viscosity and ionic conductivity of an ionic liquid without any assumption or models, provided that the positions, velocities and interaction potentials are known as a function of time; this is only available in computer simulations. 4.2 Models of diffusion in random media Brownian motion model

When a small particle is immersed in a liquid, the particle will be hit by a large number of surrounding molecules and will move around randomly in the liquid; see Fig.4-1 (a). A deep insight of A. Einstein predicted such a random motion in 1905, which had been already found by Robert Brown in 1827. The motion of the Brownian particle was described by P.Langevin by the following equation, which is now called the Langevin equation;

1 ( )f tm

υ γυ+ = (4.15)

where υ is the velocity of the Brownian particle, γ is a friction coefficient. f (t) is a random force acting on the particle characterized by a Gaussian white noise as,

f (t) f (t') = 2Dδ(t − t ') (4.16)


Then, the diffusion coefficient is calculated from eq.(4-10) as, D =

kTmγ

=kT

6πaη (4.17)

where, the Stokes law of mγ = 6πaη is used representing the hydrodynamic friction force acting on the particle whose diameter is a. The eq.(4-17) is connecting the diffusion coefficient of a particle to the viscosity of the surrounding medium. More sophisticated way to evaluate γ is given by Zwanzg-Mori projection operator formalism as [146, 147],

γ =1

kTf (t) f (t')

0

∞∫ dt

(4.18),

Substituting (4-15) to (4-17) gives (4-16). The Langevin equation approach is also used for the conformational relaxation of polymer chains by Rouse model [150].

At very high frequency in THz or infrared region, the ionic motion is not a simple Brownian motion even in a liquid but is similar to a harmonic oscillator in sold since the ion feels the local potential made by the surrounding counter ions. Thus the dynamics of ions in liquid is a combination of the short-time local vibrational mode and the long-time diffusive mode. Such a short time local vibrational motion is called Cage effect and is observed by inelastic neutron scattering in molten salt even in the high temperature liquid state; see Fig. 4-1(b).

(a) Brownian motion (c) Jump motion(b) Vibrational-diffusion

Figure 4.1. Schematic picture of the motion of an ion in liquid and glassy states.

The cage effect is a consequence of the non-random nature of the "random force" in eq.(4-15), which can be introduced into Generalized Langevin equation (GLE) formalism by extending the friction term by a memory function form as,


1( ) ( ) ( )t

M t t' t ' dt' f tm

υ γυ υ−∞+ + ∫ − = (4.19)

Here, M(t) is a memory function which can be evaluated by the projection

operator technique to extract the correlating part of the random force to the observable, in this case the velocity [146,149]. Multiply <v(t)| to eq.(4-19) to calculate velocity auto-correlation function and make a Fourier-Laprace transform, then one can obtain the following form for diffusion coefficient by using eq.(4-10) [146,149,151,120], D[ω] =

1−iω + γ + ˜ M [ω]

(4.20)

where ˜ M [ω] is the Fourier-Laprace transform of M(t). This formula is quite generally applicable to calculate frequency dependence of conductivity and other dynamical responses at short time region near the local oscillation frequencies from GHz to THz, such as QENS, QELS and NMR relaxation, optical Kerr effect etc. Mode-coupling theory of glass transition

Brownian motion model shown above is focusing on the atomic scale single ionic motion in liquid. On the other hand, one can look at another limit of long range hydrodynamic modes such as, density ρ(k, t), concentration ci(k, t), current J(k, t) etc., where k is the wave number [147,152]. All these quantities are fluctuating about their mean values even in an equilibrium liquid state. The relaxation and the time correlation of these quantities will also slow down when approaching to the glass transition point from liquid side. Bengtzelius, Gotze and Sjogren [153,154] used Mode-Coupling approximation for the generalized Langevin equation for density correlation function φ(t) as,

(4.21). As for the memory function M(t), Leutheusser [155] used the following approximation; M(t) = γδ(t) + λΩ0

2φ 2(t) (4.22), and, Bengtzelius et al. [156] as,


M(t) =1

(2π )3 V (k,q)∫ φ(k, t)φ(k − q, t)dq (4.23),

and vertex function V(k,q) was calculated from static structure factor. Gotze and Sjogren demonstrated an Ergodic to Non-Ergodic transition in the solution of eq. (4-21) at large coupling case, although the transition temperature Tb from MCT is much higher than the experimentally determined glass transition temperature Tg [69,157].

The MCT is also applied for the current fluctuations in molten salts [158,159], which can be used to calculate the ac conductivity in liquid and glass transition region. From such a calculation, Bosse and Kaneko demonstrated the decoupling of the smaller ions from the larger ions in glass transition [160]. The application of MCT to the glass transition of polymers are discussed in the references in comparison with recent computer simulations [161]. 5. Ionic transport in glass transition region 5.1 VTF law and strong vs. fragile

Temperature dependence of transport quantities such as diffusion coefficient, ionic conductivity and viscosity are often expressed by Arrhenius type law at high temperature region. However, if the temperature is lowered below Tmp and approaching to Tg, then the transport quantities exhibit non-Arrhenius more drastic temperature dependence, which is often expressed by so called, Vogel-Tammann-Fulcher (VTF) or Williams-Landel-Ferry (WLF) equation4. For the viscosity η , it is expressed as,

η = η0 exp B

T − T0

⎛

⎝ ⎜

⎞

⎠ ⎟ (5.1),

where η0, B, and T0 are empirical parameters. In the limit of T0 = 0 K, the eq.(5-1) reduces to a normal Arrhenius law, which is called a strong liquid. On the other hand, if T0 is finite value the η depends non-linearly in Arrhenius plot is called fragile liquid [92-94,162]; the corresponding ionic conductivity is plot on Fig. 5-1 from eq. 4-17 and 4-7.

The equation (5-1) has been explained by a free-volume theory [141-143,162], or a configurational entropy theory [139].

4Other forms for the temperature dependences in supercooled liquid state are discussed in ref. [55]


Figure 5.1. Effect of fragility on ionic conductivity of supercooled liquid.

The original free-volume theory [141,142] is based on the idea that molecular transport occurs only when the voids having a volume greater than some critical volume υ* are formed by the redistribution of the free volume υf in the liquid. Then, the diffusion coefficient is expressed by the following form:

D = D0 exp −γυ *

υ f

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ (5.2)

Here, the temperature dependence of the average free volume υf can

approximately be written as υ f = υ0α f T − T0( ) (5.3)

where αf is the thermal expansion coefficient of the free volume in the liquid state, T0 is the temperature at which the free volume will vanish, and υ0 is the volume of the liquid at T0. Substituting equation (5-3) into (5-2) gives the empirical VTF equation (5-1). On the other hand, in the configurational entropy theory of Adam and Gibbs [139] the transition state was supposed to be formed by correlational fluctuation of atomic (or ionic) configuration; the configurational entropy SC is


accompanied with the formation of the transition state, and the transition probability per unit time can be written as

νi = ν

i0 exp TS

C

b

(5.4).

If the SC is approximated as SC~ ∆Cp/T +const., then the same temperature dependence as in equation (5-1) can also be derived.

No energy barrier effect is taken into account in the free-volume theory, but some modification is possible like as the hybrid theory by Macedo and Litobitz [163]. They considered a minimum energy barrier Ev* for an atom migrating from an equilibrium position to a neighboring position by following the rate theory of Eyring, and also took into account the probability of finding a vacancy at the destination using the free-volume theory. Instead of equation (5-2) in the free-volume theory, the resultant expression for the hybrid theory is expressed simply as

D = D0 exp −γυ*

υ f

+Ev

kT

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

(5.5)

Similar formulations have been used for ionic conductivity of polymers

[164,165] and inorganic glasses as a unified model by J.L.Souquet [166,167], which takes into account the formation energy of defect in a polymer or glass by an enthalpy term similar to Ev in eq.(5-5) to explain temperature and composition dependences of ionic conductivity. Decoupling index

The electrical conductivity σ of an ionic liquid is inversely proportional to the shear viscosity η of the liquid, which can be derived from eqs. (4-7) and (4-17) as,

σ = (ze)2 N

6πaη (5.6)

Thus, the first guiding principle for developing better ionic conductor is to

seek for the liquid of low viscosity, which can be realized by lowering the glass transition temperature Tg and/or increase the fragility. This strategy is actually useful for searching good solvents or polymers for electrolytes.


However, the eq (5-6) was found to break more than 10 decades in superionic conductor glasses such as AgI-oxide systems etc., where the theoretical prediction of eq.(5-6) gives 10-14 S/cm at Tg but actually have the ionic conductivity 10-3 S/cm [29,120,121,168-171]. The deviation from equation (5-6) is discussed by using another key concept of "coupling vs. decoupling", which was proposed by C.T.Moynihan at first [172] and widely applied by A.C.Angell [168,169,173,174]. The "decoupling index", R τ= τs/τσ, (5.7), which is the ratio of the electrical relaxation time τσ=ε∞/σ and the mechanical relaxation time τs = η/G∞ is found a good measure of the ion dynamics in supercooled liquids and glasses. By using this index, C.A.Angell exhibited the Rτ~ 1012 for superionic conductor glasses such as silver oxyhalide glass and even a sodium silicate glasses, and named them as decoupling systems[168,169]. On the other hand, some glass forming systems such as KNO3-Ca(NO3)2, LiCl-6H2O and polymer electrolytes as PEO-LiClO4 has only below Rτ~ 104 at Tg; they are called coupling systems (note that recent results suggest that the lithium is coupled but proton is decoupled in LiCl-6H2O [175-177]).

Figure 5.2. Decoupling of the ionic conductivity from the viscosity mode in superionic conductor glasses, in comparison with the coupling system expected from eq.5-6.


Difference of polymer electrolytes and superionic conductor glasses One important conclusion of these fundamental considerations is that the

ionic transport mechanism of inorganic glass is completely different from that of polymer electrolyte; the former is a decoupling system and the latter is a coupling one [173,174,178-180]. It means, the ionic diffusion in the inorganic glass is decoupled from the oxide glass framework, on the other hand, the diffusion of ions in polymer electrolyte is strongly coupled with the motion of polymer chains. From this difference, the inorganic glass can be used as a solid electrolyte below Tg, despite the "solid" polymer electrolyte is useful only above Tg, in rubber state where the macroscopic rigidity is maintained by network entanglement. From the thermodynamic point of view discussed in §3, the "solid" polymer electrolytes are used in a supercooled liquid state, which is stabilized by network entanglement to form a rubber state.

It should be noted here that even a coupling system such as K(NO3)-Ca(NO3)2, a deviation of viscosity from VTF to Arrhenius like behavior is seen in supercooled liquid state above Tg, which is discussed to be due to a possible hopping motion of ions. Excess-free-volume (EFV) theory for coupling-decoupling

This phenomenological concept of coupling-decoupling was explained by a statistical model using an excess-free-volume (EFV) theory [29,120,121,170], where the free-volume theory was extended to the non-equilibrium glassy state by introducing the excess-free-volume υex which is defined as υex = υf - υeq (5.8) where υf is the total free-volume and υeq the equilibrium free-volume. Then the diffusion coefficient Dα of species α is then given by

Dα = Dα0 exp −υα

*

υ eq + υ ex

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ (5.9).

The excess free volume υex is an order parameter characterizing the deviation from the quasi-equilibrium liquid state and was assumed to follow the structural relaxation equation as,

dtdδ

= -

τS

δ - ∆αq

(5.10)


where δ = υex/υeq and ∆α is the difference in thermal expansion coefficients, τs is the structural relaxation time, and q is the cooling-heating rate. This is a special case of eq. 3-7 modified to include fluctuation generation term by quenching. By solving numerically the eq. 5-11, the conductivity was calculated as a function of temperature and thermal histories to give a semi-quantitative explanation of the decoupling nature of the superionic conductor glasses [120,121]. The point of this model is to consider the critical volume υα

* for each species, which is almost equal to the ionic volume. Then the shearing of the total free-volume by different constituting ions gives rise to the multiple glass transitions; first Tg for the largest ions, second Tg for the smaller ions.

The EFV approach of ionic transport in glass is quite simple but useful to explain various dynamic properties near the glass transition temperature, although the non-linear character of the structural relaxation equation forced us to use numerical calculations. The EFV theory assumes that the ionic transport mechanism is essentially a free-volume distribution even in the glassy state below Tg; in a sense the EFV model is regarded as a two-liquid model which assumes liquid like motion of mobile ions in the glass matrix consists of immobile ions. This is too much simplified idea not to consider the energy term for ion migration and Souquet et al. extended the theory to include energy term as was discussed above [166,167].

However, the excess free volume concept may be useful even in the glassy state especially for superionic conductor glasses, which was beautifully demonstrated by Swenson and Borjesson [181]. They observed a good correlation between the volume expansion of the network and the ionic conductivity of the glass. The correlation is best fitted by

σ/σ0 ~ [(Vd-Vm)/Vm]3 (5.11), where Vd and Vm are the volumes of the doped and undoped glasses respectively. Thus, the Vd-Vm is an excess volume introduced by the dopant salt. Although the reason of the cubic dependence is unknown it is an indication of the importance of the open space in glasses for ionic conduction. Recently, Aniya indicated a correlation between the volume expansion and average electronegativity of the glasses [182].

The EFV theory predicts 2nd glass transition of mobile ions in glassy state, since the remaining excess free volume in the glass will shrink to less than a critical value at a lower temperature to result in the localization of the mobile ions. This expectation was not known at the publication of the original paper, but is actually observed in the precise heat capacity measurements on xAgI-(1-x)AgPO3 superionic conductor glasses in the range of x>0.3


[183,184]. This means the silver ions in superionic conductor glasses can be regarded as a liquid in a restricted space at least in high concentration region. Moreover, the EFV theory predicts non-Arrhenius temperature dependence of conductivity even in the glassy state, which is in good accordance with recent precise measurements of conductivity at very low temperatures [185,186]. A glass transition due to the localization of mobile ion was also observed in crystalline superionic conductor Na-β-alumina at ca. 120 K [187].

It should be noted that some strong counter arguments were proposed after the publish of the original paper [121] based on the reported unusual increase in ionic conductivity under high pressure [188,189]. However, the revised experiments confirmed the normal decrease in conductivity even in the superionic conductor glasses [190-192] that gives the activation volume almost equal to the volume of the mobile ion. 6. Solid like models 6.1 Jump diffusion model

On the contrary to the Brownian motion picture for ionic transport in liquid, jump diffusion models are often used, in which the diffusion of ion is viewed as a succession of a jump process from one position to another separated from the original position by a distance d.; see Fig. 4-1(c). In the jump model, the probability Pi(t) of finding an ion on site i at time t is expressed by the following Master equation [193-196];

dPi(t)

dt= Wi, jPj (t) −W j,iPi(t)

j∑ (6.1)

where Wi,j is a transition rate of ion from site j to i.

From eq. 4-10, the frequency dependent diffusion coefficient can also be written as [197-199], D[ω] = −

16

ω2 [r(t) − r(0)]2 eiωt0

∞∫ dt (6.2)

The square displacement is calculated by,

[r(t) − r(0)]2 = (s − s0)2 P(s,t | s0,0)p(s0)s,s0

∑

where, P(s,t|s0,0) is the transition probability from a site s0 at t=0 to the site s at time t equal to Wi,j in discrete lattice. As a simplest case, if one assumes Wi,j


=1/τ for adjacent sites and Wi,j =0 for others in three dimensional cubic lattice, then one can derive the following form as: D =

16

d 2

τ=

16

νd 2 (6.3).

where, d is the jump distance, τ is the residence time and ν = τ -1 is the jump frequency. The master equation (6-1) with (6-3) is equivalent to the diffusion equation (4-4) at long time limit. It should be noted that a random Marcofian process is considered in the jump model described by eq.(6-1) but no vibrational motion or cage effect is taken into account. Thus, the jump model predicts high frequency plateau in ac conductivity without resonance peak corresponding to the short-time local vibrational motion of mobile ions. This simple jump model with constant jump distance and constant jump rate is used very often for ionic diffusion in crystalline solids as will be seen later, however it should be modified for liquid and glasses considering the possible distribution of the jump distance d and the residence time τ. Continuous time random walk (CTRW) theory

Rigorous solution of the master equation (6-1) for a random system is itself a challenging theoretical problem [195,200-206]. If one looks at a tagged particle in random potential field, it is a good approximation to assume that the ion is waiting in a site for a certain period ∆t and jump to another site separated from the original position by ∆r. Then the ∆t and ∆r are considered as random variables and introduce a waiting time distribution function ψ(∆t); this model is called the continuous time random walk (CTRW) model. Then the frequency dependent diffusion coefficient is expressed as,

D[ω] = −

ω 2

6σ rms

2 iωψ(iω)1−ψ(iω)

(6.4)

where ψ(iω) is the Fourier-Laprace transform of ψ(∆t). Stating from eq.(6-4) and introducing an appropriate distribution function for ψ(∆t), one can derive the frequency dependent conductivity and also the glass transition phenomena.

CTRW model has been applied to the diffusion of ions and electrons in random medium [196,200,201,207-210]. Dyre obtained a simple form of the


frequency dependent conductivity from CTRW model using a percolation path approximation as,

σ[ω] = σ (0) iωτln(1+ iωτ )

(6.5),

which is in good agreement with experiments in spite of its simple form [76, 196, 209].

Jump diffusion model has been also used in different form for the glass transition phenomena [211]. Odagaki and Hiwatari proposed a trapping diffusion model for glass transition phenomena based on CTRW theory, which predicts Gaussian to non-Gaussian transition accompanied by the glass transition [212,213]. Jump diffusion model is also relating to the percolation theory discussed separately in §7. 6.2 Application of crystal defect theory

In §4 and §6-1, we have seen how the ionic diffusion in random medium can be described by modern stochastic dynamics theories. On the other hand, the ionic transport in inorganic glasses have been often described along with the classical defect transport theory for ionic crystals [5,33,92,93,132]. In this approach, the framework structure of the glass is viewed to be stable in the time scale of ionic diffusion, although this assumption is challenged and modified in advanced discussion including mobile ion interaction [79], relaxation of surrounding structure [82,214] and different ways by Nitzan and Ratner [215] using a dynamic percolation theory.

Since the crystal defect theory has been established very precisely [216,217], one can obtain a more detailed information on the effect of the local structure and chemical bond on the ion dynamics, although we should be careful about the fundamental question and the limit of this approach to random systems.

First, we shall start from the conventional defect diffusion in ionic crystals. Figure 6-1 is a model of the elementary process of ion diffusion in ionic crystals like NaCl or AgCl etc. [218].

In the conventional ionic crystals, only a small number of defects, vacancies or interstitial ions are mobile almost individually, but sometimes correlated way as in interstitialcy mechanism in fig. 6-1.

In the defect theory of crystal the number of mobile ions is evaluated by the mass action law for defects in the crystal. As an example, the number of mobile interstitial ions for Frenkel type defects is expressed by

n(T) = NN ' exp −GF

2kT⎛ ⎝ ⎜

⎞ ⎠ ⎟ (6.6)


vacancy interstitial ion

interstitial mechanism

interstitialcy mechanism

vacancy mechanism

+ + +

+

+

+

+++

- - -

---

- - -

Figure 6.1. Schematic models of ionic diffusion mechanism in ionic crystal. where N and N' is the numbers of stable and interstitial sites, and GF is the Gibbs free energy for defect formation. If one assume Arrhenius type thermal excitation or Eyring's rate theory for the elemental process of ionic diffusion as shown in Fig.6-2, the jump rate ν of an ion from one site to the neighbor is expressed as,

ν(T) = ν 0 exp −Gm

kT⎛ ⎝ ⎜

⎞ ⎠ ⎟ (6.7),

where Gm is the activation free energy for migration and ν0 is an attempt frequency.

a

f

f0

V

Figure 6.2. Potential barrier for ion migration from one site to the other.

Then, substituting equation (6-7) to (6-3) and using (4-7) we can derive,

σ(T) = T−1σ 0 exp −(Gm + GF /2)kT

⎛ ⎝ ⎜

⎞ ⎠ ⎟

(6.8)


Actually, the temperature dependence of the ionic conductivity of alkali oxide glasses is well expressed by the following Arrhenius type equation; σ(T) = T mσ 0 exp −∆E

kT⎛ ⎝ ⎜

⎞ ⎠ ⎟ (m=0 or -1) (6.9)

This equation is called Rasch and Hinrichsen relationship in the glass

science field [92,93]. Replacing ∆E by Gm+GF/2 and choosing m = -1 in eq.(6-9), then it reduces to eq.(6-8). Separation of number and mobility

Although this procedure is well defined in crystal defect theories, it is not so evident in glass and polymers which has a random structure. The first question is, how we can evaluate the number of the mobile ions n(T) in eq.(6-6). Does the number depend on the temperature? time-scale of observation?

A number of tracer diffusion experiments were reported [219-221] to evaluate the number of mobile ions n. For this purpose the Haven ratio HR or correlation factor f is used, which is the ratio of the tracer diffusion coefficient Dt and conductivity diffusion coefficient Ds derived from equation (4-7) assuming the ni is equal to the all alkali ions in the glass. f = HR = Dt/Ds (6.10)

According to [219-221] the Haven ration is in the range of 0.5~1.0 in many single and mixed alkali silicate glasses. So, as long as in alkali silicate glasses, the mobile ion number is almost equal to or a half of the total number of alkali ions and is not temperature dependent.

On the other hand, from integral intensity of quasielastic neutron or optical scattering and NMR spectra, one can evaluate the number of "fast" mobile ions in the glass as a function of temperature. From these experiments, it is estimated about 50% or less at room temperature for silver oxyhalide glasses by NMR spin-echo intensity [222,223], and it is strongly temperature dependent for AgI-Ag2MoO4 glass by quasi-elastic light scattering experiments [71].

A significant problem is that the evaluated number of "mobile" ions are not always coincide with the value from other experiments. From the hierarchical structure concept in §2-2, the tracer diffusion coefficient belongs the same category as the d.c. conductivity since both are the dynamical response in macroscopic space and time scale. On the other hand, the dynamical response experiments like QENS, QELS, NMR and FDC are looking at different time scales from 10-12 s to 10-6 s. The jump rate of ions in a


glass has a very broad distribution more than 6 decades [224] and temperature dependent. If one uses very high frequency response technique, it will detect only a small number of "excellently mobile" ions. Defect model by Charles-Glarum-Elliott

Interacting mobile ions can be taken into account by a pair defect as a first approximation. The correlational motion of ions are viewed as a collision of two billiard balls here. Let us consider alkali ions in an alkali silicate glass. The alkali ion locates near the non-bridging oxygen (NBO) as shown in Fig. 2-3. If an alkali ion moves from one site near NBO to another site near the other NBO, then the latter NBO is accompanied by two alkali ions. So, this pair of alkali ions can be treated as a "defect" such as in lattice defect theory. The number of the defects can be calculated by assuming the creation energy, and the motion of the "defect" is dealt with conventional defect diffusion theory. Based on this picture the ionic conductivity and dielectric relaxation of the defects can be calculated [225,226]. This model is further extended to the Diffusion Controlled Relaxation (DCR) theory by Elliott in order to explain anomalous behavior of a.c. conductivity and NMR relaxation time T1 [80,81]. Weak electrolyte model

Ravain and Souquet first applied the weak electrolyte theory well known in electrolyte solution field to ionic transport in alkali oxide glasses and confirmed good correlation between the ionic conductivity and the square root of the Na2O activity which is evaluated from the electromotive force (EMF) measurements [227,228]. This idea was also applied to explain the correlation between the optical basisity and conductivity [36] and is now included in the dissociation term in their unified model as eq. (5-5) [166,167,229]. Random potential energy model by Stevels and Taylor

The potential barriers acting on a mobile ion is randomly distributed due to the structural randomness of the glass. Stevelss and Taylor first proposed a random potential energy model, in which the mobile ion is assumed to overcome the many random potential barriers along the conduction path. The d.c. ionic conductivity is dominantly controlled by the maximum height of the potential barriers but lower barrier will affect the higher frequency response such as the a.c. conductivity and ultrasonic absorption [230-232]. This rather intuitive concept is now caused partly by percolation path approximation for jump diffusion in random potentials [76,196,209]. The relation between the Random potential model and the weak-electrolyte model shown above was discussed by Glass and Nassau [16].


Anderson-Stuart model Microscopic origin of the activation energies for ion migration ∆E is in

principle determined from the difference in effective potential acting on the mobile ions at a stable position and a transition state. Anderson and Stuart [233] expressed the potential difference in terms of two different components; i.e. electrostatic part ∆Eb and strain term ∆Es as, ∆E = ∆Eb + ∆Es

=zizoe

2

4πε1

r + r0

−2L

⎛

⎝ ⎜

⎞

⎠ ⎟ + 4πGrD (r − rD )2

(6.11)

where, zi and zo are the valence of mobile ion and oxygen, ε is a dielectric constant, ri and ro is the radius of mobile ion and oxide anion, L is the jump distance. As for the second strain energy term, G is a shear modulus, rD is the doorway radius which the mobile ion push the surrounding ions away at transition state.

The strain energy effect can be estimated from pressure dependence of the conductivity. Oyama et al. investigated the ionic conductivity of silver iodide containing glasses under hydrostatic pressure, and conclude that the strain energy term is dominant in silver ions on the other hand the electrostatic term is larger in lithium ions [190]. The original version of Anderson-Stuart model has been modified in some extent especially on the strain energy term, see ref. [33,234] for details.

Figure 6.3. Schematic picture of Anderson-Stuart model.


Correlational motion of ions The most difficult remaining problem is concerning the correlational

effect among mobile ions during jump diffusion process in glasses. Two types of effects have been known on the many body correlation for ionic diffusion. The first one is the backward correlation of an ion which jumps from one site to another caused by the repulsive force from the another ion occupying the target site [235] or Coulombic effective potential from all the other mobile ions [79,85,236], which manifests itself in high frequency response. The second aspect of the correlation is the simultaneous jump of two or more ions which has been pointed out mainly by computer simulations [237-239] and a few theories [240,241], which is more difficult to detect in experiments but some isotope diffusion work suggests its importance [242].

If one compare the structure of crystal and glass precisely, the structural relaxation in glass is essentially different from those in crystals. As was discussed in §2, the local structure around each atom is fit to be locally most stable at glass transition. So, if one removes an ion from one site and places it to another open place, it will generate a large structural deformation or electrostatic repulsion. Only way to move an ion smoothly in the glass is to move it to another place where another same kind of ion already locates; so the latter ion should also push another same type ion to move ... It means the ionic transport in glass is inevitably correlational, or many body phenomena, which is not so significant in conventional defect diffusion theories dealing with low concentration of defects in three dimension.

Correlational effect among mobile ions is most distinctly observed in so called mixed alkali effect or mixed cation effect [243], which is the counter part of mixed anion effect discussed in §2. Ionic conductivity of a glass containing two or more cations will exhibit "generally" much smaller conductivity value than the single cation glasses as shown in fig. 6.4 [224]. This effect is pronounced only in d.c. conductivity and disappears at high frequency region [244]. The mixed cation effect is now understood as the result of the correlational motion of the same type of cations diffusing in an individual channel. During the glass transition process, the coordination structure around each cation, ex. around Na+ and Li+, is fitted to the size of the each cation to minimize the local energy. Since the coordination environment surrounding an ion relaxes slower than the hopping of mobile ions in glassy state, the each cation seeks to move to another similar position; ie. the Li+ moves in the channel for Li+ and Na+ is in Li+ channel. In order to exchange the channels they need some extra thermal energies for structure relaxation [82,214,237,239]. Thus the ionic transport in mixed cation glass is viewed as a site percolation problem as will be discussed in the following section [245-247].


Figure 6.4. Ionic conductivity of 0.3((1-R)Li2O-RNa2O)-B2O3 glasses; an example of mixed cation effect [224]. 7. Percolation theory

All the models discussed above assume the macroscopic ionic conduction is a simple repetition (random walk sequence) of the elementary jump process in homogeneous three dimensional space. In other word, the model only takes into account the local structure of glasses. On the other hand, there is another picture that the ionic diffusion in glass is not a simple random walk process but is a migration in a specially preferable regions [248-250]. This idea is based on the significant character of glassy state, ie. the glass has a intrinsic "heterogeneous structure" created by the frozen fluctuation of density and concentration during the supercooled liquid state and glass transition process [89, 162, 251].

Ionic transport in a strongly inhomogeneous structure is dealt with a percolation theory [91,252-254]. Percolation problem is regarded as a special case of the master equation 6-1. If one assumes the transition rate Wij connecting the adjacent two sites i and j is random variable of 1 or 0, then the equation 6-1 is reduced into a bond percolation problem; see fig. 7-1 (a), where the solid line corresponding to Wij=1 is the bond through which an ion can move.

One important consequence of the percolation theory is the appearance of the percolation threshold pc. An ion located at site i at time t = 0 can escape to infinite distance if and only if the average probability of the "bonding" is more than the percolation threshold pc. The pc is calculated to be 0.5 for two dimensional square lattice and 0.25 for three dimensional simple cubic lattice


[91]. It is seen in fig.7-1(a), where the 72% of the total bonds are connected, which is larger than the threshold value of 50% for 2d-square lattice, thus an ion at a site can diffuse to long distance.

Another type is known as a site percolation problem, where the transition from site i to j is possible if and only if the site j is not blocked; see fig. 7-1(b), where black circles represent the mobile ions and white ones are the blocks, or vise versa. An ion on a black circle can diffuse to the infinity by hopping on the black circles. In case of the site percolation problem, the percolation threshold pc is 0.59 for two dimensional square lattice, and 0.31 for three dimensional simple cubic lattice.

(a) Bond Percolation (b) Site Percolation Figure 7.1. Bond (a) and Site (b) percolation models in two dimensional square lattices.

Near the percolation threshold, the conductivity obeys the following power-law dependence on the site occupation probability p ;

σ ~ (p − pc )µ

(7.1) where, µ >1 is the conductivity exponent. In view of the percolation theory, the Nernst-Einstein formula eq.(4-5) should be modified using percolation efficiency fI [247] as,

σ = (zαe)2cαuα fIαα∑ (7.2)

The percolation efficiency fI indicates the degree of deviation from the

random walk and is a measure of the effectiveness of ionic motion toward making a long distance diffusion. Even if the ions have high local mobility uα, the macroscopically observed dc conductivity may be much smaller when the percolation efficiency is small. Conversely, below the percolation threshold pc, where the fI =0, even the dc conductivity is not observed the local mobility of


the ions in a limited percolation clusters will respond to the high frequency measurements.

Percolation theory also predict the frequency exponent of ac conductivity near pc as,

σ ~ (iω)n (7.3), where n <1 is a conductivity frequency exponent [91,254,255].

Static percolation theory is clearly demonstrated in organic-inorganic hybrid glasses of AgI and alkalliammonium salts [60] as shown in fig.7-2, where the dc conductivity is almost zero at the volume fraction φ of AgI is less than 35% but is a superionic conductor glass more than 50%. This is because the silver iodide sites are connecting up to infinity at φ>35% here. Also, a power-law type frequency dependence is seen near the threshold composition.

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

10 MHz1MHzDC limit

10GHz

σ /

Scm

-1

φ [AgI]

at 25 °C

Figure 7.2. Ionic conductivity of organic-inorganic hybrid glass of AgI-alkylammoniumiodides. A clear percolation threshold is seen in dc conductivity but continuous in high frequency values.

On the other hand, no clear percolation threshold is observed in inorganic glasses as AgI-oxide and Li oxide glasses [256]; they can be rather viewed as a random mixture of oxide halide and mobile cations [90]. A percolation theory is also applied to the composition dependence of the conductivity in silver chalcogenide glasses, although no threshold is seen [257]. A very clear conductivity jump is seen in some compositions in chalcogenide glasses [39,258], where nano-scale liquid-liquid phase separation is confirmed by FE-


SEM observation, although some confusion in data is reported probably due to the difference in preparation conditions [259].

Percolation concept is also useful for polymer and gel electrolytes; ex. it is used in proton conducting polymer membranes, where the nano-sized "cluster" of water is assumed to be phase separated from perfluorinated polymer chains. The proton transfer is discussed with free volume and percolation concept [260]. In case of the lithium ion transport in polymer electrolytes, the ionic motion is not a static percolation but is coupled with the motion of the polymer chains as was discussed in §2 and 5. In order to deal with this problem, Nitzan and Ratner developed a dynamic bond percolation (DBP) theory [215,261-263], where the transition rate Wij is not a static random variable but depends on slowly varying time. The lithium ion may move only when the local bond between the lithium and the oxide is broken and reformed again with another oxygen. 8. Summary and future prospects

Ionic transport in glass and polymer has been a challenging frontier in statistical mechanics of irreversible processes, where all the aspects of supercooled liquid and glass transition phenomena are typically manifest themselves as unusual behavior of the ionic conductivity. From irreversible thermodynamics point of view, the glass transition is understood as a non-linear fluctuation freezing owing to the rapid change of the external parameter as temperature. The classical free volume approach seems still informative in understanding the essential feature of the glass transition and ionic transport in glass and polymers, although some modifications have been proposed.

Ion dynamics in random media can be analyzed by the generalized Langevin equation formalism in short time region, and master equation formalism of jump diffusions in rather long time region especially by a percolation picture in longer time scale. Hierarchical structure of glass and polymer is an origin of complex dynamics observed by different experimental techniques which have own time and space windows. Correlational effects among mobile ions have been revealed significant by recent computer simulations, however its experimental evidence is remaining as a chalenging theme.

Some important problems were omitted in this review, in which a great advancement on the non-exponential relaxation observed in a frequency dependent conductivity, NMR relaxation time, neutron an light scattering etc.; they are compiled in other reviews [55,73-76,79].

Main topics in this article is focused on the statistical dynamics aspects in relation to the ionic conductor glass and polymer. Interactions between the mobile ion and the surrounding ions are not discussed precisely here, just mentioning on a Coulomb interaction, a strain energy and exclusive volume


effect. However, some recent studies revealed that the ionic transport in liquid, crystal and glass is not a simple process of classical mechanics but is a quantum mechanical problem. Even an elementary jump of an ion is accompanied by time dependent chemical bond rearrangements [58,59,264-267]. This is a new paradigm in solid state ionics field which thanks to the recent advancement of first-principle simulations. This is similar to the advancement in chemical reaction kinetics study after 1980 th., when quantum chemical programs such as Gaussian started to be used by chemists to analyze the molecular structure and chemical reactions. The correlational motion of the mobile ions and surrounding ions may be viewed completely different ways from these new approaches [268,269]. Acknowledgement

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