1 General introduction and context of the research
1.2 Relaxation phenomena in structurally disordered materials
1.2.2 Energy landscape of glass-forming liquids
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crystallization can be induced [24,40].
It is widely accepted that at glass transition, molecular motion ceases virtually (except for thermal vibrations) [25,41,42]. This behavior explains the following several observations among many others [29]:
i) The ability to form glasses is universal and independent to atomic and molecular properties,
ii) The glass transition temperature decreases when the cooling rate is lowered, iii) Volume, enthalpy and entropy are continuous across the glass transition, and no change of the molecular structure is observed at glass transition temperature, iv) The glass expansion coefficient and specific heat are lower than those of the liquid, and
v) Hysteresis effects depend on the nature of the glass transition.
1.2.2 Energy landscape of glass-forming
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minima explored by the liquid increases with decreasing temperature. Such behavior has not been observed above this temperature.
At lower temperatures, they have noticed a sharp transition, which depends on the cooling rate as shown in Fig. 1.6, when the liquid is trapped in the deepest accessible “energy basin”. The configuration space has been partitioned into
“basins”, such that a local minimization of the potential energy maps any point to the same minimum [38]. It is of interest to note that in binary mixtures, the different size of the two components often prevent the system to crystallize, allowing its study in a broad temperature interval from below to well above the glass transition temperature.
The main information that can be extracted from Fig. 1.6 is that at lower (T 0.3) and higher (T 1.0) temperatures, the energies do not change
Figure 1.6. Temperature dependent average energies of the inherent structures, for four different cooling rates of a binary Lennard-Jones mixture AB, where 80% of the particles are of type A and 20% are type B [Sastry et al., 1998 [38]].
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significantly with temperature, while in the range 0.3T 1.0 they increase progressively with the increase in temperature. Here, T denotes the dimensionless scaled temperature Ta/Tg, where Ta refers to the absolute temperature. The term
“inherent structures” in the caption refers to the energy minimum configurations obtained by performing a local potential energy minimization for selected configurations.
Fig. 1.7 shows the individual minimum energies for the related configurations at cooling rate of 8.33105. A careful observation of this figure indicates that at intermediate and higher temperatures, the individual configuration energies cover a broad range. At a temperature above the glass transition but well below the onset of nonexponential relaxation, the barrier heights separating potential energy minima sampled by the liquid increase abruptly. In this case, kinetic energy permits access to most “basins”.
Figure 1.7. Individual minimum energies for the related configurations at cooling rate of 8.33×10-5 [Sastry et al., 1998 [38]].
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Another interesting behavior to be noted in Fig. 1.7 is that as temperature is lowered in the low temperature region, a gradual trend leading to a narrow distribution of configuration energies around the average values is observed.
However, the statistical independence of individual points varies strongly with temperature. At these lower temperatures, the sampling shifts to lower energies and mutual access among “basins” becomes subject to considerable “activation”
[38]. It is therefore clear that the relationship found here between static, topographic features of the energy landscape and glass-forming liquid dynamics leads to a better thermodynamic understanding of the glass transition phenomenon.
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Part I
Ion transport mechanisms in
structurally disordered materials
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Chapter II
Characterization of the a.c.
conductivity in superionic glasses
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Chapter II
Characterization of the a.c. conductivity in superionic glasses
Jonscher [4] has suggested that there is a “universal relaxation law” based on the indisputable common property of all condensed matter: the many-body interactions between their constituent parts. However, three decades after, the origin of this relaxation law still remains elusive even if many studies have been carried out.
In the framework of this law, the overall frequency dependence of conductivity )( is described by [4]
()()An, (2.1) where the pre-exponential factor A and the power law exponent n are experimentally determined material constants. The second term of Eq. (2.1) refers to the a.c. conductivity. Between frequencies ranging from kHz to MHz, for glasses with moderate to high alkali content, it has been found that at (or above) room temperature (RT), n0.6[49]. For various ion conducting glasses, it has been reported that the power law exponent approaches the value of 1.0 at high frequency limit, the temperature being maintained at (or above) RT. For instance,
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Cole and coworkers [50] measured the conductivity of sodium trisilicate glass from Hz up to GHz and found that at RT, n approaches unity at high frequencies. Such studies on different glassy systems abound in the literature. An interesting work is that for Durand and coworkers [51] who observed the same power law exponent behavior for a silver thiogermanate glass at low temperatures, between °C to °C, with frequency up to GHz. Moreover, many researchers found experimentally that n1 behavior can be observed by decreasing the temperature without extending the frequency range beyond tens of kHz [52-56]. By questioning on the origin of n1, Hsieh and Jain [49]
empirically found that the high temperature-high frequency and low temperature-low frequency a.c. conductivity behaviors of the lithium silicate glass 61SiO2·35Li2O·3A12O3·1P2O5 (mol%) can be regarded as the same phenomenon under certain conditions.
The pre-exponential factor A is related to the d.c. conductivity (0), i.e.
the frequency independent part of (), by
), ( .
cn
A (2.2) where c represents the onset frequency of the dispersive behavior, also considered as the hopping rate [19,57-60]. The d.c. conductivity of a random material is the sum of the partial conductivities of the ionic and electronic charge carriers. It is therefore obvious that any change in its value is inherent to a change of carrier concentration and/or carrier mobility.
The concentration of ionic defects can be increased by several ways, for instance by doping or by deviation from stoichiometry. The former method consists to the addition of aliovalent impurities which requires the generation of ionic defects with opposite charge to maintain electrical neutrality. For the latter, the reaction with the gas phase results in a reduction or oxidation of the compound and a formation of excess vacancies or interstitials. However, this process simultaneously produces electronic species, thereby leading to mixed conduction
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[61]. Another clue to increase the concentration of ionic defects in a particular class of materials (structurally disordered materials) is to increase the temperature (i.e. – transition in AgI, for instance). Let us mention that another source of enhanced carrier concentration is related to the formation of space-charge regions in the neighborhood of interfaces.
The most important factor determining the ionic mobility is the height of the potential barrier that the ion must overcome to pass from one well to an adjacent well, as it is well established that ions proceed by jumps when they move from one position to another one close by. In turn, the barrier height depends on factors such as strain energy that needs to be expended for the ion to squeeze through the bottleneck, the polarizability of the lattice, and the electrostatic interactions between the ion and its surroundings [61]. What does the frequency dependence of conductivity tell us about ion motions in disordered materials?
In structurally disordered materials, at low frequency, the mean square displacement behaves linearly with time as [60]
Dt t
r
d.c.
2( ) , (2.3)
where D denotes the diffusion coefficient. This linear time dependence is due to the random diffusion of the ions as they migrate from site to site through the disordered matrix. Such a time dependent mean square displacement is found in the classical random walk model of diffusion and is a feature of uncorrelated motions [60]. For this random walk, the diffusion coefficient is defined as
D 2c
6
1 , (2.4)
where represents the mean hopping distance. By applying the Fourier transformation to Eq. (2.3), we obtain the Nernst-Einstein relationship between the d.c. conductivity and the hopping rate [58,60]
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c BT k Nq
(0) 6
2
2 . (2.5)
Here, the symbols have usual meanings. N is the number density of “mobile ions”.
In general, for all Jonscher-like materials, the variation of the d.c. conductivity with reciprocal temperature for different compositions can be modeled by the following Arrhenius equation
) / exp(
)
( T Ea kBT
, (2.6) where Ea and denote the activation energy and a temperature independent parameter which has been found similar to almost all vitreous materials, respectively [62]. By using Nernst-Einstein relation, the pre-exponential factor
can be expressed as [63]
B m
B k
S k
a
Nq exp 6
0 2 2 0
, (2.7)
where a and 0 represent the average jump distance between the mobile ion sites and the cation vibrational frequency, respectively. Sm is the entropy of migration.
Fig. 2.1 depicts the reciprocal temperature dependence of the d.c. conductivity for a series of ion conducting glasses. We can observe that at lower temperatures, Eq. (2.6) (solid lines) fits reasonably well all the experimental data. However, at higher temperatures, for the optimized Ag conducting glasses, there is a significant departure from Arrhenius plot. This behavior has been successfully explained by Maass and coworkers [66]. There, they studied, by Monte Carlo simulations, the transport of ions in an energetically disordered structure and found that in the presence of both disorder and Coulomb interaction the departure from Arrhenius behavior observed is correctly reproduced. They shown that by decreasing the strength of the energetic disorder, the activation energy can be systematically lowered and the crossover temperature to the non-Arrhenius
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regime becomes smaller. This is not the case for traditional ionic glasses (closed marks in Fig. 2.1) in which both the activation energies and the strength of the energetic disorder are relatively large. Accordingly, the crossover temperature should be high and might not be reached below the glass transition.
In contrast to the behavior observed in Fig. 2.1, the behavior shown in Fig. 2.2 is consistent with the Arrhenius equation (2.6) for both the low and high regimes.
Fig. 2.3 illustrates the temperature dependence of the activation energy Ea for the ion conducting glasses shown in Fig. 2.1. Let’s note that the higher activation energy and poorer conducting glasses have the normal behavior of constant activation energy, where as the optimized Ag superionic glasses have rapidly decreasing activation energies as temperature increases.
At higher frequencies (at least for c), the mean square displacement becomes nonlinear and is described by [60]
n ct t
r 2 1
a.c.
2( ) ( ) , with t1/c. (2.8) Figure 2.1. Arrhenius plots of the ionic conductivity for a set of ion conducting glasses. For the poorer conducting glasses, the Arrhenius plots have a straight slope, whereas the optimized Ag conducting glasses have significant curvature at highest temperatures [Kincs et al., 1996 [64]].
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Figure 2.3. Temperature dependence of the “apparent” activation energy for the glasses shown in Fig. 2.1 [Kincs et al., 1996 [64]].
Figure 2.2. Arrhenius plots of the ionic conductivity for some silver-ion conducting glasses. Solid lines represent the least squares straight line fits [Bhattacharya et al., 2008 [65]].
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Eq. (2.8) implies that at higher frequencies, the mean square displacement is subdiffusive. In contrast to the d.c. regime, this situation indicates an ion motion that is nonrandom or temporally correlated. This temporally correlated motion may result from the anomalous diffusion of a random walk limited to a fractal geometry [66]. Let’s note that, until today, the correlated motion is better pictured as motion in which the ions perform numerous “unsuccessful” backward and forward hops before any “successful” diffusive motion occurs [60,67]. After appropriate approximations of the mean square displacement (Eq. (2.8)) and the application of Fourier transformation, the following expression for the real part of a.c. conductivity is obtained [60]
c c n
B
n r n
T k
Nq 1 (2 )cos( /2) ( / )
2 ) 6
( 2
2 0 2 .
a.c
, (2.9)
with r02 2.
Here, r0 denotes the short length pertaining to the first regime of ion motion and
represents the statistical Gamma function.
Nowick and coworkers [53] have shown that in the region between constant loss regime (n1) and Jonscher regime (n0.6), it is possible to treat the data as a superposition of both regimes. In other words,
()()AnB. (2.10) The exponent value of 0.6 in Jonscher regime is considered to arise by the ion-ion interactions. It is of importance to note that in the seminal work of Jonscher [4], the distinction between these two separate polarizing processes has not been pointed out.
During the process of the hopping of the ions, even separate hopping events may have a broad distribution of relaxation times, and this effect can manifest as
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stretching of the relaxation times. Thus, the term Aωn exhibits activated temperature dependence in contrary to the linear frequency dependent term Bω which exhibits only weak temperature dependence. The exact origin of the last term is unclear. However, some authors believe that it may be the result of low energy distortions occurring in the network [56,68,69]. Moreover, experimental evidences [56] suggest that the process leading to this latter term (Bω) is distinct from the motion of ions giving rise to the former term (Aωn).
In addition to its weak temperature dependence, the term Bω displays its own
“universal” behavior in that values of B are often about sec·S·m-1 (to within a decade) [52,70]. B is referred to as a nearly constant loss (NCL) which is present in all Jonscher-like materials. It has been reported that, at sufficiently low temperatures or high frequencies, the term Bω dominates over the power law dependence of the exponent n [71-74].
2.1 A review of some structural models of ion dynamics in disordered
materials
In order to gain insights on microscopic mechanisms responsible for the a.c.
universality in all condensed matter, as shown empirically by Jonscher [4], a large number of theoretical models such as coupling model, jump relaxation model, diffusion-controlled relaxation model, asymmetric-double-well-potential model, random barrier model, MIGRATION concept, etc. have been proposed. In general, these models can be classified into three groups [75]:
i) Network models
ii) Debye relaxation models with distributions of relaxation times, and
iii) Models employing fundamentally modified atomic level relaxation processes.
In the following, we will give a brief overview of six prominent models, from among the most common.
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