Ionic transport in mixed-alkali glasses: hop through the distinctly different conduction pathways of low dimensionality

Empirically, the ac part of the conductivity Kω^β with a constant K being real was obtained by assuming that the time-dependent mean square displacement of mobile charge carriers was described by 〈r² (t)〉 ∼t^α , where β = 1 − α [15, 16]. In this section we attempt to develop a general relationship that exists between the measured function Reσ (ω ) = σ_dc + Kω^β and the time-dependent mean square displacement, 〈r² (t)〉 ∼t^α , of mobile charge carriers.

Swenson and Adams [8] suggested that the two types of alkali ions were randomly mixed and have distinctively different conduction pathways of low dimensionality by applying the bond-valance technique to reverse Monte Carlo (RMC)-produced structural models for a system of MA phosphate glasses. In addition, it was shown that the experimental ionic conductivity and activation energy could be determined directly from the volume fraction of the percolating pathway cluster, F, of the bond-valence RMC method [17, 18]:

$$\begin{equation} \ln ({\sigma _{{\rm exp}\,} T\sqrt M }) = A_1 ( {F\sqrt M } )^{1/3} +B_1 \propto - \left( {\frac{{E_{\rm{A}} }}{{k_{\rm{B}} T}}}\right), \end{equation} \noindent \tag{ 1 }$$

where A₁ and B₁ are constants, σ_exp is the experimentally determined conductivity, T is the temperature, M is the mass of a cation, E_A is the activation energy and k_B is the Boltzmann constant. The correlation between the free pathway volume, F, and the electrical conductivity was also tested for binary sodium silicate glasses [19].

In reality, conduction takes place on some complex subset of the three-dimensional percolation clusters and the aggregation of microscopic particles diffusing in a fluid medium represents a common process leading to fractal structures. In the general frame of site percolation, Mustarelli et al [20] successfully explained that in AgI-based fast ion conducting glasses, the ion transport takes place along nanochannel pathways with fractal structure. In addition, Sanson et al [21] found a general correlation between the Ag–I distance and ionic conductivity in AgI fast ion conducting glasses by extended x-ray absorption fine structure spectroscopy (EXAFS) measurements. Thereby they suggested a connection between the ionic conduction pathway volume defined by Adams and Swenson and the mean volume of the Ag–I coordination sphere. Therefore we can assume that the percolation object, which has a fractal structure, consists of the effective volume occupied by the aggregation of the conducting monomers. We shall derive the time evolution expression for the conductivity in such a conducting pathway of low dimensionality.

Now we consider the irreversible aggregation and clustering phenomena that are modeled by the Smoluchowski coagulation equation of the form [22]

$$\begin{equation} \frac{{{\rm{d}}c_k }}{{{\rm{d}}t}} = \frac{1}{2}\sum\limits_{i+j = k} {K(i,j)c_i } c_j -c_k \sum\limits_{j = 1} {K(k,j)c_j } . \end{equation} \noindent \tag{ 2 }$$

The Smoluchowski equation describes the time evolution of the cluster-size distribution {c_k (t)} and the coagulation kernel K(i,j) in equation (2) represents the rate coefficient for a specific coalescence between clusters of sizes i and j. In irreversible aggregation processes, the cluster-size distribution approaches a scaling form, in general

$$\begin{equation} c_k (t) = k^{-\theta } f(k/t^z ), \end{equation} \noindent \tag{ 3 }$$

where f is a scaling function which depends on certain characteristics of the coagulation coefficients, f(x) ≪ 1 for x ≫ 1 and f(x) ∼ exp (−x^Δ) with Δ some negative constant for x ≪ 1 [22, 23]. Using Smoluchowski's coagulation equation, it may be possible to determine that the exponents in the mean cluster size k(t) ∼ t^z when $t \to \infty$, resulting in the small- and large-x behavior of the scaling function f(x). The dynamic scaling in equation (3) is expected to describe the evolution of the process in the limit when the particle density is small (ρ_m ≪ 1 ). In addition, k ≫ 1 and t ≫ 1 are required in order to be in the scaling region. There is an upper bound for the time variable t < τ_c . This upper bound is defined as the time of complete formation of the aggregates in a finite system for the aggregation process and, similarly, as the time of gel appearance in the gelation process. The cluster-size distribution can be measured by the first cumulant, K₁, of the autocorrelation function of the quasielastic light scattering experiments [24]. For the cluster-size distribution, c_k, the first cumulant is given by

$$\begin{equation} K_1 = \frac{1}{{I(q,0)}}\int {k^2 } c_k (t)S_k (q)(D_k q^2 +A)\,{\rm{d}}k, \end{equation} \tag{ 4 }$$

where q is the scattering vector, D_k and A are, respectively, the translational and rotational distribution constants of mass k, S_k(q) is the scattering factor of a k-cluster and I(q,0) is the time-averaged total scattered intensity [23, 25].

By using the scaling assumption in equation (3), we can determine the effective volume, V(t) , occupied by the cluster as follows [23]:

$$\begin{equation} V(t) = \sum\limits_k {V_k } c_k (t)\sim\rho _{\rm{m}} \int_1^\infty {k^{d/D_{\rm{f}} -\theta } } f(k/t^z )\,{\rm{d}}k\sim\rho _{\rm{m}} t^{(d/D_{\rm{f}} -\theta +1)z} , \end{equation} \tag{ 5 }$$

where V_k = k^d/D_f is the effective volume occupied by a k-site cluster and the prefactor is chosen so that V(1) = ρ_m , since V(1) is equal to the volume fraction of the diffusing particles at the first time step. Here, ρ_m is the density of particles, d is the spatial dimension and D_f is the fractal dimension of the aggregates.

As a consequence of mass conservation in the diffusion-limited cluster–cluster aggregation (DLCA) model, conservation of the total number of particles enforces a particular value on exponent θ = 2 and D_f = 1.75 in equations (3) and (5) [23, 26]. The fractal dimension D_f can be determined by using light scattering and small angle x-ray scattering measurements. In the context of conservation of the total number of particles in the system, the DLCA model is more appropriate for the ionic conduction system. However, it should be noted that it is characterized by different scaling exponents θ and z for many different aggregation phenomena, especially with the non-conserved polymer systems [27].

A time-dependent conductivity relation yields from equations (1) and (5) the following:

$$\begin{equation} \ln ({\sigma T\sqrt M }) \propto ( {F\sqrt M } )^{1/3} \sim [ {\rho _{\rm{m}} \sqrt M }]^{1/3} t^{(d/D_{\rm{f}} -\theta +1)z/3} , \end{equation} \noindent \tag{ 6 }$$

where σ is the ac conductivity, T is the temperature, ρ_m is the mass density of mobile particles and M is the mass of a cation. This equation reveals that the percolation cluster of mobile ions in the materials has formed with the fractal pathway structure. The supporting evidence for this is the fact that the increase of characteristic cluster size with time is reasonably well described by exp (c't) for an extremely slow aggregation of colloidal particles, as well as for the conducting mobile ions of the alkali–metal glass systems, where c' depends on the experimental conditions [18, 24]. Consequently, our model represents that the percolation takes place along the low-dimensional conduction pathways with fractal structure. It is noteworthy that the ions are moving through the constructed structural pathways in geometry and, correspondingly, the fractal dimension of the pathways is the same for all ionic elements for a given system.

The exponent z is equal to 1 in equations (5) and (6) in the case of the DLCA and therefore we may rewrite the expression of the ac conductivity, σ(t) , in three dimensions in equation (6):

$$\begin{equation} \sigma (t)T = C\,{\rm{exp}}\left( {\frac{t}{{\tau _{\rm{c}} }}} \right)^\alpha , \end{equation} \tag{ 7 }$$

where $\alpha = \frac{1}{{D_{\rm{f}} }}-\frac{1}{3}$, constant C, and a characteristic time $\tau _{\rm{c}} = [\rho _{\rm{m}} \sqrt M ]^{-1/3\alpha }$. The obtained time-dependent conductivity in equation (7) exhibits the implicit contributions from all significant interactions with anions and the effects of local structure in the glass systems, since the structural correlations for various types of mobile ions are combined to a unified empirical relationship by the cation mass as a scaling factor in glass systems [18]. The different structural energy landscapes can be taken into account for explaining the various contributions from all significant interactions such as site relaxation, a selective hopping mechanism and static disorder in the network. Therefore, equation (7) may permit an empirical description of the interplay between glass structure and ionic conductivity for various systems, such as the SA and MA alkali glasses. It is interesting to note that the exponent α of ac conductivity in equation (7) depends on the fractal structure in geometry. As mentioned above, we reiterate here that the ac conductivity in equation (7) is due to the movement of conducting ions in the percolation cluster with the fractal structure pathway.

With the expansion of ${\rm exp}[x] = 1+x+\frac{{x^2 }}{{2!}}+\frac{{x^3 }}{{3!}}+ \cdot \cdot \cdot$, equation (7) can be rewritten as

$$\begin{equation} \sigma (t)T = C\sum\limits_{n = 0}^\infty {\frac{1}{{\Gamma (1 + n)}}} \left( {\frac{t}{{\tau _{\rm{c}} }}} \right)^{n\alpha } , \end{equation} \tag{ 8 }$$

where Γ (n) refers to the Euler gamma function and C is a constant. The first term with n = 0 in equation (8) corresponds to the conductivity independent of time. We note that the second term of the series expansion with n = 1 in equation (8) is identical to the mean square displacement of the mobile ions in the linear response theory, 〈r² (t)〉 ∼t^α , which is empirically given in [15, 16]. The term with n = 1 and 2 in equation (8) leads to the expression for the conductivity of SA and MA ions, respectively. We will show it later in an explicit manner.

In the linear response theory, the characteristic time τ_c is related to the activation energy as follows [16]: $\tau _{\rm{c}} = \tau _0 \,{\rm{exp}}( {\frac{{E_{\rm{A}} }}{{k_{\rm{B}} T}}})$. From a fit to the bond-valance RMC, the activation energy of ionic conducting glasses is given by $\ln (\sigma T\sqrt M ) \propto -({\frac{{E_{\rm{A}} }}{{k_{\rm{B}} T}}})$ in equation (1). That is, the conductivity obeys the Arrhenius relation. Thus, we may obtain the corresponding activation energy of alkali glass, E_A = −k_B T ${\rm In}\,\tau _0 (\rho _{\rm{m}} \sqrt M )^{1/3\alpha }$, by adopting the characteristic time $\tau _{\rm{c}} = [\rho _{\rm{m}} \sqrt M ]^{-1/3\alpha }$ in equation (7).

We shall develop the processes of transport events by a simplified model, as illustrated in figure 2, of the mobile ions that undergo an anomalously slow transport through a self-similar conducting pathway within an intra-cluster and then hop to an adjacent inter-cluster. After undergoing sub-transport in a cluster, the mobile ions hop from one percolation cluster to an adjacent cluster due to an external driving force. It is presumably considered, for example, that the potential energy at branch i of a cluster in figure 2 is the same as that at branch i of an adjacent cluster. This type of equipotential branches can exist either in the inter-clusters or in an intra-cluster. Thus, the hopping process may be possible both in the inter-cluster and in an intra-cluster, since it is easy to hop where the potential energy is the same. We note that the exponent n in equation (8) represents successive fractal branches in the percolation object. The dc conduction may arise from ions moving through the many inter-cluster pathways and the ionic conducting character in n = 0 pathway necessarily requires ion displacements that are uncorrelated and non-fractal.

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We shall now derive the microscopic foundation of dispersive transport close to thermal equilibrium starting with the explanation by Metzler and Klafter [28]. That is, from the continuous time version of the Champman–Kolmogorov equation combined with the Markovian Langevin equation of a damped particle in an external force field, a fractional Klein–Kramers equation is derived whose velocity averaged high-friction limit reproduces the fractional Fokker–Planck equation, and explains the occurrence of the generalized transport coefficient η * . Based on the continuous time version of the Chapman–Kolmogorov equation in the long-time limit t ≫ max{ τ ,τ *} , one obtains the fractional Klein–Kramers equation [28, 29]

$$\begin{equation} \fl \frac{{\partial P(x,v,t)}}{{\partial t}} = _0 D_t^{1-\alpha } \left[ {-v^{*}\frac{\partial }{{\partial x}}+\frac{\partial }{{\partial v}}\left( {\eta ^{*}v-\frac{{F^{*}(x)}}{m}} \right)+\frac{{\eta ^{*}k_{\rm{B}} T}}{m}\frac{{\partial ^2 }}{{\partial v^2 }}} \right]P(x,v,t), \end{equation} \noindent \tag{ 9 }$$

where v* ≡ vτ */τ^α is the velocity with $[v^{*}] = {\rm{cm}}\,{\mathop{\rm s}\nolimits} ^{-\alpha }$, η * ≡ η τ */τ^α denotes the friction constant with $[\eta ^{*}] = {\mathop{\rm s}\nolimits} ^{-\alpha }$, F*(x) ≡F(x)τ */τ^α is an external force field in phase (position–velocity) space with $[F^{*}] = {\rm{cm}}\,{\rm{g}}\,{\mathop{\rm s}\nolimits} ^{-1-\alpha }$, τ * is the inter-trapping time scale, and τ is the internal waiting time scale. They obtained equation (9) by means of a non-Markovian generalization of the Chapman–Kolmogorov equation. The relation between the Klein–Kramers equation and the Langevin equation is given in [29, 30]. On the right-hand side of equation (9) in brackets, the first term describes the spatial drift due to the velocity, the second term accounts for the friction and external force feedback to the velocity as expressed through the corresponding Langevin equation, and the third term represents the entropy-based velocity diffusion. The Langevin picture rules the Markov motion parts in between successive trapping states, indicating that the result gives some insight into the fractional Fokker–Planck equation for multiple trapping systems [29]. The integration of the fractional Klein–Kramers equation (9) over the position coordinate leads in the force-free limit to the fractional Rayleigh equation [28, 29]

$$\begin{equation} \frac{{\partial P(v,t)}}{{\partial t}} = _0 D_t^{1-\alpha } \eta ^{*}\left[ {\frac{\partial }{{\partial v}}v+\frac{{k_{\rm{B}} T}}{m}\frac{{\partial ^2 }}{{\partial v^2 }}} \right]P(v,t). \end{equation} \tag{ 10 }$$

The fractional Rayleigh equation is equivalent to the fractional Fokker–Planck equation with a linear force and therefore corresponds to the sub-diffusive Ornstein–Uhlenbeck process [31]. Here, the Riemann–Liouville fractional operator, ₀ D_t^p , is defined by the convolution [28]

$$\begin{equation} _0 D_t^p f(t) = \frac{1}{{\Gamma (n-p)}}\frac{{{\rm{d}}^n }}{{{\rm{d}}t^n }}\int_0^t {\frac{{f(t')}}{{(t-t')^{1-n+p} }}} \,{\rm{d}}t', \end{equation} \tag{ 11 }$$

for n ⩾ Re(p) >n − 1 . In the case of the fractional Rayleigh equation (10), the mean velocity obeys

$$\begin{equation} \frac{{{\rm{d}}\langle v(t)\rangle }}{{{\rm{d}}t}} = -_0 D_t^{1-\alpha } \eta ^{*}\langle v(t)\rangle , \end{equation} \tag{ 12 }$$

with η * ≡ η τ */τ^α , as is readily verified by $\langle v(t)\rangle = \int\!{vP(v,t)\,{\rm{d}}v}$, followed by an integration by parts of the right-hand side in equation (10). We note that the solution of the fractional relaxation equation in equation (12) is given in terms of the Mittag–Leffler function, $E_\alpha (-\eta ^{*}t^\alpha ) = \sum\nolimits_{n = 0}^\infty {\frac{{(-\eta ^{*}t^\alpha )^n }}{{\Gamma (1+\alpha n)}}}$ [28, 29]. The right-hand side of dynamic equation (12) is negative and therefore the acting force in the system is attractive.

Based on the experimental studies from EXAFS [32, 33], NMR [34] and infrared spectroscopy [35, 36], x-ray and neutron scattering measurements [5], it is commonly accepted that the origin of MAE has structural character, associated with a mismatch effect [37, 38]. A mismatch arises when one type of cation tries to enter a site created earlier by another in the glassy network. The mismatch effect in hopping systems can be modeled by site energies that are different for different types of ions [38]. Swenson et al [5] showed that the two different kinds of alkali ions are randomly mixed and are located in essentially one-dimensional pathways and the presence of B ions in the pathways of the A ions must result in an effective blocking of many of the energetically most favorable pathways for the A ions, and vice versa. Again, the blocking of migration pathways is caused by the site mismatch effect, and correspondingly the gradient of site energies drives the ions to move along pathways with a low-energy site for one type of ion and a high-energy site for the other type. Therefore, the momentum of ions can be changed because of the blocking of migration pathways. It is emphasized that there is neither a crucial role of repulsive Coulomb blockage nor a site mismatch for the SA glass and the various modified alkali–metal ion conducting glasses. Maass also suggested that Coulomb interaction between mobile ions with strength q² /av = 20 have to be taken into account for explaining the differing behavior of activation energies in single modified and mixed ion glasses, where q is the effective charge of the mobile ions, a is the distance between the ions and the parameter v in the energy functions defines an energy scale [10]. The Coulomb interactions cause a significant spread in the potential energies of the ions [38]. Similar to Maass' suggestions [10, 37], the external potential $U = -\int\! {F\,{\rm{d}}x}$ in equation (9) can be adopted to consider the repulsive effects due to such a Coulomb blockage and a site mismatch effect, so the result being with a type of potential $U = -\int\! {F\,{\rm{d}}x} = -\eta (1+b)mv$ leads to the modified fractional Rayleigh equation

$$\begin{equation} \frac{{\partial P(v,t)}}{{\partial t}} = _0 D_t^{1-\alpha } \eta ^{*}\left[ {-b\frac{\partial }{{\partial v}}v+\kappa _\alpha \frac{{\partial ^2 }}{{\partial v^2 }}} \right]P(v,t). \end{equation} \noindent \tag{ 13 }$$

Consequently, in the case of percolation governed by the modified fractional Rayleigh equation, the evolution of the mean velocity of moving ions through the fractal-structured pathway obeys the equation

$$\begin{equation} \frac{{{\rm{d}}\langle v(t)\rangle }}{{{\rm{d}}t}} = \eta _\alpha b\,_0 D_t^{1-\alpha } \langle v(t)\rangle , \end{equation} \tag{ 14 }$$

with a positive constant η_αb ≡ η bτ*/τ^α . The right-hand side of dynamic equation (14) is positive and therefore the acting force in the system is repulsive.

The conductivity of cations moving through the fractal structured pathway may be explicitly calculated by j(t) = σ (t)E , where the current density is j(t) = Nq〈v(t)〉 . Here N is the number of mobile ions, q is the charge of the ions and E is the electric field. Therefore, the solution to equation (14) can give us the dynamics of the conducting ions with a repulsive interaction in the fractal structured pathway medium of glasses.

Since $_0 D_t^{1-\alpha } t^p = \frac{{\Gamma (1+p)}}{{\Gamma (p+\alpha )}}t^{p+\alpha -1}$ and $_0 D_t^q 1 = \frac{1}{{\Gamma (1-q)}}t^{-q}$, we have

$$\begin{equation} _0 D_t^{1-\alpha } {\rm exp}\,(at^\alpha ) = \sum\limits_{n = 1}^\infty {\frac{{a^{n-1} \Gamma [1+(n-1)\alpha ]}}{{\Gamma (n)\Gamma (n\alpha )}}} t^{n\alpha -1} , \end{equation} \tag{ 15 }$$

where $a = \tau _{\rm{c}}^{-\alpha } = [\rho _{\rm{m}} \sqrt M ]^{1/3}$.

From equations (14) and (15), we thus obtain the average velocity of the conducting ions undergoing anomalously slow transport in a fractal structured cluster with a repulsive interaction

$$\begin{equation} \langle v(t)\rangle = \langle v(0)\rangle +\sum\limits_{n = 1}^\infty {\frac{{b\eta _\alpha a^{n-1} \Gamma [1+(n-1)\alpha ]}}{{\Gamma (n)\Gamma (1+n\alpha )}}} t^{n\alpha } . \end{equation} \tag{ 16 }$$

If the density of particles is small and, correspondingly, parameter a in equation (15) is less than 1, then very few ions move through the successive fractal branches of the pathway in the cluster. This indicates that the number density of particles rapidly decreases if the mobile ions are proceeding into the deeper branches of the cluster. Consequently, the characteristic time will be longer at deeper branches in the percolation object. This implies that the mobile ions are trapped for a very long time in a deep branch of the pathway. The result is consistent with the fact that an MA slowdown is observed in the longer residence time [9]. The corresponding conductivity decreases rapidly. We will discuss it later.

We now consider that the mobile ions are propagating from the cluster at $(\vec{r} _0 ,t_0 )$ to the adjacent cluster at $(\vec{r} ,t)$ in the absence of interactions [39] by the hopping process after transport through the fractal structured pathway in the percolation object. Fourier transform of the σ (t) at (t − t₀ ) > 0 allows one to obtain the frequency-dependent conductivity in equation (16) via

$$\begin{eqnarray} &&\fl \sigma _{{\rm hops}} (\omega ) = \left[ \delta (\vec{r} - \vec{r}_0)-\frac{1}{2}\left(\frac{\skew3\vec{F}_{0}}{M} \right)(t-t_0 )^2\right]\frac{1}{\sqrt {2\pi}}\int_0^\infty \sigma (t_0 )\,{\rm{e}}^{{\rm{i}}\omega t_0 } \,{\rm{d}}t_0 ,\quad t > t_0 \nonumber\\&&\ms\ms = \,{\rm{constant}}\,\mathop {\lim }\limits_{\varepsilon \to 0} \,(\varepsilon +{\rm{i}}\omega )^2 \int_0^\infty {\sigma _{{\rm{hops}}} (t)\,{\rm{e}}^{-(\varepsilon +{\rm{i}}\omega )t} } \,{\rm{d}}t, \end{eqnarray} \tag{ 17 }$$

where δ is the Dirac delta function, $\vec{F_0 } \,{\rm{e}}^{{\rm{i}}\omega t_0 }$ is the external driving force due to applying an electric field and M is the mass of mobile ions. The time-dependent hopping conductivity σ_hops (t) is proportional to Nq〈v(t)〉 at the position $\vec{r}$ and the average velocity is given by equation (16). Using this we obtain the real part of σ_hops (ω ) at t < τ_c , followed by integration by parts [40]:

$$\begin{eqnarray} &&\fl {\rm Re}\,\sigma _{{\rm{hops}}} (\omega )\sim\frac{{Nq}}{{\sqrt {2\pi } }}\sum\limits_{n = 1}^\infty {\frac{{b\eta _\alpha a^{n-1} \Gamma [1+(n\!-\!1)\alpha ]}}{{\Gamma (n)\Gamma (1+n\alpha )}}} \left\{ {\Gamma (1\!+n\alpha )\sin \left( {\frac{{n\alpha }}{2}\pi } \right)\omega ^{1-n\alpha } +\! B\omega } \right\},\quad \!\! n\alpha < 1.\nonumber\\ \end{eqnarray} \tag{ 18 }$$

Here, $B = -\lim _{t \to \infty } (\langle v(0)\rangle +t^{n\alpha } )\sin \,\omega t$ is the undetermined diverging value but the time t has an upper limit of the characteristic time $\tau _{\rm{c}} = [\rho _{\rm{m}} \sqrt M ]^{-1/3\alpha }$, and the integration in equation (18) is valid for nα < 1 .

Recently, the possibility of existence of fractional kinetic equations containing non-integer power-law exponents has followed from the general theory of dielectric relaxation that has been suggested by the work of Nigmatullin et al [41] on the polymerization reaction of polyvinylpyrrolidone and by the work of Coffey et al [42] on the non-inertial rotational diffusion. Equation (12) for the mean drift velocity is similar to the kinetic equation for the relaxation process based on the decoupling of a memory function on a self-similar structure [41]. In this paper, however, the physical situation is explicitly featured by the dynamic process, including the empirical contributions from all significant interactions, on the spatial picture of a self-similar structured pathway. The slow transport in an intra-cluster is basically due to the trapping ions in the fractal structure and the fractal time process.

Finally, we show that the modified fractional Rayleigh equation in the fractal structured pathway can be used to reproduce the expression of universal macroscopic conductivity in glass systems. That is,

$$\begin{equation} {\rm Re} \, \sigma (\omega ) = \sigma _{{\rm{dc}}} +K\omega ^\beta = \sigma _{{\rm{dc}}} [1+(\omega /\omega _{\rm{h}} )^\beta ], \end{equation} \noindent \tag{ 19 }$$

where σ_dc = constant , $K\sim\frac{{Nq}}{{\sqrt {2\pi } }}\sum\nolimits_{n = 1}^\infty {\frac{{b\eta _\alpha a^{n-1} \Gamma [1+(n-1)\alpha ]}}{{\Gamma (n)}}} \sin \left( {\frac{{n\alpha }}{2}\pi } \right)$, and β = 1 − nα . The value of exponent $\alpha = \frac{1}{{D_{\rm{f}} }}-\frac{1}{3}$ is 0.24 for a cluster model of DLCA with D_f = 1.75 and thereby β = 1 − α = 0.76 for n = 1. The exponent 0.5 < β < 0.9 is observed for a wide variety of structurally different ion-conducting glasses. It confirms the demonstration of Sidebottom [11] that for ionic materials the power-law exponent decreases with decreasing dimensionality of the ion conduction pathway. It is worth noting that the transport phenomena in systems exhibiting sub-diffusive should have a condition with nα < 1 and the integer number n represents the pathway branch having fractal structure in the percolation cluster. For many SA glass materials, it turns out that the ions can move only in the first branch (n = 1) of the pathway. However, the ions may possibly move through the multiple branches of the trapping pathway if there is an additional repulsive interaction such as a Coulomb blockade in the case of MA glasses. We reasoned this since the Coulomb interactions cause a significant spread in the potential energies of the ions [38].

If we take a long time scale of t (t < τ_c ) with (2m − 1)π < ω t < 2mπ for integer m, then the Bω term in equation (18) may correspond to the nearly constant loss (NCL) [43, 44]. In other words, the mobile ions, which remain in their respective sites for a long time but less than τ_c , seem to be responsible for the NCL. The characteristic time will be very long if the density of mobile particles is very low in the percolation cluster since $\tau _{\rm c} = [\rho _{\rm m} \sqrt M ]^{-1/3\alpha}$.

Two different viewpoints for NCL behavior have been proposed in the literature. One is that NCL and Jonscher behaviors originate from clearly distinct dynamic processes as described by the local vibrational relaxation [44, 45]. The other is that all the conductivity spectra are caused by hopping motions of the mobile ions as described by the random barrier model or by the concept of mismatch and relaxation [46, 47]. There is another report on a new MAE that the non-hopping contributions to the ac conductivity involve both the mobile ions and the glassy network [48]. Our result for the NCL regime exhibits a viewpoint that the mobile ions remain in their respective sites for a long time in the multiple trapping systems. Therefore, the long time duration is basically due to the trapping ions in the fractal structure and the fractal time process. However, the trapping effects can take place in the presence of asymmetric potential wells in the percolation cluster.