Ionic Conductors: Effect of Temperature on Conductivity and Mechanical Properties and Their Interrelations

The ionic transport and the mechanical properties in solids are intimately related. However, few studies have been done to elucidate the background of that relation. With the objective to fill this gap and gain further understanding on the fundamental properties of ion conducting materials, we are studying systematically the mechanical properties of different materials. In the present study, after showing briefly our previous results obtained in crystalline materials, results regarding the relation between ionic conduction and mechanical properties in superionic glasses is presented. All these results indicate the intimate relation between the mechanical and ionic conduction. The results also indicate that the Grüneisen parameter and the Anderson–Grüneisen parameter of ionic conductors exhibit large temperature dependence and increase with temperature.


Introduction
The ionic transport and the mechanical properties in solids are intimately related. The existence of such relation is understandable, because, when an ion diffuses in a solid material it is accompanied by a local deformation of the medium, and reciprocally, the ion can be moved by deforming mechanically the ambient that surround the moving ion. Despite this fact, the number of studies done to elucidate the relation between the mechanical and other physical properties in ionic conductors are limited . By searching the literature, we find diverse studies that are related with the subject in consideration. Some of them exbibit interesting but complex physical properties [6,12,23,24], whose mechanistic explanation are awaited. For instance, usually the sound velocity in solids decreases by increasing temperature. However, in β-Ag 3 SI, a well-known ionic conductor, the sound velocity increases with temperature in the temperature interval between 157 K and 360 K [6]. Although the temperature range is limited, the increase of the sound velocity with temperature is also observed in the superionic phase of AgI [15]. To the best of our knowledge, no satisfactory explanation has been given to those observations. Interestingly, there have been reported that the thermal conductivity in β-Ag 3 SI [34] and superionic AgI [35] increase with temperature. These behaviors are opposite to those observed usually. Theoretical study showed that such a behavior of thermal conductivity can be accounted for by the increase of hard-core like collision between the mobile ions and the surrounding lattice ions [36]. However, a different model is needed to understand the origin of the behavior obtained experimentally.
In recent years, much attention has been paid to heterostructures and interfaces to enhance the ionic conductivity [37][38][39]. At the interface region of the heterostructure, the mechanic stress is different from that in the bulk region. This difference originates the increase of the ionic mobility. On the other hand, studying the effect of multiple external perturbances to ion dynamics is effective for the understanding of the mechanism of superionic conduction. For instance, light was used to study the influence of elastic stress on ionic transport in a heterojunction formed by a superionic crystal and an electrode [12]. The result provided a hint to understand the role of electrons in the ionic transport mechanism, a subject of fundamental importance but not thoroughly investigated [40][41][42]. Regarding this point, it will be enlightening to point out that the phenomenon reported in [12] has much in common with the photodoping effect known in the field of amorphous semiconductors [43][44][45]. Indeed, the mechanism of ionic conduction discussed in [12] is similar to that described in [44]. However, regarding the details on how the mechanical stress enters into the mechanism of light induced ionic transport remains to be solved. A related phenomena to that presented here is the acoustoionic and acoustoelectronic effect which has been investigated for many years [7,15]. By analyzing the data obtained there, we may extract useful information regarding the interaction between sound wave and charge carriers. The acoustoionic and acoustoelectronic interactions are manifested macroscopically in the sound velocity and attenuation. However, depending on the material in consideration and on the thermodynamic state, the observed phenomenon is quite complex. As an example, we may cite the behavior of internal friction reported in the superionic crystal Cu 6 PS 5 Br [23]. In this system, the effect of two different transitions, the superionic and ferroelastic transitions are reflected in the observed internal friction. Systematic investigation in these types of materials is expected to provide insights to understand the nature of interionic interactions of ionic conductors.
Understanding the mechanical properties of ionic conductors are also important from the application point of view [22,26,29,[46][47][48]. From the side of materials synthesis, we may cite for instance, the mechanically milling process which are widely used in the preparation of solid electrolytes used in batteries [46]. The material properties that are gained from this method of synthesis depends on the mechanical properties of the materials. In recent years, much attention has been paid to develop all-solid-state lithium secondary batteries. To enhance the electrochemical performance of bulk-type batteries, an intimate solid-solid contact between electrode active materials and solid electrolytes is required in addition to high ionic conductivity [22]. It has been also recognized that the mechanical properties of the materials play a large role in the processing and assembly of batteries [26]. The mechanical strength of solid electrolytes plays an important role in suppressing the growth of dendrites, by increasing the interfacial stability and avoiding the propagation of cracks [47]. However, despite its importance, information on fundamental mechanical properties of battery materials is limited. A short summary of recent works on elastic, plastic, and fracture properties of crystalline oxide-based Li-ion solid electrolytes is available in [26]. As noted in many studies [22,26,29,[46][47][48], the role of mechanical properties in a real battery system is multiple and complex. It extends from fundamental materials properties to assembly of devices.
With the objective to gain a further understanding on the fundamental properties of superionic materials, we are studying systematically the mechanical properties of different materials [17][18][19][20]. In the present report, after showing briefly the previous works done in our group, results regarding the relation between ionic conduction and elastic properties in superionic glasses will be presented. In particular, the results on temperature dependence of the Grüneisen and Anderson-Grüneisen parameters will be shown. Despite the fact that these quantities are intimately related with the anharmonicity of lattice vibrations, data of such quantities in ionic conductors are very scarce in the literature. Our study indicates that these quantities exhibit large temperature dependence and increase with the increase of temperature, results which are in accord with the ion transport properties, but not recognized throughout.

The Relation between Elastic Property and Ionic Conductivity
Usually, the elastic constants of the materials decrease by increasing temperature. It is known that the decrease is almost linear up to a temperature near to the melting Crystals 2021, 11, 1008 3 of 15 temperature. To see the peculiarity of ionic conductors, a comparison of the temperature dependence of the elastic constant C 11 of ionic conductors and non-ionic conductors is shown in Figure 1 [49]. For the case of ionic conductors such as PbF 2 and BaF 2 , the deviation from the linearity starts at a temperature much lower than the melting temperature. AgCl and AgBr are also compounds that exhibit relatively high ionic conductivity. For the case of AgCl, the deviation from the linearity is not so evident in Figure 1. However, it has been shown that its behavior contrasts with the behavior of typical ionic crystals, NaCl [19]. The other materials shown in Figure 1, Cu 2 O, Al, and alkali halides are no ionic conductors.
pendence of the elastic constant of ionic conductors and non-ionic conduc shown in Figure 1 [49]. For the case of ionic conductors such as PbF2 and BaF2, the tion from the linearity starts at a temperature much lower than the melting tempe AgCl and AgBr are also compounds that exhibit relatively high ionic conductivity. F case of AgCl, the deviation from the linearity is not so evident in Figure 1. However been shown that its behavior contrasts with the behavior of typical ionic crystals [19]. The other materials shown in Figure 1, Cu2O, Al, and alkali halides are no ion ductors.
The temperature dependence of the elastic constant arises from different fac we consider a crystal without defects, it arises from the variation of lattice potential e associated with the anharmonicity. Therefore, the comparison of the temperature de ence of the elastic constants of different materials shown in Figure 1 provides a s point to discuss the mechanical properties of ionic conductors.  [49]. T perature is scaled by the melting temperature .
The behavior shown in Figure 1 is related with the ionic transport properties materials, because crystal defects and lattice vibration anharmonicity are factor mately related with the ionic conduction. This relation is seen more clearly in Fig where the elastic constants and the ionic conductivity of PbF2 are put in a single [17]. PbF2 is one of the well-known superionic materials which exhibit a diffuse transition to the superionic state at = 705 K. The figure shows clearly the corr between the temperature dependencies of the ionic conductivity and the elastic con The ionic conductivity in PbF2 starts to increase at around 600 K. At around the temperature, the elastic constants start to deviate from the almost linear behavi served at low temperature. At around 700 K, an abrupt increase of the ionic condu and large change in the elastic constants is discernible. The large increase of the ion ductivity is commonly understood to arises from a development of extensive Frenk  [49]. The temperature is scaled by the melting temperature T m .
The temperature dependence of the elastic constant arises from different factors. If we consider a crystal without defects, it arises from the variation of lattice potential energy associated with the anharmonicity. Therefore, the comparison of the temperature dependence of the elastic constants of different materials shown in Figure 1 provides a starting point to discuss the mechanical properties of ionic conductors.
The behavior shown in Figure 1 is related with the ionic transport properties of the materials, because crystal defects and lattice vibration anharmonicity are factors intimately related with the ionic conduction. This relation is seen more clearly in Figure 2, where the elastic constants and the ionic conductivity of PbF 2 are put in a single figure [17]. PbF 2 is one of the well-known superionic materials which exhibit a diffuse phase transition to the superionic state at T c = 705 K. The figure shows clearly the correlation between the temperature dependencies of the ionic conductivity and the elastic constants. The ionic conductivity in PbF 2 starts to increase at around 600 K. At around the same temperature, the elastic constants start to deviate from the almost linear behavior observed at low temperature. At around 700 K, an abrupt increase of the ionic conductivity and large change in the elastic constants is discernible. The large increase of the ionic conductivity is commonly understood to arises from a development of extensive Frenkel disorder in the anion sublattice [50]. The behavior shown in Figure 2 suggests that the microscopic mechanisms that originate the mechanical and electrical properties have common elements. Indeed, in previous studies it has been shown that the elastic constants of ion conducting materials AgCl and AgBr deviate from the linear behavior at high temperature, whereas the elastic constants of a typical ionic crystal NaCl retain their linearity up to a temperature close to the melting point [18,19]. Thus, we may say that the temperature dependence of the elastic constant provides much insight to understand ionic conductors.
The temperature dependence of the sound velocity in ionic conductors CuG and AgGeAsSe3 [11], show a behavior that supports the result shown in Figure 2. O other hand, for the case of Cu2Se, Cu2Te, Ag2Se and Ag2Te, the temperature depen of the Young's modulus, shear modulus and Poisson's ratio behave linearly, and no ations or anomalies were detected at the superionic transition temperature [8]. The r of this difference in the behavior is not clear. However, it will be useful to point ou further study, that the transport properties of these chalcogenide materials depend o details of chemical composition [51]. On the other hand, studies in the liquid ph Ag2+δSe indicate that the sound velocity changes smoothly at the stoichiometric com tion [52,53]. From Figure 2, we note that in PbF2, the temperature dependence and the dev from the linear behavior of and are much larger than that of . Similar b ior has been also observed in other fluorite type ionic conductors such as CaF2, SrF BaF2 [54]. However, for these compounds, the degree of deviation from the linear is comparable to that of . An explanation to the different behavior of the tem ture dependence of the elastic constants was given by arguing that is determ from the contribution of Coulomb and short-range forces that have the same whereas for and they have opposite signs, and that the effect of crystal d on and cancel each other [4,54]. Similar behavior of the temperature dependence of elastic constants to that sho Figure 2 has been also observed in AgCl and AgBr [10,13,16,18,19]. These crystals the rock-salt structure. However, the physical properties of these compounds are dif from the alkali halide compounds that have the same crystal structure. Particularl mechanical and ionic transport properties at high temperature are quite different. T ver halides exhibit a large increase of thermal expansion [55] and ionic conductivit Figure 2. Temperature dependence of the elastic constants C ij (C 11 , C 12 and C 44 ) and the ionic conductivity σ i in PbF 2 [17].
The temperature dependence of the sound velocity in ionic conductors CuGeAsS 3 and AgGeAsSe 3 [11], show a behavior that supports the result shown in Figure 2. On the other hand, for the case of Cu 2 Se, Cu 2 Te, Ag 2 Se and Ag 2 Te, the temperature dependence of the Young's modulus, shear modulus and Poisson's ratio behave linearly, and no deviations or anomalies were detected at the superionic transition temperature [8]. The reason of this difference in the behavior is not clear. However, it will be useful to point out for a further study, that the transport properties of these chalcogenide materials depend on the details of chemical composition [51]. On the other hand, studies in the liquid phase of Ag 2+δ Se indicate that the sound velocity changes smoothly at the stoichiometric composition [52,53].
From Figure 2, we note that in PbF 2 , the temperature dependence and the deviation from the linear behavior of C 11 and C 12 are much larger than that of C 44 . Similar behavior has been also observed in other fluorite type ionic conductors such as CaF 2 , SrF 2 and BaF 2 [54]. However, for these compounds, the degree of deviation from the linearity of C 12 is comparable to that of C 44 . An explanation to the different behavior of the temperature dependence of the elastic constants was given by arguing that C 11 is determined from the contribution of Coulomb and short-range forces that have the same signs, whereas for C 12 and C 44 they have opposite signs, and that the effect of crystal defects on C 12 and C 44 cancel each other [4,54].
Similar behavior of the temperature dependence of elastic constants to that shown in Figure 2 has been also observed in AgCl and AgBr [10,13,16,18,19]. These crystals have the rock-salt structure. However, the physical properties of these compounds are different from the alkali halide compounds that have the same crystal structure. Particularly, the mechanical and ionic transport properties at high temperature are quite different. The silver halides exhibit a large increase of thermal expansion [55] and ionic conductivity [56] at high temperature. Therefore, we may argue that similar materials properties interrelation to that shown in Figure 2 can be drawn. The peculiar behavior of AgCl and AgBr is considered to arise from a rapid increase in the concentration of Frenkel defects at high temperatures, which results from a temperature dependent decrease in the Gibbs free energy for defect formation [13,56]. Some authors suggest that the large softening of the elastic constants observed in silver halides is a manifestation of pre-melting phenomenon [56]. Others discuss that the temperature dependence of the elastic constants of AgCl and AgBr indicate the starting of the transition to the superionic state [13]. Such an interpretation seems consistent with a theoretical study on wave number dependent static dielectric function [57], which indicates that the superionic material in the liquid phase has a strong tendency to sublattice crystallization when compared to usual molten salts.
Among the elastic constants of AgCl and AgBr, C 11 exhibits the largest temperature variation. The temperature dependence of C 44 is small, and C 12 exhibit an intermediate variation. In addition, AgBr exhibits a stronger deviation from the linearity than AgCl [10,13,16,18,19]. As discussed for the case of PbF 2 , these observations indicate that specific force constants are more effectively interrelated with the atomic transport properties. This observation relates also with the different behavior of the Anderson-Grüneisen parameter discussed later.
The strong temperature dependence of the elastic constant of ionic conductors are usually interpreted to arise from the creation of large number of crystals defects [50,54,56]. On the other hand, it is also known that the elastic constants reflect the bonding nature of the materials [58]. In real situations, the plausible interpretation is that both factors and others if any, are reflected in the observed behavior. In the following, a result related with the bonding nature point of view is shown [19].
One of the quantities related with the directionality of the chemical bond is given by the Cauchy relation. For the case of cubic crystals, the Cauchy relation is given by C 12 = C 44 . If the interatomic interaction in a material consists only of radial interaction such as Coulomb interaction, the Cauchy relation is obeyed. In other words, by studying the deviation from the Cauchy relation, we can get information on the degree of nonradial interaction between the atoms. Figure 3 shows the temperature dependence of the ratio C 44 /C 12 in NaCl, AgCl and AgBr. Firstly, for the case of Ag halides we note a large deviation from C 44 /C 12 = 1, whereas NaCl starts to deviate from 1 at a temperature higher than 300 K. We also observe that C 44 /C 12 of Ag halides increase towards 1 with the increase of temperature. This temperature dependence is opposite to that observed in NaCl. It is worth mentioning at this point that the behavior shown in Figure 3 can be related with the bond fluctuation model of ion transport [40][41][42]. Although AgCl and AgBr have the rock-salt structure, their chemical bonding is not purely ionic as suggested from Figure 3. According to the bond fluctuation model, when an ion moves from one site to another site, it is accompanied by a local distortion of the electronic cloud that surrounds the mobile ion [40]. Since such distortions occur locally at different sites of the material and at different times, they can be visualized as local fluctuations of the chemical bond. The origin of the correlated ionic motions, which is a characteristic of superionic conductors is explained naturally from this model [40][41][42]. Results of ab-initio molecular dynamics study [59] and other recent studies support the view proposed in the bond fluctuation model [60][61][62][63][64]. From the point of view of this model, the result shown in Figure 3, in particularly the case of AgBr, suggests that the degree of central forces in the determination of elastic constants increases with the increase of temperature.
Regarding the temperature dependence of C 44 /C 12 in NaCl, a comment should be given. The behavior show in Figure 3 has been obtained by using the data reported in [65], where C 12 > C 44 . On the other hand, there are studies that report an opposite behavior, C 12 < C 44 [66,67]. The difference in magnitude between C 12 and C 44 is not large, and in addition, the temperature variation of C 12 is not monotonous [66,68]. Although small, its temperature derivative changes the sign from positive to negative at around 500 K [68]. As far as the authors are informed, there is no direct experimental study that judge which behavior is correct.
Crystals 2021, 11, 1008 6 of 15 rystals 2021, 11,1008 As far as the authors are informed, there is no direct experimental study tha behavior is correct. The elastic anisotropy influences a variety of physical phenomena. The in Figures 2 and 3 suggest the possibility that the ionic conduction is related tic anisotropy of the materials. This hypothesis has been investigated by ap pression for the index of anisotropy [69] proposed some years ago. Here, is the so-called Zener's anisotropy, which is defined as the two quantities and 2 ⁄ . The elastic constant , represents to deformation with respect to a shear stress applied across the (100) plan direction, whereas 2 ⁄ represents the resistance to shear deform across the (110) plane in the 11 0 direction. The expression of given in has been introduced in such a way to give a single index of anisotropy, an zero for an isotropic crystal [69]. Figure 4, shows the temperature dependence of calculated for Na AgBr [18]. From the figure, we note that the of AgBr exhibits very large dependence. Particularly, we note that its elastic anisotropy starts to incre around 550 K. For AgCl, its elastic anisotropy is larger al low temperature served that AgCl and AgBr exhibit minimal values at around 600 K and 5 tively. For the case of NaCl, the value of and its temperature depend That is, NaCl is almost elastically isotropic over a wide temperature rang shown in Figure 4 suggests clearly that the elastic anisotropy is intimately re The elastic anisotropy influences a variety of physical phenomena. The results shown in Figures 2 and 3 suggest the possibility that the ionic conduction is related with the elastic anisotropy of the materials. This hypothesis has been investigated by applying an expression for the index of anisotropy [69] proposed some years ago. where Here, A Z is the so-called Zener's anisotropy, which is defined as the ratio between two quantities C 44 and (C 11 − C 12 )/2. The elastic constant C 44 , represents the resistance to deformation with respect to a shear stress applied across the (100) plane in the [010] direction, whereas (C 11 − C 12 )/2 represents the resistance to shear deformation applied across the (110) plane in the 110 direction. The expression of A U given in Equation (1) has been introduced in such a way to give a single index of anisotropy, and its value is zero for an isotropic crystal [69]. Figure 4, shows the temperature dependence of A U calculated for NaCl, AgCl and AgBr [18]. From the figure, we note that the A U of AgBr exhibits very large temperature dependence. Particularly, we note that its elastic anisotropy starts to increase largely at around 550 K. For AgCl, its elastic anisotropy is larger al low temperature. It is also observed that AgCl and AgBr exhibit minimal values at around 600 K and 500 K, respectively. For the case of NaCl, the value of A U and its temperature dependence is small. That is, NaCl is almost elastically isotropic over a wide temperature range. The result shown in Figure 4 suggests clearly that the elastic anisotropy is intimately related with the defect creation and the ionic conduction property of the materials. This is a new and interesting observation, because till very recently, no study on elastic anisotropy of solid electrolytes has been reported [29], excepting our preliminary study [18]. We are planning to extend the analysis of A U to other ionic conducting materials to gain further information on this subject and draw a decisive conclusion.

Temperature Dependence of the Anderson-Grüneisen Parameter
An understanding of the anharmonic effects and the temperature dependent thermo mechanical properties is of primordial importance in ionic conducting materials. In Figure  1, it was shown that the elastic constants vary linearly at low temperatures. Such behavior is described by a temperature independent Anderson-Grüneisen (AG) parameter which is given by [70]: where is the thermal expansion coefficient, is the isothermal bulk modulus, is the volume, is the temperature, and is the pressure. The AG parameter has been introduced as an attempt to improve the description of the temperature dependen Young's modulus [70]. Nowadays, it is used as an important tool for investigating differ ent aspects related to thermoelastic properties of solids. Although there are some studies reporting its temperature dependence [20], usually, the AG parameter is considered as a constant of the material or depends only weakly on temperature [71]. However, for the case of ionic conducting materials, we have found that the AG parameter depends strongly on temperature [20].
The result shown in Figure 5 indicates clearly the difference between the behavior o ion conducting materials AgCl, AgBr and CaF2 and the other non-ion conducting materi als KBr, Mg2SiO2 and Ag. For the cases of AgBr and CaF2 we observe a peak. Such kind o peaked behavior has been also observed in other ionic conducting materials investigated PbF2 and BaF2 [20]. The peak of the AG parameter at certain temperature arises from the different relative contributions of temperature derivatives of , and the therma expansion coefficient. For instance, for the case of fluorine compounds investigated, it was discussed that when the temperature dependence of dominates, the AG curve shows an upward curvature, whereas when the temperature dependences of and therma expansion coefficient dominate, the AG curve shows a downward curvature. On the other hand, the peak of AG parameter observed in AgBr arises mainly from the thermal expan sion coefficient which exhibit a large temperature dependence. Indeed, if the temperature dependence of the thermal expansion coefficient is ignored, the peaked behavior is no observed [20]. The result of the analysis mentioned, points that the increase of the anhar monicity which is quantified by the AG parameter has different components. In other words, although the different materials parameters are interrelated, the analysis provides

Temperature Dependence of the Anderson-Grüneisen Parameter
An understanding of the anharmonic effects and the temperature dependent thermomechanical properties is of primordial importance in ionic conducting materials. In Figure 1, it was shown that the elastic constants vary linearly at low temperatures. Such behavior is described by a temperature independent Anderson-Grüneisen (AG) parameter which is given by [70]: where α is the thermal expansion coefficient, B T is the isothermal bulk modulus, V is the volume, T is the temperature, and P is the pressure. The AG parameter has been introduced as an attempt to improve the description of the temperature dependent Young's modulus [70]. Nowadays, it is used as an important tool for investigating different aspects related to thermoelastic properties of solids. Although there are some studies reporting its temperature dependence [20], usually, the AG parameter is considered as a constant of the material or depends only weakly on temperature [71]. However, for the case of ionic conducting materials, we have found that the AG parameter depends strongly on temperature [20]. The result shown in Figure 5 indicates clearly the difference between the behavior of ion conducting materials AgCl, AgBr and CaF 2 and the other non-ion conducting materials KBr, Mg 2 SiO 2 and Ag. For the cases of AgBr and CaF 2 we observe a peak. Such kind of peaked behavior has been also observed in other ionic conducting materials investigated, PbF 2 and BaF 2 [20]. The peak of the AG parameter at certain temperature arises from the different relative contributions of temperature derivatives of C 11 , C 12 and the thermal expansion coefficient. For instance, for the case of fluorine compounds investigated, it was discussed that when the temperature dependence of C 11 dominates, the AG curve shows an upward curvature, whereas when the temperature dependences of C 12 and thermal expansion coefficient dominate, the AG curve shows a downward curvature. On the other hand, the peak of AG parameter observed in AgBr arises mainly from the thermal expansion coefficient which exhibit a large temperature dependence. Indeed, if the temperature dependence of the thermal expansion coefficient is ignored, the peaked behavior is not observed [20]. The result of the analysis mentioned, points that the increase of the anharmonicity which is quantified by the AG parameter has different components. In other words, although the different materials parameters are interrelated, the analysis provides a hint to design a material with enhanced ionic conductivity. In a previous section, it was pointed out that not all the elastic constants exhibit the same temperature dependence. This observation is also reflected in the behavior of the Anderson-Grüneisen parameter. An example is shown in Figure 6 for the case of ion conducting material AgBr. We note that and that are associated with and exhibit strong temperature dependence. On the other hand, the temperature dependence of associated with is weak. The result indicates again that and are more deeply involved than in the appearance of the ionic conductivity. Since the elastic constants can be understood in principle from the interatomic interaction point of view, the result provides an important clue to understand the microscopic mechanism of ion transport. For instance, in a previous report [20] it was pointed out that the scaled value of 300 K ⁄ 1.2 correlates with the value of Frenkel defect formation energy at temperature . The results shown above were focused on temperature dependence of elastic constants. As pointed elsewhere [40], the crystalline ionic conductors exhibit a structural phase transformation under the action of relatively low pressure. Experimental studies confirm this conjecture [63,72]. A reason for this behavior resides in the weak angular In a previous section, it was pointed out that not all the elastic constants exhibit the same temperature dependence. This observation is also reflected in the behavior of the Anderson-Grüneisen parameter. An example is shown in Figure 6 for the case of ion conducting material AgBr. We note that δ 11 and δ 12 that are associated with C 11 and C 12 exhibit strong temperature dependence. On the other hand, the temperature dependence of δ 44 associated with C 44 is weak. The result indicates again that C 11 and C 12 . are more deeply involved than C 44 in the appearance of the ionic conductivity. Since the elastic constants C ij can be understood in principle from the interatomic interaction point of view, the result provides an important clue to understand the microscopic mechanism of ion transport. For instance, in a previous report [20] it was pointed out that the scaled value of δ 11 (T R )/δ 11 (300 K) = 1.2 correlates with the value of Frenkel defect formation energy at temperature T R . In a previous section, it was pointed out that not all the elastic constants exhibit the same temperature dependence. This observation is also reflected in the behavior of the Anderson-Grüneisen parameter. An example is shown in Figure 6 for the case of ion conducting material AgBr. We note that and that are associated with and exhibit strong temperature dependence. On the other hand, the temperature dependence of associated with is weak. The result indicates again that and are more deeply involved than in the appearance of the ionic conductivity. Since the elastic constants can be understood in principle from the interatomic interaction point of view, the result provides an important clue to understand the microscopic mechanism of ion transport. For instance, in a previous report [20] it was pointed out that the scaled value of 300 K ⁄ 1.2 correlates with the value of Frenkel defect formation energy at temperature . The results shown above were focused on temperature dependence of elastic constants. As pointed elsewhere [40], the crystalline ionic conductors exhibit a structural phase transformation under the action of relatively low pressure. Experimental studies confirm this conjecture [63,72]. A reason for this behavior resides in the weak angular The results shown above were focused on temperature dependence of elastic constants. As pointed elsewhere [40], the crystalline ionic conductors exhibit a structural phase transformation under the action of relatively low pressure. Experimental studies confirm this conjecture [63,72]. A reason for this behavior resides in the weak angular force constant that the ionic conducting materials have. For the case of zinc-blend structure compounds, the elastic constants can be written in terms of bond stretching and bond bending force constants and Coulombic forces [73,74]. Typical ionic conductors such as γ-AgI and Cu halides have the zinc-blend type structure. The pressure dependence of the force constant of CuCl, CuBr and CuI indicated that the bond stretching force constant increases, whereas the bond bending force constant decreases with the increase of pressure [75]. These studies suggest that weakening of angular force constant is favorable for the ionic conduction. However, it is important to recognize that a complete absence of angular force constant is not favorable for ionic conduction. For the promotion of a correlated ionic motion, the presence of certain degree of angular force is indispensable. The bond fluctuation model mentioned above [40][41][42] provides the physical background to the discussion given here.

Grüneisen Parameter and Ionic Conductivity in Glasses
In conjunction with the AG parameter mentioned above, the Grüneisen parameter which describes the relationship between phonon frequency and volume, is a quantity widely used in the discussion of anharmonic lattice vibrations. The Grüneisen equation of state which includes the Grüneisen parameter is considered as one of the most valuable thermodynamic equation of state [76,77], because it links the phonon energy to the potential energy of the material. In the thermodynamic description, the Grüneisen parameter is written as: where α is the thermal expansion coefficient, B T is the isothermal bulk modulus, V is the molar volume, and C V is the molar heat capacity. Despite their importance, data of δ and γ in ionic conducting materials are quite limited [18,20,42,[78][79][80][81][82][83]. In view of this situation, we started a program to evaluate the Grüneisen parameter of ionic conductors. However, for the evaluation of γ we need to know the values of various quantities appearing in Equation (4), which is not easy to find. Therefore, developing a different way of evaluation is highly desirable. Some years ago, a method of evaluation of γ that uses only the sound velocity was proposed [84]. According to that work, the Grüneisen parameter is given by: where V L and V T are the longitudinal and transverse sound velocities, respectively. It has been shown that the evaluation based on Equation (5) shows a satisfactory agreement with the evaluation based on Grüneisen equation [84]. The expression given in Equation (5) is convenient in the analysis of mechanical properties of glassy materials, because for isotropic materials the sound velocities can be written as: and where B T , G and ρ are the bulk modulus, shear modulus and density, respectively. Figure 7 shows the temperature dependence of the Grüneisen parameter γ of ion conducting glasses xAg 2 S-(1 − x)AgPO 3 . We note that the Grüneisen parameter increases with the increase of temperature. As in the case of AG parameter mentioned above, usually, the Grüneisen parameter is considered as a constant or weakly temperature dependent quantity [77]. This type of behavior is illustrated in Figure 8, for the case of chalcogenide glasses that are not ionic conductors. The values of γ shown in Figures 7 and 8 were evaluated from sound velocity data reported in [81,85], following Equation (5). The comparison between the behavior shown in Figures 7 and 8 shows clearly the difference between the γ of ion-conducting and non-ion-conducting glasses.
Crystals 2021, 11, 1008 10 of 15 ent quantity [77]. This type of behavior is illustrated in Figure 8, for the case of chalcogenide glasses that are not ionic conductors. The values of shown in Figures 7 and 8 were evaluated from sound velocity data reported in [81,85], following Equation (5). The comparison between the behavior shown in Figures 7 and 8 shows clearly the difference between the of ion-conducting and non-ion-conducting glasses.  The glassy systems can be considered to consist from an agglomeration of structural units. On the other hand, as can be inferred from the discussion given in previous sections, the Grüneisen parameter of glassy materials contains information on the softness of the connectivity between the structural units. From this point of view, the result shown in Figure 8 indicates that the connectivity remains almost the same in the temperature range shown. On the other hand, the result shown in Figure 7 indicates that the connectivity varies largely with temperature. This way of understanding is in line with a recent model for the viscosity of glass forming liquids [86,87]. There, the viscous flow and the structural relaxation are considered to result from thermally activated bond-breaking and bondswitching processes. According to this model, the temperature dependence of the viscosity is described by the mean values of the bond strength and the coordination number, and their fluctuations of the structural units that form the melt. Within this model, it is envisaged that in ion conducting glassy materials, the fluctuation between the structural units plays an important role. Recently, some studies regarding the interrelation between ent quantity [77]. This type of behavior is illustrated in Figure 8, for the case of chalcogenide glasses that are not ionic conductors. The values of shown in Figures 7 and 8 were evaluated from sound velocity data reported in [81,85], following Equation (5). The comparison between the behavior shown in Figures 7 and 8 shows clearly the difference between the of ion-conducting and non-ion-conducting glasses.  The glassy systems can be considered to consist from an agglomeration of structural units. On the other hand, as can be inferred from the discussion given in previous sections, the Grüneisen parameter of glassy materials contains information on the softness of the connectivity between the structural units. From this point of view, the result shown in Figure 8 indicates that the connectivity remains almost the same in the temperature range shown. On the other hand, the result shown in Figure 7 indicates that the connectivity varies largely with temperature. This way of understanding is in line with a recent model for the viscosity of glass forming liquids [86,87]. There, the viscous flow and the structural relaxation are considered to result from thermally activated bond-breaking and bondswitching processes. According to this model, the temperature dependence of the viscosity is described by the mean values of the bond strength and the coordination number, and their fluctuations of the structural units that form the melt. Within this model, it is envisaged that in ion conducting glassy materials, the fluctuation between the structural units plays an important role. Recently, some studies regarding the interrelation between The glassy systems can be considered to consist from an agglomeration of structural units. On the other hand, as can be inferred from the discussion given in previous sections, the Grüneisen parameter of glassy materials contains information on the softness of the connectivity between the structural units. From this point of view, the result shown in Figure 8 indicates that the connectivity remains almost the same in the temperature range shown. On the other hand, the result shown in Figure 7 indicates that the connectivity varies largely with temperature. This way of understanding is in line with a recent model for the viscosity of glass forming liquids [86,87]. There, the viscous flow and the structural relaxation are considered to result from thermally activated bond-breaking and bondswitching processes. According to this model, the temperature dependence of the viscosity is described by the mean values of the bond strength and the coordination number, and their fluctuations of the structural units that form the melt. Within this model, it is envisaged that in ion conducting glassy materials, the fluctuation between the structural units plays an important role. Recently, some studies regarding the interrelation between the mechanical properties and ionic conductivity in mixed network former glasses [30] and gel polymer electrolytes [31] have been reported. It will be interesting to investigate if the model mentioned above [86,87] conforms with the experimental findings reported in [30,31].
In view of the result shown in Figure 2, and the discussion given above, it is of considerable interest to investigate the relation between the Grüneisen parameter and the ionic conductivity. A result for the case of xAg 2 S-(1 − x)AgPO 3 glasses is shown in Figure 9. Here, the ionic conductivities corresponding to low temperatures were evaluated by extrapolating the data reported in [88,89]. As expected, we observe that the ionic conductivity increases with the value of γ. As mentioned above, the observed behavior can be understood to arise from the weakening of the connectivity between the structural units that form the glass by increasing the temperature. However, it does not imply that a structural unit moves as whole. The structural unit forms a network and the ion that breaks the local constraint originates the ionic conduction. That is, the ionic conductivity results from the decoupling between the network and the mobile ions [86,87,90,91]. the mechanical properties and ionic conductivity in mixed network former glasses [30] and gel polymer electrolytes [31] have been reported. It will be interesting to investigate if the model mentioned above [86,87] conforms with the experimental findings reported in [30] and [31]. In view of the result shown in Figure 2, and the discussion given above, it is of considerable interest to investigate the relation between the Grüneisen parameter and the ionic conductivity. A result for the case of xAg2S-(1-x)AgPO3 glasses is shown in Figure 9. Here, the ionic conductivities corresponding to low temperatures were evaluated by extrapolating the data reported in [88,89]. As expected, we observe that the ionic conductivity increases with the value of . As mentioned above, the observed behavior can be understood to arise from the weakening of the connectivity between the structural units that form the glass by increasing the temperature. However, it does not imply that a structural unit moves as whole. The structural unit forms a network and the ion that breaks the local constraint originates the ionic conduction. That is, the ionic conductivity results from the decoupling between the network and the mobile ions [86,87,90,91]. In this report we have focused in to show how the AG parameter and the Grüneisen parameter of glassy materials are related with the ionic conduction. The elastic properties of glasses such as the Young's modulus and Poisson's ratio allows to get insight into the short and medium range orders existing in glasses [92]. Regarding the role of medium range structure on ionic transport, it has been shown that in AgI-containing superionic glasses, the activation energy of ion transport decreases with the decreases in the wave number of first-sharp-diffraction-peak (FSDP) [93]. The FSDP reflects the presence of an intermediate range ordering within the glass.

Conclusions
In the present report, it was shown that the temperature dependence of the elastic constants in ionic conductors are intimately related with the temperature dependence of the ionic conductivity. The analysis of the temperature dependence of elastic constants reveals that in cubic materials such as PbF2 and AgBr, the force constants associated with and are related strongly with the atomic transport properties. It was also shown that the elastic anisotropy is intimately related with the defect creation and ionic transport property of the materials. Data of Grüneisen and Anderson-Grüneisen parameters were presented. Despite these quantities reflect the degree of anharmonicity of lattice vibrations, they are rarely reported in the literature. Our result indicates that in ion conducting materials, the temperature dependence of the Grüneisen and Anderson-Grüneisen parameters are large. Result in ion conducting glasses revealed that the ionic conductivity In this report we have focused in to show how the AG parameter and the Grüneisen parameter of glassy materials are related with the ionic conduction. The elastic properties of glasses such as the Young's modulus and Poisson's ratio allows to get insight into the short and medium range orders existing in glasses [92]. Regarding the role of medium range structure on ionic transport, it has been shown that in AgI-containing superionic glasses, the activation energy of ion transport decreases with the decreases in the wave number of first-sharp-diffraction-peak (FSDP) [93]. The FSDP reflects the presence of an intermediate range ordering within the glass.

Conclusions
In the present report, it was shown that the temperature dependence of the elastic constants in ionic conductors are intimately related with the temperature dependence of the ionic conductivity. The analysis of the temperature dependence of elastic constants reveals that in cubic materials such as PbF 2 and AgBr, the force constants associated with C 11 and C 12 are related strongly with the atomic transport properties. It was also shown that the elastic anisotropy is intimately related with the defect creation and ionic transport property of the materials. Data of Grüneisen and Anderson-Grüneisen parameters were presented. Despite these quantities reflect the degree of anharmonicity of lattice vibrations, they are rarely reported in the literature. Our result indicates that in ion conducting materials, the temperature dependence of the Grüneisen and Anderson-Grüneisen parameters are large. Result in ion conducting glasses revealed that the ionic conductivity increases with the increase of Grüneisen parameter. All these observations indicate that studies of mechanical properties provide much insight to understand the materials properties of ionic conductors. However, for its deep understanding further studies on diverse materials is necessary. In the present report, we have concentrated in to present results on