Ionic transport mechanisms in oxide based glasses in the supercooled and glassy states

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1 Solid State Ionics 05 (998) 37 4 Ionic transport mechanisms in oxide based glasses in the supercooled and glassy states * Jean Louis Souquet, Michel Duclot, Michel Levy Laboratoire d Electrochimie et de Physicochimie des Materiaux et des Interfaces (UMR CNRS INPG No. 563, associee a` l Universite Joseph Fourier Grenoble), ENSEEG, BP 75, F3840 Saint Martin d Heres, ` France Abstract A model for ionic transport in glasses is proposed assuming that, like for ionic crystals, the displacement of cations would result from the migration of interstitial pairs. The formation of these pairs is supposed to be an activated function of the temperature and their migration to be dominated by an activated jump mechanism at low temperature and by a free volume co-operative mechanism above the vitreous temperature. Using numerical fits for conductivity data in the whole temperature range, that change of mechanism enables the determination of the defect formation enthalpy and its migration enthalpy in the low temperature domain. Keywords: Oxide glasses; Ion conductivity; Vitreous transition temperature. Introduction molten silicates [ 3]: For glass forming mixtures in the solid or super- E s? T 5 A exp ]]] cooled liquid state, ionic transport due to alkali R(TT 0) () cations strongly depends on temperature, T. The variations of the conductivity temperature product Such a behaviour for sodium and potassium s? T in an Arrhenius representation show two disilicates is illustrated in Fig.. distinct behaviours. At the lowest temperatures, that The purpose of this paper is to develop a microproduct follows an activated relationship: scopic model which could ascribe a physical mean- ing to the different parameters (A and E in Eq. () E and A, E and T0 in Eq. ()). Predictable values s?t5a exps ]D. () RT obtained using this model will be then compared to those resulting from experimental data. At higher temperatures, experimental data obey an empirical V.T.F. relationship originally established to describe the viscosity temperature dependence of S. Ionic transport below the glass transition temperature (T ) g * Corresponding author. Tel.: ; fax: In this temperature range, and for all ionic conduc tive glasses, the representation of experimental data / 98/ $ Elsevier Science B.V. All rights reserved. PII S (97) D

2 38 J.L. Souquet et al. / Solid State Ionics 05 (998) 37 4 Fig.. Schematic representation of ionic conductivity dependence on temperature below T. g cationic site. Nevertheless, thermal vibrations allow some partial dissociation which leads to the formation of point defects as it is the case in crystalline structures. When a cation leaves its normal posi- tion, it will be in what could be considered an interstitial position and its previously occupied site will become a vacancy. This defect formation in the glass structure is equivalent to the formation of a Frenkel type defect in an ionic crystal. In oxide Fig.. Ionic conductivity as a function of temperature for sodium and potassium disilicates [4]. of the conductivity temperature product, s?t, in Arrhenius coordinates results in straight lines whose extrapolation towards infinite temperature converges based glasses for which alkali concentration, C, is to corresponding values of the pre-exponential term over 0% atom, normal sites are close together (4 4 5 lying between 0 and 0 S cm K. On the other A to 5 A) and a leaving cation is now expected to hand, room temperature conductivities spread over 8 share a neighbouring non bridging oxygen with orders of magnitude only due to the variation of the another cation. This combination of two cations conductivity activation energy (0.5 ev#e # ev) surrounding a non bridging oxygen can be described depending upon the nature and the concentration of as an interstitial pair defect. the added alkali cation. This is schematically illus- The concentration of these interstitial pairs, C, trated in Fig.. may be very small compared with the total con- For cationic glasses and regardless of the conduc- centration of alkali cations, C. In other words, tion mechanism, electrical transport can be expressed glasses are weak electrolytes. In that case, the as the product of three terms, the charge carriers chemical equilibrium between alkali cations in reguconcentration, C, their electrical mobility, m and lar sites and interstitial positions leads to the followtheir charge, F: ing relationship: s5 FCm (3) DG S]]D f C 5 C exp RT (4) Since the relative dielectric constant r of inor- ganic glasses is low (5# r#5) ionic species are where DGf5DHfTDSf is the free energy associ- strongly associated. For instance most monovalent ated with the simultaneous formation of an interstications will be associated with non bridging oxygen tial pair and a cationic vacancy. The mobility, m atoms. Such an associated cation should be regarded of the charge carrier (the interstitial cation) in the as in a normal position, hence defining a regular electric field is expressed by the general relationship:

3 J.L. Souquet et al. / Solid State Ionics 05 (998) F l calculate a theoretical value for the pre-exponential m 5]] n0p (5) term, A. Assuming an homogeneous distribution of RT 6 the alkali cations through the glass, the jump distance, l, is the mean distance between two non- In this expression, l is the mean distance between two cationic sites, n the attempt frequency and P is bridging oxygen atoms and is related to the total 0 /3 the probability of a successful jump. alkali concentration [l 5 (/C) ]. For classical At low temperatures, below T, this probability P oxide glasses, this distance typically varies from 4 A g is only a function of the migration energy DH, i.e., to 5 A. The attempt frequency, n 0, may be obtained m the necessary energy for an interstitial cation to jump from the far-infrared absorption which occurs be- to a neighbouring position: tween 500 cm and 00 cm in oxide glasses [5]. 3 Corresponding values of n0 are around 0 Hz. DH m P 5 exp S]]]D (6) Assuming a low value for the entropy term, DS f, RT reasonable values of the pre-exponential term, A, lie 4 5 Using Eqs. (3) (6), the cationic conductivity in the in the 0 0 (S cm K) range in remarkable low temperature range is given by: agreement with most experimental data [6]. For oxide glasses, the activation energies E range S DH D f from 0.5 ev to ev. From the model (Eq. (9)), this F Cl n DS ]] DH 0 f m parameter is the sum of two enthalpy terms. Contrary s? T 5]]] exps]d exp ]]]]] 6R R RT to what can be done for ionic conductive materials (7) presenting temperature-dependent intrinsic and extrinsic domains, the migration and formation en- This equation may be identified with an Arrhenius thalpies cannot be separated without further experitype law (Eq. ()) with: ments and assumptions. F Cl n0 DSf ]]] S]D A 5 exp (8) 6R R 3. Ionic transport over the glass transition and temperature DH f E 5]] DH m. (9) At high temperature, that is above T g, in addition to the low temperature process, another displacement Eq. (7) is the usual expression proposed for the mechanism is observed, which is schematised in Fig. intrinsic ionic conductivity temperature dependence 4a and 4b. Local deformations of the macromolecuin ionic crystals. The successive steps of a cationic lar chains enable the transfer of the defect to a displacement according to this description are illus- neighbouring position. This deformation requires trated in Fig. 3. local fluctuations of the free volume allocated to the Eq. (8) interestingly gives the opportunity to chain segments [7]. Let us define V * as the minif Fig. 3. Representation of a cationic displacement in a glass below the vitreous transition temperature. (a) The local perfect structure. (b) Interstitial pair formation. (c) Interstitial pair migration.

4 40 J.L. Souquet et al. / Solid State Ionics 05 (998) 37 4 temperature depends on the relative values of P and P. According to Eq. (), a free volume migration mechanism is possible as soon as the temperature reaches T. Nevertheless, it is detectable only for 0 higher temperatures because the low temperature activated mechanism prevails below a particular temperature, T, which can be seen as the electrical g vitreous temperature. It is to say that, in the Fig. 4. Representation of an interstitial pair migration assisted by a particular temperature range (T 0,T,T g), P relocal movement of the macromolecular network (a and b). mains very small compared to P, and the mobility expression (Eq. (4)) is reduced to Eq. (5). Hence, mum free volume required in order that the chain the conductivity expression is the same as the movement results in a defect displacement. Hence, activated form proposed in Eq. (7). On the contrary, the probability, P, that this minimum free volume is over T g, P becomes preponderant and the conreached, is: ductivity expression now reads: D S Vf 0 f 0 V * DH V* ] f f f P5expS (0) s? T 5 A exps ]]D exp ]]]] RT V a (T T ) where V is the mean free volume allocated to a (5) f chain segment. Its temperature dependence can be or expressed by the following expression: DHf B V 5V a (TT ) () s? T 5 A exps ]]D exps ]]] D. f 0 f 0 RT R(T T 0) where af is the thermal expansion coefficient of the (6) free volume. P can then be rewritten as: This expression of s?t, is the product of two V * f exponential terms where the first is representative of P 5 exps ]]]] D. () V a (T T ) an activated mechanism and where the second 0 f 0 describes a V.T.F. behaviour. This kind of relation- This expression confirms that this second mecha- ship has already been proposed to improve the nism only appears above the ideal vitreous transition viscosity temperature fits for BO3 or alkali silicate temperature, T 0, at which the free volume disap- above their vitreous transition temperature [8]. We pears. Above this temperature, the cationic displace- may also note that the pre-exponential term in this ment may occur either by an activated jump (P )or relationship contains the same microscopic characby an entropic free volume mechanism (P ). As teristic parameters as in Eq. (8). It means that the these mechanisms are exclusive, the total probability limit of the conductivity temperature product when of a successful displacement reads: P 5 P P PP 5P P( P ) 5P P ( P ). (3) the temperature tends to infinity does not depend on the nature of the displacement mechanism. 4. Numerical fits and discussion Under such conditions, the cationic mobility becomes: As shown in Fig. 5, this approach has been used to F l fit experimental data for a silver based glass m 5]] n fp P PP g. (4) RT 6 0 (0.7AgI 0.3AgMoO 4) for which a full set of conductivity measurements in a wide temperature range, The variation of this mobility as a function of below and above its vitreous transition temperature, D

5 J.L. Souquet et al. / Solid State Ionics 05 (998) pre-exponential factor A (A and A for respectively, the low and high temperature data) in agreement with theoretical values, confirming that this term should have the same physical meaning in the full temperature range. From the fit of the high temperature data, we can gain access to values of the parameters defined in Eq. (6), i.e. A, B, T0 and particularly DH f, which can now be obtained independently. Assuming that the formation of the defect does not depend on the nature of the displacement mechanism, the knowledge of the formation enthalpy, supposed to be the same below and above T g, enables the determination of the migration enthalpy in the low temperature domain. The corresponding values have been added in the rightmost column in Table. We may note that the respective orders of magnitude of the two enthalpies are reasonably in agreement with those generally observed in crystalline materials presenting extrinsic and intrinsic conductivity domains which allow, in that case, the separation of the two contributions. The best agreement between the low and high Fig. 5. Variations of the s?t product as a function of temperature temperature fits for the preexponential term is obin Arrhenius coordinates for AgI (0.7) AgMoO 4(0.3). Dashed curves tained for the 0.7AgI 0.3AgMoO4 data which were represent the extrapolations of the fits to Eqs. () and (6) for conductivity data from [9]. obtained from conductivity measurements carried out on the same sample in the whole temperature range. has been published [9]. We have also calculated the The presented conductivity results for disilicates values of the parameters involved in Eqs. () and were obtained from two sets of measurements re- (6) for our own conductivity measurements carried spectively performed on two different samples at low out on sodium and potassium disilicates. The results and high temperature. That may explain the slight are reported in Table. Fit to Eq. () is obtained by discrepancy between the two calculated preexponena simple linear regression while fit to Eq. (6) results tial terms in that case. from a least square analysis procedure in which all We may now emphasize that Eq. (6) contains the four parameters (A, B, T0 and DH f) have no a priori usual V.T.F. contribution (Eq. ()). The multiplying fixed values. term takes into account the formation of defects. In these three examples, the low and the high Fitting conductivity data exclusively using Eq. (), temperature fits result in comparable values for the i.e., assuming DHf50, is also possible. The results Table Numerical values resulting from the fits of conductivity data to Eq. () at low temperature and to Eq. (6) for high temperature T,T T.T DH (ev) g g m log A E (ev) log A DH (ev) B (ev) T (K) 0 0 f 0 NaO SiO KO SiO AgI 0.3AgMoO The charge carrier migration enthalpy is obtained by DH 5E (DH )/. m f

6 4 J.L. Souquet et al. / Solid State Ionics 05 (998) 37 4 Table our model because it includes a fraction of DHf when Results of the numerical identification to Eq. () for the different Eq. () is used instead of Eq. (6). parameters log A B (ev) T (K) 0 0 NaO SiO References KO SiO AgI 0.3 AgMoO [] H. Vogel, Phys. Z. (9) 645. [] G. Tamman, W. Hesse, Z. Anorg Allg. Chem. 56 (96) 45. of such numerical identifications to the pure V.T.F. [3] G.S. Fulcher, J. Am. Ceram. Soc. 8 (95) 339. equation are reported in Table. [4] E. Caillot, M. Duclot, J.L. Souquet, F.G.K. Baucke, R.D. We may see that, in that last fit, the pre-exponen- Werner, Phys. Chem. Glasses 35 (994). tial terms for the disilicate glasses are lower than the [5] E.I. Kamitsos, M.A. Karakassides, G.D. Chryssikos, J. Phys. values extrapolated from the Arrhenius relationship Chem. 9 (987) [6] G.D.L.K. Jayasinghe, D. Coppo, P.W.S.K. Bandarayake, J.L. (A ) in the low temperature domain. The ideal Souquet, Solid State Ionics 76 (995) 97. vitreous transition temperatures, T 0, are also much [7] M.H. Cohen, D. Turnbull, Chem. Phys. (959) 64. lower than those which could be expected from the [8] P.B. Macedo, T.A. Litovitz, J. Chem. Phys. 4 (965) 45. empirical rule T T / [0], which are respectively, [9] J. Kawamura, M. Shimoji, J. Non-Cryst. Solids 88 (986) 0 m 574 K and 659 K for the sodium and potassium 95. [0] E. Caillot, M. Duclot, J.L. Souquet, C.R. Acad. Sc. Paris 3 disilicate. Correspondingly, the free-volume parame- (Serie II) (99) 447. ter is much higher than that obtained from the fit to