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Molar volume, glass-transition temperature, and ionic conductivity of Na- and Rb-borate glasses in comparison with mixed Na–Rb borate glasses
Journal of Non-Crystalline Solids 351 (2005) 3816–3825 www.elsevier.com/locate/jnoncrysol
Molar volume, glass-transition temperature, and ionic conductivity of Na- and Rb-borate glasses in comparison with mixed Na–Rb borate glasses Frank Berkemeier a
a,*
, Stephan Voss
´ rpa´d W. Imre a, Helmut Mehrer ,A
a,b
a
Institut fu¨r Materialphysik, Universita¨t Mu¨nster, Wilhelm-Klemm Straße 10, D-48149 Mu¨nster, Sonderforschungsbereich 458, Germany b Infineon Technologies AG, P.O. Box 80 09 49, 81609 Munich, Germany Received 18 March 2005; received in revised form 5 October 2005 Available online 18 November 2005
Abstract Mass densities, molar volumes, glass-transition temperatures, and ionic conductivities are measured in series of YNa2O Æ (1 Y)B2O3 glasses, with Y = 0.00, 0.04, 0.08, 0.12, 0.16, 0.20, 0.25, 0.30 and YRb2O Æ (1 Y)B2O3 glasses, with Y = 0.00, 0.12, 0.16, 0.20, 0.25, 0.30. Measurements of the molar volumes indicate that the incorporation of rubidium ions leads to a considerable expansion of the network, which is not observed for sodium ions. The glass-transition temperature increases with increasing alkali content and reaches a maximum near Y = 0.25 for both glass systems. These trends are attributed to changes in the glass network. For each glass composition an Arrhenius-activated increase of the product of dc conductivity and temperature is observed. The activation enthalpy decreases with increasing number density of ions. A comparison between the binary sodium- and rubidium-borate glasses from this work, with the ternary sodium–rubidium borate glasses studied earlier in our laboratory, provides interesting insights in the influence of the glass structure on ionic transport processes and the mixed-alkali effect. 2005 Elsevier B.V. All rights reserved. PACS: 61.43.Fs; 66.10.Ed
1. Introduction Network glasses represent a special class of disordered materials and are categorised by the types of network formers: e.g. silicate, germanate, phosphate, aluminate, or borate glasses. The ionic conductivity of these glasses is due to the addition of so-called network modifiers, such as alkali oxides (A2O). The motion of mobile ions in materials with disordered structures is one of the basic questions of solid-state ionics. From the viewpoint of application, ion-conducting materials have high technological potential
*
Corresponding author. Tel.: +49 (0) 251 8333587; fax: +49 (0) 251 8338346. E-mail address: [email protected] (F. Berkemeier). 0022-3093/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2005.10.010
for electrochemical devices such as solid–oxide fuel cells, solid-state batteries, or chemical sensors [1]. In this paper we consider alkali-borate glasses and for this purpose we remind the reader of some structural features of these glasses: adding alkali oxide (A2O) to boron oxide changes the coordination number of some boron atoms from three to four and leads to a transformation of trigonal BO3/2 units into negatively charged tetrahedral BO 4=2 units. Both BO3/2 and BO4=2 units are connected by bridging-oxygen atoms and represent the smallest structural units of the glass network. The formation of two boron–oxygen tetrahedra consumes the additional oxygen provided by the alkali oxide A2O. Since each tetrahedron is charge-deficient by minus one elementary charge, two alkali ions compensate two anionic BO 4=2 units. An increase in A2O concentration results in a further transfor-
F. Berkemeier et al. / Journal of Non-Crystalline Solids 351 (2005) 3816–3825
mation of boron from three- to four-fold coordination. Beyond a concentration of about 25–30 mol% alkali oxide content the formation of non-bridging oxygen atoms becomes a relevant process [2,3]. The alkali ions are largely decoupled from the mostly static boron-oxide network and are mobile even far below the glass-transition temperature. Several models have been developed to describe the ionic motion in network glasses (e.g. [4,5]). These models treat many features of ionic motion, but nevertheless, a general model which explains all known experimental phenomena is still missing and further experimental work is needed. In this paper we present a systematic study of mass density, molar volume, glass-transition temperature, and ionic transport in binary sodium-borate and rubidium-borate glasses of various alkali concentrations. Ionic transport in mixed sodium–rubidium borate glasses was investigated recently in our laboratory [6,7], and in the discussion we compare single-alkali and mixed-alkali borate glasses. 2. Experimental details 2.1. Glass preparation We prepared a series of binary YNa2O Æ (1 Y)B2O3 glasses, with Y = 0.04, 0.08, 0.12, 0.16, 0.20, 0.25, 0.30 and binary YRb2O Æ (1 Y)B2O3 glasses, with Y = 0.12, 0.16, 0.20, 0.25, 0.30.1 A pure B2O3 glass was prepared as well. Appropriate amounts of dried powders of sodium carbonate (Merck, 99.999%) and diboron trioxide (Merck, 99.9995%) for the sodium-borate glasses, and rubidium carbonate (Alfa Aesar, 99.975%) and diboron trioxide for the rubidium-borate glasses were mixed and melted in a platinum crucible at a temperature of usually 1273 K. During heating the alkali carbonates decompose into alkali oxide and gaseous CO2, which is released from the melt. To avoid inhomogeneities in the glass, the melt was kept at 1273 K for 3 h, swivelled several times, and subsequently poured into a cylindrical carbon mould (diameter = 10 mm, height = 30 mm) kept at room temperature. For alkali contents of Y > 0.2 the melting temperature was reduced to about 1150 K and a stainless-steel mould, pre-heated to about 573 K, was used, to avoid the bursting of the glass cylinders (as a consequence of internal stresses). After the preparation the glasses were heat-treated at a temperature of about 20 K below their calorimetric glasstransition temperature for 5 h and then slowly cooled down to room temperature with 6 K/h to remove internal stresses.
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2.2. Glass characterisation In earlier work of Schoo et al. [8] borate glasses also prepared in our laboratory were analyzed by the analytic laboratory of Schott AG (Mainz). Deviations from the batch compositions were less than 2.5%. In addition we compared dc-conductivity and glass-transition temperature of two glasses prepared independently with the same nominal composition. Both deviate from each other less than 10%. We also determined the ratio of sodium and rubidium in mixed-alkali glasses by microprobe analysis and found an agreement within the accuracy of the analysis in comparison with the batch composition. Since boron and oxygen cannot be measured in our microprobe the compositions of the binary borate glasses were not determined. Since water can influence the physical properties, in particular the mobility of ions, in hygroscopic glasses [9], we checked the concentration of hydrogen in the glasses by means of NMR. The NMR experiments do not reveal any hydrogen contamination of the samples indicating the absence of water [10,11]. To check the non-crystallinity of our glass samples X-ray measurements were performed using a Siemens D5000 X-ray diffractometer. No sharp Bragg peaks were found, indicating the absence of crystalline phases (for details see [12]). The distribution of the alkali ions and of BO3/2 and BO 4=2 units of the network of our glasses was investigated by nuclear-magnetic resonance (NMR) experiments at the Institute of Physical Chemistry of our University, using the 23Na spin echo and the 11B magic-angle spinning techniques. The results suggest an almost homogeneous distribution of alkali ions and of BO3/2 and BO 4=2 units in the glasses [13,10]. The so-called N4-value is defined as N4 ¼
cðBO 4=2 Þ cðBO3=2 Þ þ cðBO 4=2 Þ
.
ð2:1Þ
Here cðBO 4=2 Þ denotes the concentration of BO4=2 units and c(BO3/2) the concentration of BO3/2 units. The N4value can be deduced from NMR experiments (see, e.g. [3]). For our glasses the measured N4-value coincidences within the experimental errors with the value expected from the molar fractions of A2O and B2O3 given by Y/(1 Y). The densities of the glasses were determined by an Archimedes method. For this purpose the mass of a glass sample mglass and the mass of a piece of semiconductorgrade silicon mSi – as a reference for calibration – was measured in air and in n-dodecane. Then the density of the glass is obtained from air . mair mSi mdod glass Si q ¼ qair þ air ðqSi qair Þ ; ð2:2Þ . mglass mdod mair glass Si
1
Rubidium–borate glasses with Y = 0.04 and Y = 0.08 where not prepared because the dc conductivities of those glasses are below the lower limit of our impedance analyser and cannot be measured.
where qair and qSi are the densities of air and silicon, respectively.
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The calorimetric glass-transition temperatures of the samples were measured using differential-scanning calorimetry (DSC). A small piece of each glass was crushed in an agate mortar, dried at 383 K for 24 h, and encapsulated into a copper container. The measurements were performed using a Perkin–Elmer DSC 7 device employing several heating rates between 5 and 100 K/min. A typical DSC thermogram of these glasses contains three regions, which belong to three different states: the glassy state at low temperatures, the undercooled liquid state at intermediate temperatures, and the crystalline state at high temperatures. The corresponding calorimetric glass-transition temperature Tg is defined as the mid-point temperature of the extensions of the pre- and the post-transition baselines of the transition from glassy- to undercooled liquid-state. 2.3. Conductivity measurements The conductivity measurements were performed by impedance spectroscopy. Disks of 2 mm thickness cut from the glass cylinders served as samples. Their two flat surfaces were ground parallel and the sample diameter was reduced to 9 mm by ultrasonic drilling. In the next step the surfaces were polished with 6 lm diamond paste and finally metallic contacts, consisting of a silver bottom and a platinum top layer, were deposited on both surfaces using a sputter coater (Quorum Technologies SC 7640). Frequency-dependent conductivities were measured in a frequency range between 5 Hz and 1.3 MHz using a HP Agilent 4192A LF impedance analyser. The glass samples
were fixed between two silver electrodes to provide a good electrical contact. The connecting wires were arranged in a four-point probe configuration to exclude an influence of the wire resistance on the measurement. The sample holder was placed inside a quartz-glass tube filled with argon. A horizontal furnace (Gero SR 70–500/11) and a calibrated Ni–NiCr thermocouple were used to control the temperature of the sample. In a measuring cycle, which was controlled by a computer using the Novocontrol WinDETA 4.0 software, the samples were first heated up to a temperature about 20 K below their glass-transition temperature and subsequently cooled down with 6 K/h to reduce internal stresses introduced during sample preparation. Subsequently, the ac-conductivities were measured at a series of pre-defined temperatures by heating up the samples with 6 K/h to a maximum temperature of about 20 K below Tg. Since one frequency scan takes only about one minute each measurement can be regarded as isothermal. 3. Results 3.1. Density and glass transition The measured densities q of the sodium- and the rubidium-borate glasses are listed in Tables 1 and 2. In Fig. 1 these values are shown as a function of the total alkali concentration Y. The mass density increases in a non-linear way with increasing alkali content and the rubidium-borate glasses show higher densities than the sodium-borate glasses of the same alkali concentration.
Table 1 Densities q, molar volumes Vm, molar volumes normalised to the number of boron atoms V Bm , number densities of the sodium ions N, activation enthalpies DH, and pre-exponential factors C0 of the dc conductivity for the sodium-borate glasses Composition Y
q (g cm3)
Vm (cm3 mol1)
V Bm (cm3 mol1)
N (m3)
0.30
2.37 ± 0.01
28.4 ± 0.1
20.3 ± 0.1
1.27 · 1028
78.2 ± 0.2
5 ð1:07þ0:04 0:04 Þ 10
19.7 ± 0.1
28
87.5 ± 0.2
4 ð1:99þ0:92 0:88 Þ 10
27
98.7 ± 0.4
5 ð1:10þ0:10 0:10 Þ 10
27
114.8 ± 0.5
5 ð1:82þ0:19 0:17 Þ 10
27
128.5 ± 2.7
4 ð1:72þ1:22 0:71 Þ 10
27
143.9 ± 2.3
1:67 ð2:420:99 Þ 105
27
194.8 ± 8.4 –
5 ð3:70þ1:98 0:31 Þ 10 –
0.25 0.20 0.16 0.12 0.08 0.04 0.00
2.30 ± 0.01 2.23 ± 0.01 2.15 ± 0.01 2.09 ± 0.01 2.03 ± 0.01 1.95 ± 0.01 1.85 ± 0.01
29.5 ± 0.2 30.6 ± 0.2 31.8 ± 0.2 32.8 ± 0.2 34.0 ± 0.2 35.6 ± 0.2 37.7 ± 0.2
19.1 ± 0.1 18.9 ± 0.1 18.7 ± 0.1 18.5 ± 0.1 18.5 ± 0.1 18.8 ± 0.1
DH (kJ mol1)
1.02 · 10 7.87 · 10 6.07 · 10 4.40 · 10 2.83 · 10 1.35 · 10 0
C0 (X1 cm1 K)
Table 2 Densities q, molar volumes Vm, molar volumes normalised to the number of boron atoms V Bm , number densities of the sodium ions N, activation enthalpies DH, and pre-exponential factors C0 of the dc conductivity for the rubidium-borate glasses Composition Y
q (g cm3)
Vm (cm3 mol1)
V Bm (cm3 mol1)
N (m3)
0.30
2.93 ± 0.01
35.8 ± 0.2
25.6 ± 0.1
1.01 · 1028
84.4 ± 0.3
4 ð6:11þ0:44 0:41 Þ 10
24.1 ± 0.1
27
94.8 ± 0.3
4 ð6:30þ4:07 3:82 Þ 10
27
0.25
2.74 ± 0.01
36.1 ± 0.2
DH (kJ mol1)
8.35 · 10
C0 (X1 cm1 K)
0.20
2.54 ± 0.01
36.6 ± 0.2
22.9 ± 0.1
6.58 · 10
102.5 ± 0.4
4 ð2:70þ0:20 0:20 Þ 10
0.16
2.44 ± 0.01
36.3 ± 0.2
21.6 ± 0.1
5.31 · 1027
127.0 ± 0.9
4 ð7:20þ1:58 1:30 Þ 10
20.4 ± 0.1 18.8 ± 0.1
27
145.2 ± 3.8 –
4 ð3:85þ5:09 2:19 Þ 10 –
0.12 0.00
2.33 ± 0.01 1.85 ± 0.01
35.9 ± 0.2 37.7 ± 0.2
4.03 · 10 0
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Fig. 1. Densities of sodium- and rubidium-borate glasses as functions of the glass composition Y. The solid lines are fits with polynomials of third order to guide the eye.
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Fig. 3. Conductivity spectra of the 0.16Na2O Æ 0.84B2O3 glass. Only some selected isotherms are shown.
sodium-borate glass 0.16Na2O Æ 0.84B2O3. Isotherms of r 0 (m) are plotted as a function of the frequency m in a log–log representation for several temperatures. The spectra exhibit the following salient features:
Fig. 2. Calorimetric glass-transition temperature Tg of both binary alkaliborate glass systems as functions of the glass composition Y measured at a heating rate of 20 K/min. Both solid lines are fits with polynomials of third order to guide the eye.
The glass-transition temperature Tg as a function of the glass composition Y is shown in Fig. 2 for both binary glass systems. For each system Tg increases with increasing alkali content and reaches a shallow maximum near Y = 0.25. For borate glasses many of the property/composition trends reverse their directions with increasing alkali content. For example, a minimum in the thermal expansion coefficient and a maximum in Tg occurring at higher alkali oxide contents are observed. This behavior, which is also observed in our Tg-measurements, is denoted as the borate anomaly [3].
• a conductivity plateau with a frequency-independent range of the conductivity r 0 at sufficiently low frequencies. The plateau represents the long-range transport of the ions. Its mean value is taken in the following as the dc conductivity rdc. The product of this dc conductivity and temperature is Arrhenius activated. • a conductivity-dispersion region at higher frequencies in which the conductivity r 0 increases monotonically with increasing frequency. There are several approaches in the literature to explain this behavior (e.g. [14–17]). One possible explanation is given by Funke [18] who attributes the dispersion region to correlated forward–backward jumps of the ions. For a pure random walk of the ions without any correlations dispersion would be absent. In Fig. 4 the dc conductivity of both glass systems is plotted as a function of the total alkali concentration Y at a fixed temperature of 573 K. For both systems a
3.2. Ion dynamics To investigate the ion dynamics we measured the complex conductivities r of our glass series as functions of frequency m. At low frequencies the real part of the conductivity, r 0 , yields the dc conductivity rdc. Via the Nernst–Einstein equation rdc is connected to a charge-diffusion coefficient of the mobile ions (see below). Fig. 3 gives examples of typical conductivity spectra of the
Fig. 4. dc conductivities rdc of sodium- and rubidium-borate glasses as functions of the glass composition Y at 573 K. Filled circles: this work; open circles: work of Hunter and Ingram [21]. The solid lines are fits with polynomials of third order to guide the eye.
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significant increase of the dc conductivity with increasing alkali concentration is observed. This increase in conductivity covers almost six orders of magnitude (see also Tables 1 and 2). Similar conductivity results have been reported for other glass systems, too [19,20]. It can also be seen in Fig. 4 that at a fixed alkali concentration the dc conductivity of a sodium-borate glass is between one and two orders of magnitude higher than that of the corresponding rubidium-borate glass, indicating that the smaller sodium ions move faster than the larger rubidium ions. Hunter and Ingram also investigated sodium-borate glasses by means of conductivity measurements [21]. Their conductivity values are also presented in Fig. 4 and show good agreement with the present results. The measured ionic dc conductivity, rdc, can be related to a charge-diffusion coefficient of the ions, Dr, via the Nernst–Einstein equation rdc k B T . ð3:3Þ Dr ¼ q2 N Here kB is the Boltzmann constant, T the absolute temperature, q the charge of the ions (+1 elementary charge for alkali ions), and N the number density of ions. To calculate Dr, we assume that N equals the total number density in Tables 1 and 2, calculated from the mass density q, the molar mass M, and the alkali content Y of the glass via N = 2 Æ Y Æ NA Æ q/M, where NA is the Avogadro constant. In crystalline solids and in glasses below Tg the Fickian diffusion coefficient D exhibits Arrhenius-activated behavior D = D0 Æ exp(DH/(kB Æ T)), with an activation enthalpy DH and a pre-exponential factor D0. Assuming that Dr follows Arrhenius behavior Dr = Dr,0 Æ exp(DH/(kB Æ T)), we expect for the temperature dependence of the product rdc Æ T DH rdc T ¼ C 0 exp ; ð3:4Þ kB T with Dr;0 q2 N . ð3:5Þ C0 ¼ kB In Figs. 5 and 6 the Arrhenius diagrams of rdc Æ T of both binary glass systems are presented. The measured conductivity data can well be approximated by Eq. (3.4) and thus allow the determination of an activation enthalpy DH and a pre-exponential factor C0 for each glass composition.2 4. Discussion 4.1. Density and glass transition To get more information about the glass network we determined the molar volumes Vm of the glasses, which 2
All Arrhenius diagrams show a slight deviation from a linear behavior. In what follows, we do not consider this slight upward curvatures, which might be due to the migration of the ions in an energy-landscape [6].
Fig. 5. Arrhenius diagrams of rdc Æ T for sodium-borate glasses YNa2O Æ (1 Y)B2O3. (a) Y = 0.30, (b) Y = 0.25, (c) Y = 0.20, (d) Y = 0.16, (e) Y = 0.12, (f) Y = 0.08, (g) Y = 0.04.
Fig. 6. Arrhenius diagrams of rdc Æ T for rubidium-borate glasses YRb2O Æ (1 Y)B2O3. (a) Y = 0.30, (b) Y = 0.25, (c) Y = 0.20, (d) Y = 0.16, (e) Y = 0.12.
are obtained from the density measurements by Vm = M/q. M denotes the molar mass of the glass which is calculated from the molar masses of diboron-trioxide M B2 O3 and the respective alkali oxide MA2O via M ¼ Y M A2 O þ ð1 Y Þ M B2 O3 . Because the total alkali content influences both, M and q, Vm does not give specific information about the glass network. Since the boron atoms are the central atoms of the BO3/2 and BO 4=2 units, we consider the volume V Bm , which is obtained by dividing Vm by the number of boron atoms contained in one mol glass V Bm ¼
Vm . 2 ð1 Y Þ
ð4:6Þ
V Bm corresponds to the volume which contains one mol of boron atoms within the given glass structure. It provides a measure for the average boron–boron separation hdB–Bi according to
F. Berkemeier et al. / Journal of Non-Crystalline Solids 351 (2005) 3816–3825
sffiffiffiffiffiffiffi B 3 V m ; hd B–B i ¼ NA
ð4:7Þ
where NA denotes the Avogadro constant. In Fig. 7 the molar volume of the boron atoms V Bm is shown as a function of the glass composition for four different alkali-borate glasses – the sodium- and the rubidium-glasses of our work and the lithium- and the potassium-glasses calculated from the densities obtained by Bresker and Evstropiev [22]. V Bm depends significantly on the cation species and thus on the radius of the alkali ions. With increasing radius of the ions (rLi < rNa < rK < rRb) the glass network shows an increasing tendency for expansion. In the case of the lithium glasses the volume per mole of boron first decreases with increasing alkali content and then reaches a shallow minimum. For the sodium glasses V Bm has a minimum around Y = 0.04 and then increases with increasing Na2O content. The incorporation of the Li2O or Na2O leads to a densification of the network at lower alkali concentrations and only at higher alkali concentrations the boron–boron distance increases with Y. For glasses containing larger alkali ions the expansion dominates and for potassium- and rubidium-glasses a densification is absent. The increase of the glass-transition temperature Tg with increasing alkali concentration, shown in Fig. 2, can be attributed to modifications of the network structure: as already mentioned in the introduction, in vitreous boron oxide the boron atoms are three-fold coordinated by oxygen atoms. Their coordination changes to four oxygens when alkali oxide is incorporated. According to a molecular dynamic study of Verhoef and Hartog [23] the number of BO 4=2 units reaches a maximum at about Y = 0.25. The specific position of the maximum depends slightly on the kind of alkali ions present in the glass. Above approximately Y = 0.25 the formation of non-bridging oxygen atoms (NBO) becomes a relevant process and decreases the number of BO 4=2 units. This may result in a reversal
Fig. 7. Molar volumes normalised to the number of boron atoms V Bm of sodium- and rubidium-borate glasses as functions of the glass composition Y. The data are fitted with polynomials of third order to guide the eye. V Bm for the lithium- and the potassium-borate glasses were calculated from literature data [22].
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of the dependence of Tg as a function of the alkali-content. Ratai et al. [24] and Epping et al. [13,10] studied the structure of borate glasses by means of NMR spectroscopy. In contrast to Verhoef and Hartog they found that below Y = 0.30 nearly all oxygen atoms are consumed for converting BO3/2 into BO 4=2 units and nearly no non-bridging oxygen atoms exist below this concentration of alkali oxide. 4.2. Ion dynamics The strong alkali-concentration dependence of the dcconductivity is a fingerprint of ion-conducting glasses as already shown in Fig. 4. If we assume that the total number of alkali ions can be identified with the number of mobile charge carriers, the increase in alkali concentration cannot be the major reason for the observed increase of the dc conductivity. Instead, the major effect must be attributed to an increase of the ionic mobility. The viewpoint that the same proportion of ions is mobile independent of the total alkali content is supported by scaling properties of conductivity spectra [25]. Like the ionic conductivities, the activation enthalpies of the glasses depend on the alkali concentration, as well. According to Monte Carlo simulations of the dynamicstructure-model by Bunde, Maass and Ingram [4,26], the activation enthalpy of rdc of vitreous electrolytes depends logarithmically on the concentration of mobile ions n DH log n.
ð4:8Þ
In [4] the authors show several examples for alkaliborate, silicate and phosphate glasses which confirm the prediction of their model. Fig. 8 displays the activation enthalpies of the present glass systems as a function of the logarithm of the total number density of ions N. An almost linear dependence of DH is indeed observed, from which we may conclude at least a proportionality, N n, between total and mobile number density of ions.
Fig. 8. Activation enthalpies DH of sodium- and rubidium-borate glasses as functions of the total number densities of the alkali ions, N. The solid lines are fits according to Eq. (4.8).
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4.3. Comparison of single and mixed-alkali glasses In previous studies from our laboratory [6,7,27] conductivity measurements on mixed sodium–rubidium borate glasses (Y = 0.2) of the compositions 0:2½X Na2 O ð1 X ÞRb2 O 0:8B2 O3 ; with X = 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0, were reported. The black-filled symbols in Fig. 9 show the charge-diffusion coefficient calculated via Eq. (3.3) of these glasses at a temperature of 553 K as a function of the composition parameter X. A minimum in charge diffusion occurs for intermediate compositions which is typical for mixed-alkali glasses. From tracer diffusion measurements of 22Na and 86 Rb on this system performed in our laboratory [7,11,27] – also presented in Fig. 9 – it is known that for X P 0.4 the sodium diffusivity DNa is much faster than the rubidium diffusivity DRb . The ratio DNa =DRb increases from about 10 at X = 0.4 to about 104 at X = 1.0. Thus we conclude that for X P 0.4 the conductivity in the mixed-alkali glasses is largely determined by the sodium ions. Since at X = 0.2 both diffusivities are the same, but the number density of rubidium ions is four times larger than the number density of sodium ions, about 80% of the dc conductivity at this composition must be attributed to the rubidium ions. The composition dependence of DNa and DRb provides a qualitative explanation of the conductivity minimum. For a quantitative comparison of conductivity and tracer diffusion it is necessary to consider the Haven ratio (see below). Fig. 10 shows a comparison between the dc conductivities of the mixed-alkali glasses and the two binary glasses at a temperature of 553 K. The plot is constructed in such a way that a ternary glass of a given composition is compared with two binary glasses which contain together the same total amount of sodium- plus rubidium-oxide, respectively. In other words, a ternary glass of the composition
Fig. 10. Comparison between the dc conductivities rdc of the binary alkali-borate glasses of this work and the dc conductivities of the ternary sodium–rubidium borate glasses of Refs. [6,7] at 553 K. The abscissa exhibits three different labellings. Top label: sodium-oxide fraction in the sodium-borate glasses, middle label: rubidium-oxide fraction in the rubidium-borate glasses, bottom label: relative sodium fraction of the ternary sodium–rubidium borate glasses with a totally alkali-oxide content of 20 mol%. The solid lines are fits with polynomial functions.
0:2½X Na2 O ð1 X ÞRb2 O 0:8B2 O3 is compared with the two binary glasses of the total alkali concentrations Y Na ¼ 0:2 X
and
Y Rb ¼ 0:2 ð1 X Þ.
Accordingly, at each position of the abscissa the sum of YNa and YRb equals 0.2, which is the total alkali content of the mixed sodium–rubidium borate glasses. A new subnetwork diffusion concept to this novel view on the mixedalkali effect is in preparation [29]. Fig. 11 compares the number densities of sodium and rubidium in both, the binary and the ternary glasses. It indicates that the number density of one type of alkali atoms is practically the same in the single and in the pertaining mixed glass. Hence the comparison in Fig. 10 opposes in each case also glasses of approximately the same number density of sodium and rubidium, respectively. In addition to the three measured conductivity curves in Fig. 10, two hypothetical curves are shown as dashed lines: (i) the upper dashed line represents the dc conductivity of the ternary glasses when a linear-mixing-rule is assumed. In this case the dc conductivity of the mixed glass, rdc,lin., is calculated from rdc;lin. ¼ X rdc ðX ¼ 1Þ þ ð1 X Þ rdc ðX ¼ 0Þ;
Fig. 9. Conductivity diffusion coefficients Dr (black-filled symbols) and tracer-diffusion coefficients DNa , DRb (light grey symbols) of the mixedalkali borate glasses from the work of Imre et al. [6,7,28] as functions of the composition parameter X at a temperature of 553 K. The diffusivities in brackets are extrapolated to a temperature of 553 K from values measured at a slightly higher temperature (see Ref. [28]). The solid lines are fits with polynomials of third order.
where rdc(X = 0) is the dc conductivity of the pure rubidium-borate glass and rdc(X = 1) that of the pure sodiumborate glass.3 From this point of view the conductivity minimum of the mixed borate-glasses – the so-called mixed-alkali effect – appears as a negative deviation from the linear mixing rule. (ii) The lower dashed curve
3 Note that a linear function in a logarithmic plot is not represented by a straight line.
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Fig. 11. Comparison between the number densities of binary and ternary glasses. Top: number densities of sodium ions, NNa, of the binary sodiumborate glasses and the mixed sodium–rubidium borate glasses. Bottom: number densities of rubidium ions, NRb, of the binary rubidium-borate glasses and the mixed glasses.
represents the sum of the dc conductivities of both binary glasses. In the mixed composition range a deviation is observed between the sum-curve and the curve representing the measured mixed-alkali conductivity values. From this point of view the mixed-alkali effect appears as a positive deviation from the hypothetical expectation. Comparing the dc conductivities of a ternary glass of a given composition and the dc conductivity-sum of the two binary glasses of the same sodium- and rubidium-concentrations, implies comparing glasses of different oxygen content. From the experimental findings we conclude, that the higher oxygen content, i.e. the higher number density of BO 4=2 -units in the case of the ternary glasses, increases the ionic mobility. This is true, even if only one species, e.g. sodium for sodium-rich mixed glasses, determines the conductivity (see Fig. 9). In other words: a glass with the same number density of sodium ions has a higher conductivity when additionally a small amount of Rb2O is present in the glass, whereas the mobility of these rubidium ions is negligible compared to the mobility of the sodium ions and thus their movement does not contribute to the ionic conductivity. To gain a better insight in the origin of this conductivity increase we consider how the Arrhenius parameters – asso-
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ciated with the ionic jump processes – are affected when unlike ions are present or not. Voss et al. [11] and Imre et al. [7] determined the activation enthalpies of 22Na diffusion DH(22Na), of 86Rb diffusion DH(86Rb), and of the product of dc conductivity and temperature DH(rdc Æ T) from the ternary 0.2[XNa2O Æ (1 X)Rb2O] Æ 0.8B2O3 glasses, by means of the radiotracer technique and impedance spectroscopy. In contrast to impedance spectroscopy the radiotracer technique allows the determination of an element-specific activation enthalpy for 22Na- and 86Rb-diffusion. In the following we compare these values with the activation enthalpies of ionic conduction in the corresponding binary glasses, which have the same number density of sodium or rubidium ions, respectively, as the mixed glasses. Fig. 12 compiles the activation enthalpies DH(rdc Æ T) and DH(22Na) of the ternary sodium–rubidium borate glasses as well as DH(rdc Æ T) of the binary sodium-borate glasses as functions of the number density of sodium ions. In the ternary glass system DH(rdc Æ T) passes through a maximum when the number density of sodium increases from NNa = 0 (pure rubidium-borate glass) to NNa = 7.8 · 1027 m3 (pure sodium-borate glass). The activation enthalpy of 22Na-diffusion in the ternary glasses and the activation enthalpy of the ionic conduction in the binary sodium-borate glasses increases with decreasing sodium content. From Fig. 12 we observe that these three activation enthalpies are similar for sodium-number densities of NNa > 4 · 1027 m3. This observation shows that the activation enthalpy is no longer influenced by the presence of Rb2O. For rubidium-rich glasses, i.e. NRb > 4 · 1027 m3, the situation is more complicated. Fig. 13 shows the activation enthalpies of ionic conductivity DH(rdc Æ T) and of 86Rbdiffusion DH(86Rb) of the ternary sodium–rubidium borate
Fig. 12. Comparison between the activation enthalpies of the dc conductivity and of the tracer diffusion of 22Na: (d) conductivity measurements on sodium-borate glasses (this work), ( ) conductivity measurements on sodium–rubidium borate glasses (Refs. [6,7]), (s) 22Na-diffusion measurements on sodium–rubidium borate glasses (Ref. [6,7]). Additionally, the pre-exponential factors C0 of the dc conductivities for the binary and the ternary glasses with a sodium number density of NNa > 4 · 1027 m3 are shown in the inset graph.
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the activation enthalpy of sodium diffusion. We thus may conclude that the higher ionic conductivities of mixed sodium–rubidium borate glasses as compared to the single-sodium borate glasses with NNa > 4 · 1027 m3 is due to different pre-exponential factors C0 (see the inset graph in Fig. 12). 5. Summary and conclusions
Fig. 13. Comparison between the activation enthalpies of dc conductivity and of 86Rb diffusion: (d) conductivity measurements on rubidium-borate glasses (this work), ( ) conductivity measurements on sodium–rubidium borate glasses (Refs. [6,7]), (s) 86Rb-diffusion measurements on sodium– rubidium borate glasses (Refs. [6,7]).
glasses and the activation enthalpies DH(rdc Æ T) of the binary rubidium-borate glasses as functions of NRb. These values are plotted in such a way that the rubidium concentration increases from right to left. A difference between DH(rdc Æ T) and DH(86Rb) occurs in the single rubidium-borate glass. This finding is expressed by a temperature-dependent Haven ratio HR. HR relates the tracer-diffusion coefficient D* to the charge-diffusion coefficient Dr by HR ¼
D . Dr
ð4:9Þ
Voss et al. [11] found for the temperature dependence of HR in this glass 20:9 kJ mol1 H R ¼ 20:8 exp . ð4:10Þ kBT For NRb 5.5 · 1027 m3 22Na and 86Rb diffusion in the ternary glass system have similar activation enthalpies. Therefore sodium ions also contribute to the ionic conductivity. Due to the temperature dependence of the Haven ratio and to the significant contributions of the sodiumions to the ionic conductivity on the rubidium-rich side we cannot unambiguously conclude how the activation barriers for the rubidium-diffusion on the rubidium-rich side are affected by the presence of sodium-ions. Last but not least we discuss the dc-conductivity of Na rich borate glasses. From tracer diffusion measurements it is known that rubidium diffusion in Na–Rb borate glasses with NNa > 4 · 1027 m3 is more than two orders of magnitude slower than sodium diffusion (see Fig. 9). Therefore we conclude that the ionic dc-conductivity in mixed sodium–rubidium borate glasses on the sodium-rich side is due to diffusion of sodium ions. Concomitantly, the activation enthalpy of ionic conductivity is determined by the activation enthalpy of sodium diffusion: in this composition range the presence of rubidium ions does not affect
The molar volume of single alkali-borate glasses, when normalised to one mol boron atoms, show a clear trend with the ionic radii of the alkali ions. The large rubidium ions lead to a considerable expansion of the glass network with increasing rubidium concentration. The glass-transition temperatures of sodium- and rubidium-borate glasses increase with increasing alkali content and reach a shallow maximum between Y = 0.20 and Y = 0.30. This can be attributed to a variation of the ratio between BO3/2 and BO 4=2 units of the network and of the non-bridging oxygen content. Temperature dependent measurements of the dc conductivity of sodium- and rubidium-borate glasses with various alkali contents show a strong dependence of long-range ion transport and of the pertinent activation enthalpy on the alkali concentration. The comparison of the binary glass systems with ternary sodium–rubidium borate glasses indicates that an increased content of the BO 4=2 units of the glass network strongly enhances the ionic mobility. However, an addition of up to 8 mol% rubidium oxide to sodium-borate glass does not affect significantly the activation enthalpy of sodium diffusion. Acknowledgements We are grateful to Dr C. Cramer, Dr Y. Gao and Dr S. Murugavel (Institut fu¨r Physikalische Chemie) for their help performing some of the conductivity measurements, to Professor G. Schmitz, and Dr N.A. Stolwijk (both Institut fu¨r Materialphysik) for valuable comments on the manuscript. This work was financially supported by the Deutsche Forschungsgemeinschaft within the Sonderforschungsbereich 458. References [1] H.M. Schaeffer, J. Non-Cryst. Solids 67 (1984) 19. [2] A.K. Vershneya, Fundamentals of Inorganic Glasses, Academic, San Diego, 1994. [3] J.E. Shelby, Introduction to Glass Science and Technology, The Royal Society of Chemistry, Cambridge, 1997. [4] A. Bunde, P. Maass, M.D. Ingram, J. Non-Cryst. Solids 172–174 (1994) 1222. [5] M.D. Ingram, B. Roling, J. Phys.: Condens. Matter 15 (2003) 1595. [6] A.W. Imre, S. Voss, H. Mehrer, Phys. Chem. Chem. Phys. 4 (2002) 3219. [7] A.W. Imre, S. Voss, H. Mehrer, J. Non-Cryst. Solids 333 (2004) 231. [8] U. Schoo, C. Cramer, H. Mehrer, Solid State Ionics 138 (2000) 105. [9] G.H. Frischat, Ionic Diffusion in Oxide Glasses, Trans Tech, Zu¨rich, 1975.
F. Berkemeier et al. / Journal of Non-Crystalline Solids 351 (2005) 3816–3825 [10] J.D. Epping, H. Eckert, A.W. Imre, H. Mehrer, J. Non-Cryst. Solids 351 (43–45) (2005) 3521, doi:10.1016/j.jnoncrysol.2005.08.034. [11] S. Voss, A.W. Imre, H. Mehrer, Phys. Chem. Chem. Phys. 6 (2004) 3669. [12] S. Voss, F. Berkemeier, A.W. Imre, H. Mehrer, Z. Phys. Chem. 218 (2004) 1353. [13] J.D. Epping, NMR-Untersuchungen zur ra¨umlichen Kationenverteilung in Alkali-Boratgla¨sern, PhD thesis, Universita¨t Mu¨nster, 2004. [14] S.D. Baranovskii, H. Cordes, J. Chem. Phys. 111 (1999) 7546. [15] M. Vogel, Phys. Rev. B 70 (2004) 094302. [16] G. Garcia-Belmonte, J. Bisquert, J. Non-Cryst. Solids 337 (2004) 272. [17] J.C. Dyre, T.B. Scho¨der, Rev. Mod. Phys. 72 (2000) 873. [18] K. Funke, Prog. Solid State Chem. 22 (1993) 111. [19] M.D. Ingram, Phys. Chem. Glasses 28 (1987) 215. [20] C. Cramer, S. Bru¨ckner, Y. Gao, K. Funke, Phys. Chem. Chem. Phys. 4 (2002) 3214.
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[21] C.C. Hunter, M.D. Ingram, Solid State Ionics 14 (1984) 31. [22] R.I. Bresker, K.S. Evstropiev, Zh. Prikl. Khim. (1952) 905. [23] A.H. Verhoef, A.W. Hartog, J. Non-Cryst. Solids 182 (1995) 235. [24] E. Ratai, M. Janssen, H. Eckert, Solid State Ionics 105 (1998) 25. [25] B. Roling, A. Happe, K. Funke, M.D. Ingram, Phys. Rev. Lett. 78 (1997) 2160. [26] P. Maass, J. Non-Cryst. Solids 255 (1999) 35. [27] H. Mehrer, A.W. Imre, S. Voss, Advances in Science and Technology 37, Mass and Charge Transport in Inorganic Materials II – CIMTEC 2002, in: P. Vincenzini, V. Buscaglia (Eds.), TECHNA Srl, 2003. [28] A.W. Imre, S. Voss, H. Mehrer, Defect Diffus. Forum 370 (2005) 273. [29] A.W. Imre, S. Divinsky, S. Voss, F. Berkemeier, H. Mehrer, J. Non-Cryst. Solids, submitted for publication.
Molar volume, glass-transition temperature, and ionic conductivity of Na- and Rb-borate glasses in comparison with mixed Na–Rb borate glasses Frank Berkemeier a
a,*
, Stephan Voss
´ rpa´d W. Imre a, Helmut Mehrer ,A
a,b
a
Institut fu¨r Materialphysik, Universita¨t Mu¨nster, Wilhelm-Klemm Straße 10, D-48149 Mu¨nster, Sonderforschungsbereich 458, Germany b Infineon Technologies AG, P.O. Box 80 09 49, 81609 Munich, Germany Received 18 March 2005; received in revised form 5 October 2005 Available online 18 November 2005
Abstract Mass densities, molar volumes, glass-transition temperatures, and ionic conductivities are measured in series of YNa2O Æ (1 Y)B2O3 glasses, with Y = 0.00, 0.04, 0.08, 0.12, 0.16, 0.20, 0.25, 0.30 and YRb2O Æ (1 Y)B2O3 glasses, with Y = 0.00, 0.12, 0.16, 0.20, 0.25, 0.30. Measurements of the molar volumes indicate that the incorporation of rubidium ions leads to a considerable expansion of the network, which is not observed for sodium ions. The glass-transition temperature increases with increasing alkali content and reaches a maximum near Y = 0.25 for both glass systems. These trends are attributed to changes in the glass network. For each glass composition an Arrhenius-activated increase of the product of dc conductivity and temperature is observed. The activation enthalpy decreases with increasing number density of ions. A comparison between the binary sodium- and rubidium-borate glasses from this work, with the ternary sodium–rubidium borate glasses studied earlier in our laboratory, provides interesting insights in the influence of the glass structure on ionic transport processes and the mixed-alkali effect. 2005 Elsevier B.V. All rights reserved. PACS: 61.43.Fs; 66.10.Ed
1. Introduction Network glasses represent a special class of disordered materials and are categorised by the types of network formers: e.g. silicate, germanate, phosphate, aluminate, or borate glasses. The ionic conductivity of these glasses is due to the addition of so-called network modifiers, such as alkali oxides (A2O). The motion of mobile ions in materials with disordered structures is one of the basic questions of solid-state ionics. From the viewpoint of application, ion-conducting materials have high technological potential
*
Corresponding author. Tel.: +49 (0) 251 8333587; fax: +49 (0) 251 8338346. E-mail address: [email protected] (F. Berkemeier). 0022-3093/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2005.10.010
for electrochemical devices such as solid–oxide fuel cells, solid-state batteries, or chemical sensors [1]. In this paper we consider alkali-borate glasses and for this purpose we remind the reader of some structural features of these glasses: adding alkali oxide (A2O) to boron oxide changes the coordination number of some boron atoms from three to four and leads to a transformation of trigonal BO3/2 units into negatively charged tetrahedral BO 4=2 units. Both BO3/2 and BO4=2 units are connected by bridging-oxygen atoms and represent the smallest structural units of the glass network. The formation of two boron–oxygen tetrahedra consumes the additional oxygen provided by the alkali oxide A2O. Since each tetrahedron is charge-deficient by minus one elementary charge, two alkali ions compensate two anionic BO 4=2 units. An increase in A2O concentration results in a further transfor-
F. Berkemeier et al. / Journal of Non-Crystalline Solids 351 (2005) 3816–3825
mation of boron from three- to four-fold coordination. Beyond a concentration of about 25–30 mol% alkali oxide content the formation of non-bridging oxygen atoms becomes a relevant process [2,3]. The alkali ions are largely decoupled from the mostly static boron-oxide network and are mobile even far below the glass-transition temperature. Several models have been developed to describe the ionic motion in network glasses (e.g. [4,5]). These models treat many features of ionic motion, but nevertheless, a general model which explains all known experimental phenomena is still missing and further experimental work is needed. In this paper we present a systematic study of mass density, molar volume, glass-transition temperature, and ionic transport in binary sodium-borate and rubidium-borate glasses of various alkali concentrations. Ionic transport in mixed sodium–rubidium borate glasses was investigated recently in our laboratory [6,7], and in the discussion we compare single-alkali and mixed-alkali borate glasses. 2. Experimental details 2.1. Glass preparation We prepared a series of binary YNa2O Æ (1 Y)B2O3 glasses, with Y = 0.04, 0.08, 0.12, 0.16, 0.20, 0.25, 0.30 and binary YRb2O Æ (1 Y)B2O3 glasses, with Y = 0.12, 0.16, 0.20, 0.25, 0.30.1 A pure B2O3 glass was prepared as well. Appropriate amounts of dried powders of sodium carbonate (Merck, 99.999%) and diboron trioxide (Merck, 99.9995%) for the sodium-borate glasses, and rubidium carbonate (Alfa Aesar, 99.975%) and diboron trioxide for the rubidium-borate glasses were mixed and melted in a platinum crucible at a temperature of usually 1273 K. During heating the alkali carbonates decompose into alkali oxide and gaseous CO2, which is released from the melt. To avoid inhomogeneities in the glass, the melt was kept at 1273 K for 3 h, swivelled several times, and subsequently poured into a cylindrical carbon mould (diameter = 10 mm, height = 30 mm) kept at room temperature. For alkali contents of Y > 0.2 the melting temperature was reduced to about 1150 K and a stainless-steel mould, pre-heated to about 573 K, was used, to avoid the bursting of the glass cylinders (as a consequence of internal stresses). After the preparation the glasses were heat-treated at a temperature of about 20 K below their calorimetric glasstransition temperature for 5 h and then slowly cooled down to room temperature with 6 K/h to remove internal stresses.
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2.2. Glass characterisation In earlier work of Schoo et al. [8] borate glasses also prepared in our laboratory were analyzed by the analytic laboratory of Schott AG (Mainz). Deviations from the batch compositions were less than 2.5%. In addition we compared dc-conductivity and glass-transition temperature of two glasses prepared independently with the same nominal composition. Both deviate from each other less than 10%. We also determined the ratio of sodium and rubidium in mixed-alkali glasses by microprobe analysis and found an agreement within the accuracy of the analysis in comparison with the batch composition. Since boron and oxygen cannot be measured in our microprobe the compositions of the binary borate glasses were not determined. Since water can influence the physical properties, in particular the mobility of ions, in hygroscopic glasses [9], we checked the concentration of hydrogen in the glasses by means of NMR. The NMR experiments do not reveal any hydrogen contamination of the samples indicating the absence of water [10,11]. To check the non-crystallinity of our glass samples X-ray measurements were performed using a Siemens D5000 X-ray diffractometer. No sharp Bragg peaks were found, indicating the absence of crystalline phases (for details see [12]). The distribution of the alkali ions and of BO3/2 and BO 4=2 units of the network of our glasses was investigated by nuclear-magnetic resonance (NMR) experiments at the Institute of Physical Chemistry of our University, using the 23Na spin echo and the 11B magic-angle spinning techniques. The results suggest an almost homogeneous distribution of alkali ions and of BO3/2 and BO 4=2 units in the glasses [13,10]. The so-called N4-value is defined as N4 ¼
cðBO 4=2 Þ cðBO3=2 Þ þ cðBO 4=2 Þ
.
ð2:1Þ
Here cðBO 4=2 Þ denotes the concentration of BO4=2 units and c(BO3/2) the concentration of BO3/2 units. The N4value can be deduced from NMR experiments (see, e.g. [3]). For our glasses the measured N4-value coincidences within the experimental errors with the value expected from the molar fractions of A2O and B2O3 given by Y/(1 Y). The densities of the glasses were determined by an Archimedes method. For this purpose the mass of a glass sample mglass and the mass of a piece of semiconductorgrade silicon mSi – as a reference for calibration – was measured in air and in n-dodecane. Then the density of the glass is obtained from air . mair mSi mdod glass Si q ¼ qair þ air ðqSi qair Þ ; ð2:2Þ . mglass mdod mair glass Si
1
Rubidium–borate glasses with Y = 0.04 and Y = 0.08 where not prepared because the dc conductivities of those glasses are below the lower limit of our impedance analyser and cannot be measured.
where qair and qSi are the densities of air and silicon, respectively.
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The calorimetric glass-transition temperatures of the samples were measured using differential-scanning calorimetry (DSC). A small piece of each glass was crushed in an agate mortar, dried at 383 K for 24 h, and encapsulated into a copper container. The measurements were performed using a Perkin–Elmer DSC 7 device employing several heating rates between 5 and 100 K/min. A typical DSC thermogram of these glasses contains three regions, which belong to three different states: the glassy state at low temperatures, the undercooled liquid state at intermediate temperatures, and the crystalline state at high temperatures. The corresponding calorimetric glass-transition temperature Tg is defined as the mid-point temperature of the extensions of the pre- and the post-transition baselines of the transition from glassy- to undercooled liquid-state. 2.3. Conductivity measurements The conductivity measurements were performed by impedance spectroscopy. Disks of 2 mm thickness cut from the glass cylinders served as samples. Their two flat surfaces were ground parallel and the sample diameter was reduced to 9 mm by ultrasonic drilling. In the next step the surfaces were polished with 6 lm diamond paste and finally metallic contacts, consisting of a silver bottom and a platinum top layer, were deposited on both surfaces using a sputter coater (Quorum Technologies SC 7640). Frequency-dependent conductivities were measured in a frequency range between 5 Hz and 1.3 MHz using a HP Agilent 4192A LF impedance analyser. The glass samples
were fixed between two silver electrodes to provide a good electrical contact. The connecting wires were arranged in a four-point probe configuration to exclude an influence of the wire resistance on the measurement. The sample holder was placed inside a quartz-glass tube filled with argon. A horizontal furnace (Gero SR 70–500/11) and a calibrated Ni–NiCr thermocouple were used to control the temperature of the sample. In a measuring cycle, which was controlled by a computer using the Novocontrol WinDETA 4.0 software, the samples were first heated up to a temperature about 20 K below their glass-transition temperature and subsequently cooled down with 6 K/h to reduce internal stresses introduced during sample preparation. Subsequently, the ac-conductivities were measured at a series of pre-defined temperatures by heating up the samples with 6 K/h to a maximum temperature of about 20 K below Tg. Since one frequency scan takes only about one minute each measurement can be regarded as isothermal. 3. Results 3.1. Density and glass transition The measured densities q of the sodium- and the rubidium-borate glasses are listed in Tables 1 and 2. In Fig. 1 these values are shown as a function of the total alkali concentration Y. The mass density increases in a non-linear way with increasing alkali content and the rubidium-borate glasses show higher densities than the sodium-borate glasses of the same alkali concentration.
Table 1 Densities q, molar volumes Vm, molar volumes normalised to the number of boron atoms V Bm , number densities of the sodium ions N, activation enthalpies DH, and pre-exponential factors C0 of the dc conductivity for the sodium-borate glasses Composition Y
q (g cm3)
Vm (cm3 mol1)
V Bm (cm3 mol1)
N (m3)
0.30
2.37 ± 0.01
28.4 ± 0.1
20.3 ± 0.1
1.27 · 1028
78.2 ± 0.2
5 ð1:07þ0:04 0:04 Þ 10
19.7 ± 0.1
28
87.5 ± 0.2
4 ð1:99þ0:92 0:88 Þ 10
27
98.7 ± 0.4
5 ð1:10þ0:10 0:10 Þ 10
27
114.8 ± 0.5
5 ð1:82þ0:19 0:17 Þ 10
27
128.5 ± 2.7
4 ð1:72þ1:22 0:71 Þ 10
27
143.9 ± 2.3
1:67 ð2:420:99 Þ 105
27
194.8 ± 8.4 –
5 ð3:70þ1:98 0:31 Þ 10 –
0.25 0.20 0.16 0.12 0.08 0.04 0.00
2.30 ± 0.01 2.23 ± 0.01 2.15 ± 0.01 2.09 ± 0.01 2.03 ± 0.01 1.95 ± 0.01 1.85 ± 0.01
29.5 ± 0.2 30.6 ± 0.2 31.8 ± 0.2 32.8 ± 0.2 34.0 ± 0.2 35.6 ± 0.2 37.7 ± 0.2
19.1 ± 0.1 18.9 ± 0.1 18.7 ± 0.1 18.5 ± 0.1 18.5 ± 0.1 18.8 ± 0.1
DH (kJ mol1)
1.02 · 10 7.87 · 10 6.07 · 10 4.40 · 10 2.83 · 10 1.35 · 10 0
C0 (X1 cm1 K)
Table 2 Densities q, molar volumes Vm, molar volumes normalised to the number of boron atoms V Bm , number densities of the sodium ions N, activation enthalpies DH, and pre-exponential factors C0 of the dc conductivity for the rubidium-borate glasses Composition Y
q (g cm3)
Vm (cm3 mol1)
V Bm (cm3 mol1)
N (m3)
0.30
2.93 ± 0.01
35.8 ± 0.2
25.6 ± 0.1
1.01 · 1028
84.4 ± 0.3
4 ð6:11þ0:44 0:41 Þ 10
24.1 ± 0.1
27
94.8 ± 0.3
4 ð6:30þ4:07 3:82 Þ 10
27
0.25
2.74 ± 0.01
36.1 ± 0.2
DH (kJ mol1)
8.35 · 10
C0 (X1 cm1 K)
0.20
2.54 ± 0.01
36.6 ± 0.2
22.9 ± 0.1
6.58 · 10
102.5 ± 0.4
4 ð2:70þ0:20 0:20 Þ 10
0.16
2.44 ± 0.01
36.3 ± 0.2
21.6 ± 0.1
5.31 · 1027
127.0 ± 0.9
4 ð7:20þ1:58 1:30 Þ 10
20.4 ± 0.1 18.8 ± 0.1
27
145.2 ± 3.8 –
4 ð3:85þ5:09 2:19 Þ 10 –
0.12 0.00
2.33 ± 0.01 1.85 ± 0.01
35.9 ± 0.2 37.7 ± 0.2
4.03 · 10 0
F. Berkemeier et al. / Journal of Non-Crystalline Solids 351 (2005) 3816–3825
Fig. 1. Densities of sodium- and rubidium-borate glasses as functions of the glass composition Y. The solid lines are fits with polynomials of third order to guide the eye.
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Fig. 3. Conductivity spectra of the 0.16Na2O Æ 0.84B2O3 glass. Only some selected isotherms are shown.
sodium-borate glass 0.16Na2O Æ 0.84B2O3. Isotherms of r 0 (m) are plotted as a function of the frequency m in a log–log representation for several temperatures. The spectra exhibit the following salient features:
Fig. 2. Calorimetric glass-transition temperature Tg of both binary alkaliborate glass systems as functions of the glass composition Y measured at a heating rate of 20 K/min. Both solid lines are fits with polynomials of third order to guide the eye.
The glass-transition temperature Tg as a function of the glass composition Y is shown in Fig. 2 for both binary glass systems. For each system Tg increases with increasing alkali content and reaches a shallow maximum near Y = 0.25. For borate glasses many of the property/composition trends reverse their directions with increasing alkali content. For example, a minimum in the thermal expansion coefficient and a maximum in Tg occurring at higher alkali oxide contents are observed. This behavior, which is also observed in our Tg-measurements, is denoted as the borate anomaly [3].
• a conductivity plateau with a frequency-independent range of the conductivity r 0 at sufficiently low frequencies. The plateau represents the long-range transport of the ions. Its mean value is taken in the following as the dc conductivity rdc. The product of this dc conductivity and temperature is Arrhenius activated. • a conductivity-dispersion region at higher frequencies in which the conductivity r 0 increases monotonically with increasing frequency. There are several approaches in the literature to explain this behavior (e.g. [14–17]). One possible explanation is given by Funke [18] who attributes the dispersion region to correlated forward–backward jumps of the ions. For a pure random walk of the ions without any correlations dispersion would be absent. In Fig. 4 the dc conductivity of both glass systems is plotted as a function of the total alkali concentration Y at a fixed temperature of 573 K. For both systems a
3.2. Ion dynamics To investigate the ion dynamics we measured the complex conductivities r of our glass series as functions of frequency m. At low frequencies the real part of the conductivity, r 0 , yields the dc conductivity rdc. Via the Nernst–Einstein equation rdc is connected to a charge-diffusion coefficient of the mobile ions (see below). Fig. 3 gives examples of typical conductivity spectra of the
Fig. 4. dc conductivities rdc of sodium- and rubidium-borate glasses as functions of the glass composition Y at 573 K. Filled circles: this work; open circles: work of Hunter and Ingram [21]. The solid lines are fits with polynomials of third order to guide the eye.
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significant increase of the dc conductivity with increasing alkali concentration is observed. This increase in conductivity covers almost six orders of magnitude (see also Tables 1 and 2). Similar conductivity results have been reported for other glass systems, too [19,20]. It can also be seen in Fig. 4 that at a fixed alkali concentration the dc conductivity of a sodium-borate glass is between one and two orders of magnitude higher than that of the corresponding rubidium-borate glass, indicating that the smaller sodium ions move faster than the larger rubidium ions. Hunter and Ingram also investigated sodium-borate glasses by means of conductivity measurements [21]. Their conductivity values are also presented in Fig. 4 and show good agreement with the present results. The measured ionic dc conductivity, rdc, can be related to a charge-diffusion coefficient of the ions, Dr, via the Nernst–Einstein equation rdc k B T . ð3:3Þ Dr ¼ q2 N Here kB is the Boltzmann constant, T the absolute temperature, q the charge of the ions (+1 elementary charge for alkali ions), and N the number density of ions. To calculate Dr, we assume that N equals the total number density in Tables 1 and 2, calculated from the mass density q, the molar mass M, and the alkali content Y of the glass via N = 2 Æ Y Æ NA Æ q/M, where NA is the Avogadro constant. In crystalline solids and in glasses below Tg the Fickian diffusion coefficient D exhibits Arrhenius-activated behavior D = D0 Æ exp(DH/(kB Æ T)), with an activation enthalpy DH and a pre-exponential factor D0. Assuming that Dr follows Arrhenius behavior Dr = Dr,0 Æ exp(DH/(kB Æ T)), we expect for the temperature dependence of the product rdc Æ T DH rdc T ¼ C 0 exp ; ð3:4Þ kB T with Dr;0 q2 N . ð3:5Þ C0 ¼ kB In Figs. 5 and 6 the Arrhenius diagrams of rdc Æ T of both binary glass systems are presented. The measured conductivity data can well be approximated by Eq. (3.4) and thus allow the determination of an activation enthalpy DH and a pre-exponential factor C0 for each glass composition.2 4. Discussion 4.1. Density and glass transition To get more information about the glass network we determined the molar volumes Vm of the glasses, which 2
All Arrhenius diagrams show a slight deviation from a linear behavior. In what follows, we do not consider this slight upward curvatures, which might be due to the migration of the ions in an energy-landscape [6].
Fig. 5. Arrhenius diagrams of rdc Æ T for sodium-borate glasses YNa2O Æ (1 Y)B2O3. (a) Y = 0.30, (b) Y = 0.25, (c) Y = 0.20, (d) Y = 0.16, (e) Y = 0.12, (f) Y = 0.08, (g) Y = 0.04.
Fig. 6. Arrhenius diagrams of rdc Æ T for rubidium-borate glasses YRb2O Æ (1 Y)B2O3. (a) Y = 0.30, (b) Y = 0.25, (c) Y = 0.20, (d) Y = 0.16, (e) Y = 0.12.
are obtained from the density measurements by Vm = M/q. M denotes the molar mass of the glass which is calculated from the molar masses of diboron-trioxide M B2 O3 and the respective alkali oxide MA2O via M ¼ Y M A2 O þ ð1 Y Þ M B2 O3 . Because the total alkali content influences both, M and q, Vm does not give specific information about the glass network. Since the boron atoms are the central atoms of the BO3/2 and BO 4=2 units, we consider the volume V Bm , which is obtained by dividing Vm by the number of boron atoms contained in one mol glass V Bm ¼
Vm . 2 ð1 Y Þ
ð4:6Þ
V Bm corresponds to the volume which contains one mol of boron atoms within the given glass structure. It provides a measure for the average boron–boron separation hdB–Bi according to
F. Berkemeier et al. / Journal of Non-Crystalline Solids 351 (2005) 3816–3825
sffiffiffiffiffiffiffi B 3 V m ; hd B–B i ¼ NA
ð4:7Þ
where NA denotes the Avogadro constant. In Fig. 7 the molar volume of the boron atoms V Bm is shown as a function of the glass composition for four different alkali-borate glasses – the sodium- and the rubidium-glasses of our work and the lithium- and the potassium-glasses calculated from the densities obtained by Bresker and Evstropiev [22]. V Bm depends significantly on the cation species and thus on the radius of the alkali ions. With increasing radius of the ions (rLi < rNa < rK < rRb) the glass network shows an increasing tendency for expansion. In the case of the lithium glasses the volume per mole of boron first decreases with increasing alkali content and then reaches a shallow minimum. For the sodium glasses V Bm has a minimum around Y = 0.04 and then increases with increasing Na2O content. The incorporation of the Li2O or Na2O leads to a densification of the network at lower alkali concentrations and only at higher alkali concentrations the boron–boron distance increases with Y. For glasses containing larger alkali ions the expansion dominates and for potassium- and rubidium-glasses a densification is absent. The increase of the glass-transition temperature Tg with increasing alkali concentration, shown in Fig. 2, can be attributed to modifications of the network structure: as already mentioned in the introduction, in vitreous boron oxide the boron atoms are three-fold coordinated by oxygen atoms. Their coordination changes to four oxygens when alkali oxide is incorporated. According to a molecular dynamic study of Verhoef and Hartog [23] the number of BO 4=2 units reaches a maximum at about Y = 0.25. The specific position of the maximum depends slightly on the kind of alkali ions present in the glass. Above approximately Y = 0.25 the formation of non-bridging oxygen atoms (NBO) becomes a relevant process and decreases the number of BO 4=2 units. This may result in a reversal
Fig. 7. Molar volumes normalised to the number of boron atoms V Bm of sodium- and rubidium-borate glasses as functions of the glass composition Y. The data are fitted with polynomials of third order to guide the eye. V Bm for the lithium- and the potassium-borate glasses were calculated from literature data [22].
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of the dependence of Tg as a function of the alkali-content. Ratai et al. [24] and Epping et al. [13,10] studied the structure of borate glasses by means of NMR spectroscopy. In contrast to Verhoef and Hartog they found that below Y = 0.30 nearly all oxygen atoms are consumed for converting BO3/2 into BO 4=2 units and nearly no non-bridging oxygen atoms exist below this concentration of alkali oxide. 4.2. Ion dynamics The strong alkali-concentration dependence of the dcconductivity is a fingerprint of ion-conducting glasses as already shown in Fig. 4. If we assume that the total number of alkali ions can be identified with the number of mobile charge carriers, the increase in alkali concentration cannot be the major reason for the observed increase of the dc conductivity. Instead, the major effect must be attributed to an increase of the ionic mobility. The viewpoint that the same proportion of ions is mobile independent of the total alkali content is supported by scaling properties of conductivity spectra [25]. Like the ionic conductivities, the activation enthalpies of the glasses depend on the alkali concentration, as well. According to Monte Carlo simulations of the dynamicstructure-model by Bunde, Maass and Ingram [4,26], the activation enthalpy of rdc of vitreous electrolytes depends logarithmically on the concentration of mobile ions n DH log n.
ð4:8Þ
In [4] the authors show several examples for alkaliborate, silicate and phosphate glasses which confirm the prediction of their model. Fig. 8 displays the activation enthalpies of the present glass systems as a function of the logarithm of the total number density of ions N. An almost linear dependence of DH is indeed observed, from which we may conclude at least a proportionality, N n, between total and mobile number density of ions.
Fig. 8. Activation enthalpies DH of sodium- and rubidium-borate glasses as functions of the total number densities of the alkali ions, N. The solid lines are fits according to Eq. (4.8).
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4.3. Comparison of single and mixed-alkali glasses In previous studies from our laboratory [6,7,27] conductivity measurements on mixed sodium–rubidium borate glasses (Y = 0.2) of the compositions 0:2½X Na2 O ð1 X ÞRb2 O 0:8B2 O3 ; with X = 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0, were reported. The black-filled symbols in Fig. 9 show the charge-diffusion coefficient calculated via Eq. (3.3) of these glasses at a temperature of 553 K as a function of the composition parameter X. A minimum in charge diffusion occurs for intermediate compositions which is typical for mixed-alkali glasses. From tracer diffusion measurements of 22Na and 86 Rb on this system performed in our laboratory [7,11,27] – also presented in Fig. 9 – it is known that for X P 0.4 the sodium diffusivity DNa is much faster than the rubidium diffusivity DRb . The ratio DNa =DRb increases from about 10 at X = 0.4 to about 104 at X = 1.0. Thus we conclude that for X P 0.4 the conductivity in the mixed-alkali glasses is largely determined by the sodium ions. Since at X = 0.2 both diffusivities are the same, but the number density of rubidium ions is four times larger than the number density of sodium ions, about 80% of the dc conductivity at this composition must be attributed to the rubidium ions. The composition dependence of DNa and DRb provides a qualitative explanation of the conductivity minimum. For a quantitative comparison of conductivity and tracer diffusion it is necessary to consider the Haven ratio (see below). Fig. 10 shows a comparison between the dc conductivities of the mixed-alkali glasses and the two binary glasses at a temperature of 553 K. The plot is constructed in such a way that a ternary glass of a given composition is compared with two binary glasses which contain together the same total amount of sodium- plus rubidium-oxide, respectively. In other words, a ternary glass of the composition
Fig. 10. Comparison between the dc conductivities rdc of the binary alkali-borate glasses of this work and the dc conductivities of the ternary sodium–rubidium borate glasses of Refs. [6,7] at 553 K. The abscissa exhibits three different labellings. Top label: sodium-oxide fraction in the sodium-borate glasses, middle label: rubidium-oxide fraction in the rubidium-borate glasses, bottom label: relative sodium fraction of the ternary sodium–rubidium borate glasses with a totally alkali-oxide content of 20 mol%. The solid lines are fits with polynomial functions.
0:2½X Na2 O ð1 X ÞRb2 O 0:8B2 O3 is compared with the two binary glasses of the total alkali concentrations Y Na ¼ 0:2 X
and
Y Rb ¼ 0:2 ð1 X Þ.
Accordingly, at each position of the abscissa the sum of YNa and YRb equals 0.2, which is the total alkali content of the mixed sodium–rubidium borate glasses. A new subnetwork diffusion concept to this novel view on the mixedalkali effect is in preparation [29]. Fig. 11 compares the number densities of sodium and rubidium in both, the binary and the ternary glasses. It indicates that the number density of one type of alkali atoms is practically the same in the single and in the pertaining mixed glass. Hence the comparison in Fig. 10 opposes in each case also glasses of approximately the same number density of sodium and rubidium, respectively. In addition to the three measured conductivity curves in Fig. 10, two hypothetical curves are shown as dashed lines: (i) the upper dashed line represents the dc conductivity of the ternary glasses when a linear-mixing-rule is assumed. In this case the dc conductivity of the mixed glass, rdc,lin., is calculated from rdc;lin. ¼ X rdc ðX ¼ 1Þ þ ð1 X Þ rdc ðX ¼ 0Þ;
Fig. 9. Conductivity diffusion coefficients Dr (black-filled symbols) and tracer-diffusion coefficients DNa , DRb (light grey symbols) of the mixedalkali borate glasses from the work of Imre et al. [6,7,28] as functions of the composition parameter X at a temperature of 553 K. The diffusivities in brackets are extrapolated to a temperature of 553 K from values measured at a slightly higher temperature (see Ref. [28]). The solid lines are fits with polynomials of third order.
where rdc(X = 0) is the dc conductivity of the pure rubidium-borate glass and rdc(X = 1) that of the pure sodiumborate glass.3 From this point of view the conductivity minimum of the mixed borate-glasses – the so-called mixed-alkali effect – appears as a negative deviation from the linear mixing rule. (ii) The lower dashed curve
3 Note that a linear function in a logarithmic plot is not represented by a straight line.
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Fig. 11. Comparison between the number densities of binary and ternary glasses. Top: number densities of sodium ions, NNa, of the binary sodiumborate glasses and the mixed sodium–rubidium borate glasses. Bottom: number densities of rubidium ions, NRb, of the binary rubidium-borate glasses and the mixed glasses.
represents the sum of the dc conductivities of both binary glasses. In the mixed composition range a deviation is observed between the sum-curve and the curve representing the measured mixed-alkali conductivity values. From this point of view the mixed-alkali effect appears as a positive deviation from the hypothetical expectation. Comparing the dc conductivities of a ternary glass of a given composition and the dc conductivity-sum of the two binary glasses of the same sodium- and rubidium-concentrations, implies comparing glasses of different oxygen content. From the experimental findings we conclude, that the higher oxygen content, i.e. the higher number density of BO 4=2 -units in the case of the ternary glasses, increases the ionic mobility. This is true, even if only one species, e.g. sodium for sodium-rich mixed glasses, determines the conductivity (see Fig. 9). In other words: a glass with the same number density of sodium ions has a higher conductivity when additionally a small amount of Rb2O is present in the glass, whereas the mobility of these rubidium ions is negligible compared to the mobility of the sodium ions and thus their movement does not contribute to the ionic conductivity. To gain a better insight in the origin of this conductivity increase we consider how the Arrhenius parameters – asso-
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ciated with the ionic jump processes – are affected when unlike ions are present or not. Voss et al. [11] and Imre et al. [7] determined the activation enthalpies of 22Na diffusion DH(22Na), of 86Rb diffusion DH(86Rb), and of the product of dc conductivity and temperature DH(rdc Æ T) from the ternary 0.2[XNa2O Æ (1 X)Rb2O] Æ 0.8B2O3 glasses, by means of the radiotracer technique and impedance spectroscopy. In contrast to impedance spectroscopy the radiotracer technique allows the determination of an element-specific activation enthalpy for 22Na- and 86Rb-diffusion. In the following we compare these values with the activation enthalpies of ionic conduction in the corresponding binary glasses, which have the same number density of sodium or rubidium ions, respectively, as the mixed glasses. Fig. 12 compiles the activation enthalpies DH(rdc Æ T) and DH(22Na) of the ternary sodium–rubidium borate glasses as well as DH(rdc Æ T) of the binary sodium-borate glasses as functions of the number density of sodium ions. In the ternary glass system DH(rdc Æ T) passes through a maximum when the number density of sodium increases from NNa = 0 (pure rubidium-borate glass) to NNa = 7.8 · 1027 m3 (pure sodium-borate glass). The activation enthalpy of 22Na-diffusion in the ternary glasses and the activation enthalpy of the ionic conduction in the binary sodium-borate glasses increases with decreasing sodium content. From Fig. 12 we observe that these three activation enthalpies are similar for sodium-number densities of NNa > 4 · 1027 m3. This observation shows that the activation enthalpy is no longer influenced by the presence of Rb2O. For rubidium-rich glasses, i.e. NRb > 4 · 1027 m3, the situation is more complicated. Fig. 13 shows the activation enthalpies of ionic conductivity DH(rdc Æ T) and of 86Rbdiffusion DH(86Rb) of the ternary sodium–rubidium borate
Fig. 12. Comparison between the activation enthalpies of the dc conductivity and of the tracer diffusion of 22Na: (d) conductivity measurements on sodium-borate glasses (this work), ( ) conductivity measurements on sodium–rubidium borate glasses (Refs. [6,7]), (s) 22Na-diffusion measurements on sodium–rubidium borate glasses (Ref. [6,7]). Additionally, the pre-exponential factors C0 of the dc conductivities for the binary and the ternary glasses with a sodium number density of NNa > 4 · 1027 m3 are shown in the inset graph.
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the activation enthalpy of sodium diffusion. We thus may conclude that the higher ionic conductivities of mixed sodium–rubidium borate glasses as compared to the single-sodium borate glasses with NNa > 4 · 1027 m3 is due to different pre-exponential factors C0 (see the inset graph in Fig. 12). 5. Summary and conclusions
Fig. 13. Comparison between the activation enthalpies of dc conductivity and of 86Rb diffusion: (d) conductivity measurements on rubidium-borate glasses (this work), ( ) conductivity measurements on sodium–rubidium borate glasses (Refs. [6,7]), (s) 86Rb-diffusion measurements on sodium– rubidium borate glasses (Refs. [6,7]).
glasses and the activation enthalpies DH(rdc Æ T) of the binary rubidium-borate glasses as functions of NRb. These values are plotted in such a way that the rubidium concentration increases from right to left. A difference between DH(rdc Æ T) and DH(86Rb) occurs in the single rubidium-borate glass. This finding is expressed by a temperature-dependent Haven ratio HR. HR relates the tracer-diffusion coefficient D* to the charge-diffusion coefficient Dr by HR ¼
D . Dr
ð4:9Þ
Voss et al. [11] found for the temperature dependence of HR in this glass 20:9 kJ mol1 H R ¼ 20:8 exp . ð4:10Þ kBT For NRb 5.5 · 1027 m3 22Na and 86Rb diffusion in the ternary glass system have similar activation enthalpies. Therefore sodium ions also contribute to the ionic conductivity. Due to the temperature dependence of the Haven ratio and to the significant contributions of the sodiumions to the ionic conductivity on the rubidium-rich side we cannot unambiguously conclude how the activation barriers for the rubidium-diffusion on the rubidium-rich side are affected by the presence of sodium-ions. Last but not least we discuss the dc-conductivity of Na rich borate glasses. From tracer diffusion measurements it is known that rubidium diffusion in Na–Rb borate glasses with NNa > 4 · 1027 m3 is more than two orders of magnitude slower than sodium diffusion (see Fig. 9). Therefore we conclude that the ionic dc-conductivity in mixed sodium–rubidium borate glasses on the sodium-rich side is due to diffusion of sodium ions. Concomitantly, the activation enthalpy of ionic conductivity is determined by the activation enthalpy of sodium diffusion: in this composition range the presence of rubidium ions does not affect
The molar volume of single alkali-borate glasses, when normalised to one mol boron atoms, show a clear trend with the ionic radii of the alkali ions. The large rubidium ions lead to a considerable expansion of the glass network with increasing rubidium concentration. The glass-transition temperatures of sodium- and rubidium-borate glasses increase with increasing alkali content and reach a shallow maximum between Y = 0.20 and Y = 0.30. This can be attributed to a variation of the ratio between BO3/2 and BO 4=2 units of the network and of the non-bridging oxygen content. Temperature dependent measurements of the dc conductivity of sodium- and rubidium-borate glasses with various alkali contents show a strong dependence of long-range ion transport and of the pertinent activation enthalpy on the alkali concentration. The comparison of the binary glass systems with ternary sodium–rubidium borate glasses indicates that an increased content of the BO 4=2 units of the glass network strongly enhances the ionic mobility. However, an addition of up to 8 mol% rubidium oxide to sodium-borate glass does not affect significantly the activation enthalpy of sodium diffusion. Acknowledgements We are grateful to Dr C. Cramer, Dr Y. Gao and Dr S. Murugavel (Institut fu¨r Physikalische Chemie) for their help performing some of the conductivity measurements, to Professor G. Schmitz, and Dr N.A. Stolwijk (both Institut fu¨r Materialphysik) for valuable comments on the manuscript. This work was financially supported by the Deutsche Forschungsgemeinschaft within the Sonderforschungsbereich 458. References [1] H.M. Schaeffer, J. Non-Cryst. Solids 67 (1984) 19. [2] A.K. Vershneya, Fundamentals of Inorganic Glasses, Academic, San Diego, 1994. [3] J.E. Shelby, Introduction to Glass Science and Technology, The Royal Society of Chemistry, Cambridge, 1997. [4] A. Bunde, P. Maass, M.D. Ingram, J. Non-Cryst. Solids 172–174 (1994) 1222. [5] M.D. Ingram, B. Roling, J. Phys.: Condens. Matter 15 (2003) 1595. [6] A.W. Imre, S. Voss, H. Mehrer, Phys. Chem. Chem. Phys. 4 (2002) 3219. [7] A.W. Imre, S. Voss, H. Mehrer, J. Non-Cryst. Solids 333 (2004) 231. [8] U. Schoo, C. Cramer, H. Mehrer, Solid State Ionics 138 (2000) 105. [9] G.H. Frischat, Ionic Diffusion in Oxide Glasses, Trans Tech, Zu¨rich, 1975.
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[21] C.C. Hunter, M.D. Ingram, Solid State Ionics 14 (1984) 31. [22] R.I. Bresker, K.S. Evstropiev, Zh. Prikl. Khim. (1952) 905. [23] A.H. Verhoef, A.W. Hartog, J. Non-Cryst. Solids 182 (1995) 235. [24] E. Ratai, M. Janssen, H. Eckert, Solid State Ionics 105 (1998) 25. [25] B. Roling, A. Happe, K. Funke, M.D. Ingram, Phys. Rev. Lett. 78 (1997) 2160. [26] P. Maass, J. Non-Cryst. Solids 255 (1999) 35. [27] H. Mehrer, A.W. Imre, S. Voss, Advances in Science and Technology 37, Mass and Charge Transport in Inorganic Materials II – CIMTEC 2002, in: P. Vincenzini, V. Buscaglia (Eds.), TECHNA Srl, 2003. [28] A.W. Imre, S. Voss, H. Mehrer, Defect Diffus. Forum 370 (2005) 273. [29] A.W. Imre, S. Divinsky, S. Voss, F. Berkemeier, H. Mehrer, J. Non-Cryst. Solids, submitted for publication.