- Email: [email protected]
On the mobility of alkaline earth ions in mixed alkali alkaline earth silicate glasses
Journal of Non-Crystalline Solids 328 (2003) 157–163 www.elsevier.com/locate/jnoncrysol
On the mobility of alkaline earth ions in mixed alkali alkaline earth silicate glasses Reiner Kirchheim
*
Institut f€ur Materialphysik, Universit€at G€ottingen, Hospitalstrasse 3-7, D-37073 G€ottingen, Germany Received 20 December 2002; received in revised form 27 February 2003
Abstract Activation energies of the diffusion of alkaline earth ions in silicate glasses of the composition A2 O 2BO 4SiO2 have been compiled recently, where A is an alkali and B an alkaline earth metal. A pronounced dependence on the ratio rB =rA of radii of alkaline earth ions B2þ and alkali ions Aþ has been detected. A minimum occurs at a ratio of unity and the increase of the activation energies is much steeper for rB =rA < 1 when compared with the one for rB =rA > 1. This size effect of alkaline earth mobility can be explained qualitatively and quantitatively by a model of the mixed alkali effect, where the ionic mobility is affected by both a distribution of site energies and the packing density of oxygen. 2003 Elsevier B.V. All rights reserved.
1. Introduction The mixed alkali effect occurs in oxidic glasses with constant alkali content but a varying fraction of two different alkali ions. The replacement of one alkali ion by another one changes the mobility of both ions by orders of magnitude. Then the electrical conductivity based on the ionic conduction has a deep minimum at a composition of about half and half. This pronounced effect has attracted the attention of many scientists with most of their work compiled in [1–3]. Besides older explanations of the mixed alkali effect [4,5], there have been a variety of recent ones [6–8]. Nevertheless, these proposed models are not satisfactory so far. Even
*
Tel.: +49-551 39 5001; fax: +49-551 39 5012. E-mail address: [email protected] (R. Kirchheim).
if they are physically sound they contain too many unknown parameters to allow a reliable quantitative evaluation of the huge amount of measured data. This is not the case for a model based on a distribution of site energies and a modification of packing densities of oxygen atoms [9,10]. There the mobility of the larger ion is reduced because the presence of the smaller ion leads to an increasing packing density of oxygen atoms, i.e. the mesh size of the silicate network is reduced when compared with the one containing the larger cations only. Compared to the network which contains the smaller ions only, the small ions experience an expanded mesh size in the mixed glass and, therefore, their mobility should be increased in the mixed case. However, it can be rationalized [10] that the smaller cations are preferentially occupying sites of lower energy with a concomitant higher activation barrier. This competition for the low
0022-3093/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0022-3093(03)00474-5
R. Kirchheim / Journal of Non-Crystalline Solids 328 (2003) 157–163
energy sites reduces the mobility of small ions and it overcompensates the effect of the larger mesh size. The influence of a changing mesh size of the silicate network on ionic mobility can be treated quantitatively without a fitting parameter by comparing it with the effect an applied hydrostatic pressure has on the packing density of atoms and the mobility of ions. The only unknown parameter in this concept of a mixed glass is the width of the site energy distribution. Strictly speaking, a site exchange energy has to be considered as discussed rigorously in [10]. The new concept of a combined action of both mesh size and site energy distribution is not in contradiction with the concept of channels of high ionic concentrations [11,12]. Evidence for these channels stems from molecular dynamics simulations of silicate glasses [13]. In order to avoid blocking of ionic conduction, the channels have to contain empty sites. Restricting the analysis of a distribution of site energies and changes of the mesh size to percolating channels will not affect the equations derived in [9,10]. This has been demonstrated for the diffusion of hydrogen through nanocrystalline palladium, where the transport of hydrogen is restricted to grain boundaries [14]. It is interesting to note that the quantitative description of mixed alkali glasses was successfully achieved in [9,10], whereas no serious attempt was made to treat binary glasses within the same concept. A first attempt in [15] is not correct, because it did not account properly for the structural modification induced by an increasing alkali content. An increasing alkali concentration changes both mesh size and site energy distribution. In the present study we discuss ternary silicate glasses containing a mixture of alkaline earth and alkali ions. The same equations as derived for mixtures of alkali ions alone are used and applied to experimental results of the activation energy of alkaline earth ions QB [16] as presented in Fig. 1. The glasses had the composition A2 O 2BO 4SiO4 , where A is an alkaline and B an alkaline earth metal. In Fig. 1 values of QB are plotted versus the ratio rB =rA , where rB is the radius of the alkaline earth ions B2þ and rA the one of alkali ions. Compared to the value near rB =rA ¼ 1 QB is increasing for both rB =rA < 1 and rB =rA > 1. However the
3 activation energy [eV]
158
A=K; B=Ca, Sr, Ba A=Cs; B=Ca, Sr, Ba A=Na; B=Mg, Ca, Sr, Ba A=Li; B=Mg, Ca, Sr, Ba
2.5
2
1.5 0.6 0.8
1
1.2 1.4 1.6 1.8 rB/rA
Fig. 1. Activation energies of diffusion of alkaline earth ions B2þ in mixed A2 O 2BO 4SiO4 glasses as determined by mechanical spectroscopy or by concentration depth profiling [16]. The slopes of the straight lines are calculated in this study without using a fitting parameter.
changes are not symmetric for the right and left hand side at rB =rA ¼ 1 with a much steeper increase for rB =rA < 1.
2. Larger alkaline earth ions As mentioned before the concept of mixed alkali glasses will be applied. The activation energy Qm R for the larger alkali ion R in the mixed glass was derived to be b Qm R ¼ QR þ
2KViR ðViR ViA Þxy; VO
ð1Þ
where QbR is the activation energy of R in the binary glass, K is the bulk modulus, ViR the activation volume of diffusion of R, VO is the volume of the glass per mole oxygen, ViR is the volume of the R-ion, ViA the one of the alkali ion, x is the fraction of the total alkali content, and y is the fraction of the smaller alkali cation. Rewriting the formula of A2 O 2BO 4SiO2 as [0.5A2 O 0.5B2 O2 ]1=3 [SiO2 ]2=3 reveals that x ¼ 1=3 and y ¼ 0:5. Eq. (1) is assumed to be valid for the alkaline earth ion as well, i.e. R is replaced by B. In addition we assume that activation volume ViB and ionic volume ViB are the same. This is approximately valid for alkali ions [9] in silicate glasses
R. Kirchheim / Journal of Non-Crystalline Solids 328 (2003) 157–163
ri (nm) Liþ Naþ Kþ Rbþ Csþ Mg2þ Ca2þ Sr2þ Ba2þ
0.076 0.102 0.138 0.152 0.167 0.072 0.100 0.118 0.135
KViB ðViB ViA Þ 3VO
ð2Þ
16p2 KL2 rB3 3 ½rB rA3 ; 27VO
ð3Þ
or b Qm B ¼ QB þ
where L ¼ 6 1023 is AvogadroÕs number. In order to compare this result with the experimental data in Fig. 1, the second term on the right hand side of Eq. (1) has to be approximated by a function of rB =rA . Only Li- and Na-containing glasses fulfill the requirement rB =rA > 1 and for them corresponding values of rB3 ðrB3 rA3 Þ are plotted in Fig. 2 versus rB =rA by using radii as compiled in Table 1. Eq. (3) predicts that the larger values of rB3 ðrB3 rA3 Þ for the Na-glasses should lead to larger activation energies for these glasses when compared with the Li-glasses. This is in agreement with the data shown in Fig. 1. In addition we compare the slope of the Qm R -values for rB =rA > 1 which is predicted by Eq. (3) to be dQm 16p2 KL2 drB3 ½rB3 rA3 B ¼ ; dðrB =rA Þ dðrB =rA Þ 27VO
40 30 20 10 0
and other glasses as well [17], if ionic radii for a coordination of 6 as presented in Table 1 are used for the calculation of ViB . Then Eq. (1) becomes b Qm B ¼ QB þ
50 rB3(rB3-rA3), [10-7nm6]
Table 1 Ionic radii for alkaline and alkaline earth ions and for a coordination of 6
ð4Þ
if we assume that QbB does not depend upon the ratio of radii. The following values are inserted in Eq. (4): VO ¼ 13:6 cm3 /mol [9,18] and the slopes of the straight lines in Fig. 2 of 1.5 · 1059 m6 for Na and 9.5 · 1060 m6 for Li being used for the derivative on the right hand side of Eq. (4). Values used for the bulk modulus were K ¼ 45 GPa for
159
Na-glasses Li-glasses Li-glasses
1
1.2
1.4
1.6 rB/rA
1.8
2
Fig. 2. The polynomial rB3 ðrB3 rA3 Þ as appearing in Eqs. (3) and (4) is plotted versus the ratio of ionic radii rB =rA . The slope of the straight lines is used for the derivative on the right hand side of Eq. (4).
sodium glasses and K ¼ 55 GPa for lithium glasses. These values were calculated for similar compositions from YoungÕs modulus and PoissonÕs ratio given in [19]. With these data the following result is obtained: dQm B dðrB =rA Þ 104 kJ=mol ¼ 1:08 eV for Na-glasses; ¼ 80 kJ=mol ¼ 0:83 eV for Li-glasses: ð5Þ Straight lines with these slopes are drawn in Fig. 1 indicating good agreement with the behavior of the experimental results. 3. Width of energy distribution for ion exchange Smaller cations are assumed to sit in the sites of lowest energy and, therefore, the width of the site energy distribution wx has to be known. The result obtained for wx in [9,10] has to be modified, in order to take the double charge of the alkaline earth ions into account. In [10] the width wx of the energy distribution for an exchange of two different alkali ions was calculated by considering electrostatic contributions only. In the present context a double charged alkaline earth has to be replaced by two alkali ions, in order to maintain charge neutrality. For the sake of simplicity and the
160
R. Kirchheim / Journal of Non-Crystalline Solids 328 (2003) 157–163
The exchange energy is given by DE ¼ Eb Ea " 2 X 1 2e2 2e2 ¼ 4pe0 e k¼1 rB þ ra þ db rk rB þ ra þ da rk # 4 X e2 e2 þ : r A þ r a þ db r i r A þ r a þ da r i i¼1 ð8Þ
Fig. 3. Hypothetical configurations of two non-bridging oxygen atoms with two alkali ions or one alkaline earth ion respectively. The various separations of the surfaces of ions, i.e. dri stems from other atoms which are not shown and which hinder a closer approach of the ions.
advantage of getting a closed solution, the interaction of the cations with two non-bridging oxygen ions is considered only (cf. Fig. 3). The distances between the centers of ions are rB þ ra þ drk (k ¼ 1; 2) for the cation B2þ , where ra is the radius of the non-bridging oxygen anion. The parameter drk > 0 takes care of configurations where the cation due to blocking by other atoms is not allowed to approach the center of the negative charge as close as possible. For the two alkali ions the corresponding distances are rA þ ra þ dri (i ¼ 1, 2, 3, 4). Before the exchange of B2þ and 2Aþ the set of distance variations may be db ri or db rk respectively. Then the energy for the interaction between anions and cations is " # 2 4 X X 1 2e2 e2 b þ : E ¼ 4pe0 e k¼1 rB þ ra þ db rk r þ ra þ db ri i¼1 A ð6Þ After the ion exchange the distance variations shall be da rk and the Coulomb interaction energy becomes " # 2 4 X X 1 2e2 e2 a þ : E ¼ 4pe0 e k¼1 rB þ ra þ da rk r þ ra þ da ri i¼1 A ð7Þ
With the assumption that all dr are much smaller than rB Eq. (8) can be simplified to " 2 X 1 2e2 ðda rk db rk Þ DE ¼ 4pe0 e k¼1 ðrB þ ra Þ2 # 4 X e2 ðda ri db ri Þ þ : ð9Þ ðrA þ ra Þ2 i¼1 If for a given configuration of ions the smaller alkaline earth ion gets closer to the anions after the exchange and, therefore, the average hðda rk db rk Þi will be negative. Then the exchanged alkali ions move further apart from the adjacent anions and on the average the distance changes hðda ri db ri Þi become positive. The tacit assumption for this reasoning is a stiff amorphous network allowing elastic expansions only. The whole situation of alkaline earth ions coming closer to centers of the negative charge is approximated by an average distance change dr ¼ hda ri db ri i ¼ hda rk db rk i > 0 and an average exchange energy # " 1 4e2 dr 4e2 dr hDEi ¼ þ : 4pe0 e ðrB þ ra Þ2 ðrA þ ra Þ2
ð10Þ
ð11Þ
Because of rA > rB this average energy is negative. The reverse situation that the alkali ions come closer to the anions after the exchange, i.e. dr < 0 leads to a positive average exchange energy and, therefore, it has a much lower probability to occur. Thus in the stiff amorphous network below the glass transition temperature the smaller ions occupy sites with a smaller distance to the center of negative charge. To remove them from these sites during diffusion requires a higher activation energy when compared with the one of the larger cations sitting further apart from the anions. This
R. Kirchheim / Journal of Non-Crystalline Solids 328 (2003) 157–163
120 100 wx, kJ/Mol
b Qm A ¼ QA þ wxð1 yÞ
D(y,400C) D(y,450C) Q(y)
80
40 20 0
25
50
75
100
difference of radii (rR-rA), pm
b Qm B ¼ QB þ
Fig. 4. Width wx of the site energy distribution as a function of the difference of cation radii (rR rA ). The straight line are least square fits to the experimental points. Data obtained from diffusion coefficients do not lie on a straight line through the origin because they contain a temperature dependent (i.e. entropic) contribution (cf. [10]).
difference in activation energies of diffusion has been quantitatively described by assuming a box type distribution of exchange energies with a width wx ¼ hDEi in [10]. In accordance with this former treatment the ionic radii rA and rB in Eq. (11) will be replaced by an average one as defined by rBA ¼ ðrB þ rA Þ=2. For Dr ¼ rA rB rBA Eq. (11) becomes wx ¼ hDEi ¼
2e2 dr Dr pe0 eðrAB þ ra Þ
3
:
ð12Þ
Because of the double charge of alkaline earth ions and the interaction with two oxygen ions the width is by a factor of 4 larger than the one calculated for an alkali/alkali exchange. By plotting wx values obtained from diffusivity and conductivity data of mixed alkali glasses [10] versus the difference of radii Dr straight lines were obtained as shown in Fig. 4. Their slope yields a value of e2 dr 2pee0 ðrAB þ ra Þ
3
¼ 1:7 1012
kJ : m mol
2KViA ðViR ViA Þxð1 yÞ; VO ð14Þ
where wx is the width of the energy distribution for site exchanges between the alkalis A and R. Note that in the present context of rB =rA < 1 A in Eq. (14) has to be replaced by B and R by A and again the activation volume will be replaced by the ionic volume. Then we obtain for the mixed alkaline earth alkali glasses of this study
60
0
161
ð13Þ
4. Smaller alkaline earth ions For smaller alkali ions A in mixed alkali glasses the activation energy for diffusion was derived as
wx KViB ðViA ViB Þ: 2 3VO
ð15Þ
The value of x ¼ 1=3 was not inserted in the term wx because this term was anticipated in [9] to be proportional to x. However such a dependency cannot be proved at present. In addition, it will not affect the following reasoning. By using Eq. (12) we obtain b Qm B ¼ QB þ
e2 dr Dr pe0 eðrAB þ ra Þ
3
KViB ðViA ViB Þ: 3VO ð16Þ
Only changes of Qm B as a function of rB =rA are considered, in order to allow a comparison with the data in Fig. 1. Thus the following derivative is calculated: dQm e2 dr dðrA rB Þ B ¼ dðrB =rA Þ pe0 eðrAB þ ra Þ3 dðrB =rA Þ
16p2 KL2 d½rB3 ðrB3 rA3 Þ : dðrB =rA Þ 27VO
ð17Þ
Reasonable values for the derivatives on the right hand side of Eq. (17) are obtained from the slopes of the straight lines in Figs. 5 and 6, where rA rB and rB3 ðrB3 rA3 Þ values for relevant combinations of alkaline earth and alkali ions are plotted versus rB =rA . The corresponding slopes are )1.6 · 1010 m and )1.2 · 1059 m6 . The latter value is less reliable because of the pronounced scatter of data points in Fig. 6. This is of minor importance only, because the magnitude of the second term on the right hand side of Eq. (17) is much smaller than the first one. This is proved by using Eq. (13) and as before
162
R. Kirchheim / Journal of Non-Crystalline Solids 328 (2003) 157–163
5. Conclusions
0.1
rA-rB [nm]
0.08 0.06 0.04 0.02 0 0.5
0.6
0.7
0.8 rB/rA
0.9
1
Fig. 5. The difference rB rA as appearing in Eq. (17) is plotted versus the ratio of ionic radii rB =rA . The slope of the straight lines is used for the derivative on the right hand side of Eq. (17).
rB3(rB3-rA3), [10-7nm6]
60 50 40 30 20 10 0 0.5
0.6
0.7
0.8 rB/rA
0.9
1
Fig. 6. The polynomial rB3 ðrB3 rA3 Þ as appearing in Eq. (17) is plotted versus the ratio of ionic radii rB =rA . The slope of the straight line which is obtained from a least square fit to the data points is used for the derivative on the right hand side of Eq. (17).
K ¼ 45–55 GPa 50 GPa, VO ¼ 1:36 105 m3 / mol: dQm B ¼ 272 kJ=mol þ 93 kJ=mol dðrB =rA Þ ¼ 179 kJ=mol ¼ 1:85 eV:
The choice of the variable rB =rA for the presentation of activation energies in Fig. 1 was more or less arbitrary and not based on a physical model. In addition the error bars at the experimental data are large enough to allow other dependencies of Qm B on rB and rA to be fitted equally well as the ones developed in this study. Nevertheless the asymmetry around rB ¼ rA remains as a challenging feature to be answered qualitatively at least. This has been done in chapter 2 for rB =rA > 1 by assuming that a decreasing size of alkali ions give rise to a decreasing mesh size of the silicon/oxygen network. The larger the alkaline earth ions are the more their mobility is reduced by the narrowing of the network. For rB =rA < 1 the smaller alkaline earth ions occupy low energy configurations in the glass, because they can approach the centers of the negative charge more closely than the larger alkali ions. The larger the size difference is between these two types of ions the larger the energy difference becomes and the larger the activation energy for diffusion is. The larger changes of Qm B for rB =rA < 1 in comparison with rB =rA > 1 can be rationalized, because electrostatic interaction forces are usually stronger than elastic ones. This is also supported by the quantitative modeling in chapters 3 and 4. Of course the quantitative treatment in this study is not rigorous because of the complexity of the structure and the variety of interactions between the atoms. Major objections are the simplicity of the treatment of electrostatic interactions considering nearest neighbors only and the assumption of QbB (the activation energy of diffusion of B-ions in a binary glass) does not depend on rB =rA . Nevertheless, the agreement of the calculated slopes in Fig. 1 with the experimental data supports the model of transport in mixed glasses, the more as the agreement was achieved without adjusting a parameter.
ð18Þ Acknowledgement
A straight line with a slope given by Eq. (18) is included in Fig. 1 yielding good agreement with experimental data.
The author is grateful for financial support by the Deutsche Forschungsgemeinschaft (SFB 602).
R. Kirchheim / Journal of Non-Crystalline Solids 328 (2003) 157–163
References [1] K. Hughes, J.O. Isard, in: J. Hladik (Ed.), Physics of Electrolytes, vol. 1, Academic Press, London, 1972, p. 387. [2] G.H. Frischat, Ionic Diffusion in Oxide Glasses, TransTech, Aedermansdorf, 1975, p. 161. [3] M.D. Ingram, in: R.W. Cahn, P. Haasen, E.J. Kramer (Eds.), Materials Science and Technology, vol. 9, VCH, New York, 1991, p. 715. [4] R.L. Myuller, Zr. Tek. Fiz. 25 (1955) 236. [5] J.R. Hendrickson, P.J. Bray, Phys. Chem. Glasses 13 (1972) 43. [6] K. Funke, Prog. Solid State Chem. 22 (1993) 111; J. Non-Cryst. Solids 172–174 (1994) 1215. [7] P. Maass, A. Bunde, M.D. Ingram, Phys. Rev. Lett. 68 (1992) 3064; A. Bunde, M.D. Ingram, P. Maass, J. Non-Cryst. Solids 172–174 (1994) 1222.
163
[8] J.E. Davidson, M.D. Ingram, A. Bunde, K. Funke, J. NonCryst. Solids 203 (1996) 246. [9] R. Kirchheim, J. Non-Cryst. Solids 272 (2000) 85. [10] R. Kirchheim, D. Paulmann, J. Non-Cryst. Solids 286 (2001) 210. [11] G.N. Greaves, J. Non-Cryst. Solids 71 (1985) 203. [12] M.D. Ingram, Philos. Mag. B 60 (1989) 729. [13] P. Jund, W. Kob, R. Jullien, Philos. Mag 82 (2002) 597. [14] T. M€ utschele, R. Kirchheim, Scripta Metall. 21 (1987) 135; R. Kirchheim, Phys. Scripta T 94 (2001) 58. [15] R. Kirchheim, U. Stolz, J. Non-Cryst. Solids 70 (1985) 323. [16] F. Natrup, H. Bracht, C. Martiny, S. Murugavel, B. Roling, Phys. Chem. Chem. Phys 4 (2002) 3225. [17] P.W.S.K. Bandaranayake, C.T. Imirie, M.D. Ingram, Phys. Chem. Chem. Phys 4 (2002) 3209. [18] R. Kirchheim, Glass Sci. Technol. 75 (2002) 294. [19] O.V. Mazurin, M.V. Steltsina, T.P. Shvaiko-Shvaikovskaya, in: Handbook of Glass Data, vol. C, Elsevier, Amsterdam, 1991, p. 201ff.
On the mobility of alkaline earth ions in mixed alkali alkaline earth silicate glasses Reiner Kirchheim
*
Institut f€ur Materialphysik, Universit€at G€ottingen, Hospitalstrasse 3-7, D-37073 G€ottingen, Germany Received 20 December 2002; received in revised form 27 February 2003
Abstract Activation energies of the diffusion of alkaline earth ions in silicate glasses of the composition A2 O 2BO 4SiO2 have been compiled recently, where A is an alkali and B an alkaline earth metal. A pronounced dependence on the ratio rB =rA of radii of alkaline earth ions B2þ and alkali ions Aþ has been detected. A minimum occurs at a ratio of unity and the increase of the activation energies is much steeper for rB =rA < 1 when compared with the one for rB =rA > 1. This size effect of alkaline earth mobility can be explained qualitatively and quantitatively by a model of the mixed alkali effect, where the ionic mobility is affected by both a distribution of site energies and the packing density of oxygen. 2003 Elsevier B.V. All rights reserved.
1. Introduction The mixed alkali effect occurs in oxidic glasses with constant alkali content but a varying fraction of two different alkali ions. The replacement of one alkali ion by another one changes the mobility of both ions by orders of magnitude. Then the electrical conductivity based on the ionic conduction has a deep minimum at a composition of about half and half. This pronounced effect has attracted the attention of many scientists with most of their work compiled in [1–3]. Besides older explanations of the mixed alkali effect [4,5], there have been a variety of recent ones [6–8]. Nevertheless, these proposed models are not satisfactory so far. Even
*
Tel.: +49-551 39 5001; fax: +49-551 39 5012. E-mail address: [email protected] (R. Kirchheim).
if they are physically sound they contain too many unknown parameters to allow a reliable quantitative evaluation of the huge amount of measured data. This is not the case for a model based on a distribution of site energies and a modification of packing densities of oxygen atoms [9,10]. There the mobility of the larger ion is reduced because the presence of the smaller ion leads to an increasing packing density of oxygen atoms, i.e. the mesh size of the silicate network is reduced when compared with the one containing the larger cations only. Compared to the network which contains the smaller ions only, the small ions experience an expanded mesh size in the mixed glass and, therefore, their mobility should be increased in the mixed case. However, it can be rationalized [10] that the smaller cations are preferentially occupying sites of lower energy with a concomitant higher activation barrier. This competition for the low
0022-3093/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0022-3093(03)00474-5
R. Kirchheim / Journal of Non-Crystalline Solids 328 (2003) 157–163
energy sites reduces the mobility of small ions and it overcompensates the effect of the larger mesh size. The influence of a changing mesh size of the silicate network on ionic mobility can be treated quantitatively without a fitting parameter by comparing it with the effect an applied hydrostatic pressure has on the packing density of atoms and the mobility of ions. The only unknown parameter in this concept of a mixed glass is the width of the site energy distribution. Strictly speaking, a site exchange energy has to be considered as discussed rigorously in [10]. The new concept of a combined action of both mesh size and site energy distribution is not in contradiction with the concept of channels of high ionic concentrations [11,12]. Evidence for these channels stems from molecular dynamics simulations of silicate glasses [13]. In order to avoid blocking of ionic conduction, the channels have to contain empty sites. Restricting the analysis of a distribution of site energies and changes of the mesh size to percolating channels will not affect the equations derived in [9,10]. This has been demonstrated for the diffusion of hydrogen through nanocrystalline palladium, where the transport of hydrogen is restricted to grain boundaries [14]. It is interesting to note that the quantitative description of mixed alkali glasses was successfully achieved in [9,10], whereas no serious attempt was made to treat binary glasses within the same concept. A first attempt in [15] is not correct, because it did not account properly for the structural modification induced by an increasing alkali content. An increasing alkali concentration changes both mesh size and site energy distribution. In the present study we discuss ternary silicate glasses containing a mixture of alkaline earth and alkali ions. The same equations as derived for mixtures of alkali ions alone are used and applied to experimental results of the activation energy of alkaline earth ions QB [16] as presented in Fig. 1. The glasses had the composition A2 O 2BO 4SiO4 , where A is an alkaline and B an alkaline earth metal. In Fig. 1 values of QB are plotted versus the ratio rB =rA , where rB is the radius of the alkaline earth ions B2þ and rA the one of alkali ions. Compared to the value near rB =rA ¼ 1 QB is increasing for both rB =rA < 1 and rB =rA > 1. However the
3 activation energy [eV]
158
A=K; B=Ca, Sr, Ba A=Cs; B=Ca, Sr, Ba A=Na; B=Mg, Ca, Sr, Ba A=Li; B=Mg, Ca, Sr, Ba
2.5
2
1.5 0.6 0.8
1
1.2 1.4 1.6 1.8 rB/rA
Fig. 1. Activation energies of diffusion of alkaline earth ions B2þ in mixed A2 O 2BO 4SiO4 glasses as determined by mechanical spectroscopy or by concentration depth profiling [16]. The slopes of the straight lines are calculated in this study without using a fitting parameter.
changes are not symmetric for the right and left hand side at rB =rA ¼ 1 with a much steeper increase for rB =rA < 1.
2. Larger alkaline earth ions As mentioned before the concept of mixed alkali glasses will be applied. The activation energy Qm R for the larger alkali ion R in the mixed glass was derived to be b Qm R ¼ QR þ
2KViR ðViR ViA Þxy; VO
ð1Þ
where QbR is the activation energy of R in the binary glass, K is the bulk modulus, ViR the activation volume of diffusion of R, VO is the volume of the glass per mole oxygen, ViR is the volume of the R-ion, ViA the one of the alkali ion, x is the fraction of the total alkali content, and y is the fraction of the smaller alkali cation. Rewriting the formula of A2 O 2BO 4SiO2 as [0.5A2 O 0.5B2 O2 ]1=3 [SiO2 ]2=3 reveals that x ¼ 1=3 and y ¼ 0:5. Eq. (1) is assumed to be valid for the alkaline earth ion as well, i.e. R is replaced by B. In addition we assume that activation volume ViB and ionic volume ViB are the same. This is approximately valid for alkali ions [9] in silicate glasses
R. Kirchheim / Journal of Non-Crystalline Solids 328 (2003) 157–163
ri (nm) Liþ Naþ Kþ Rbþ Csþ Mg2þ Ca2þ Sr2þ Ba2þ
0.076 0.102 0.138 0.152 0.167 0.072 0.100 0.118 0.135
KViB ðViB ViA Þ 3VO
ð2Þ
16p2 KL2 rB3 3 ½rB rA3 ; 27VO
ð3Þ
or b Qm B ¼ QB þ
where L ¼ 6 1023 is AvogadroÕs number. In order to compare this result with the experimental data in Fig. 1, the second term on the right hand side of Eq. (1) has to be approximated by a function of rB =rA . Only Li- and Na-containing glasses fulfill the requirement rB =rA > 1 and for them corresponding values of rB3 ðrB3 rA3 Þ are plotted in Fig. 2 versus rB =rA by using radii as compiled in Table 1. Eq. (3) predicts that the larger values of rB3 ðrB3 rA3 Þ for the Na-glasses should lead to larger activation energies for these glasses when compared with the Li-glasses. This is in agreement with the data shown in Fig. 1. In addition we compare the slope of the Qm R -values for rB =rA > 1 which is predicted by Eq. (3) to be dQm 16p2 KL2 drB3 ½rB3 rA3 B ¼ ; dðrB =rA Þ dðrB =rA Þ 27VO
40 30 20 10 0
and other glasses as well [17], if ionic radii for a coordination of 6 as presented in Table 1 are used for the calculation of ViB . Then Eq. (1) becomes b Qm B ¼ QB þ
50 rB3(rB3-rA3), [10-7nm6]
Table 1 Ionic radii for alkaline and alkaline earth ions and for a coordination of 6
ð4Þ
if we assume that QbB does not depend upon the ratio of radii. The following values are inserted in Eq. (4): VO ¼ 13:6 cm3 /mol [9,18] and the slopes of the straight lines in Fig. 2 of 1.5 · 1059 m6 for Na and 9.5 · 1060 m6 for Li being used for the derivative on the right hand side of Eq. (4). Values used for the bulk modulus were K ¼ 45 GPa for
159
Na-glasses Li-glasses Li-glasses
1
1.2
1.4
1.6 rB/rA
1.8
2
Fig. 2. The polynomial rB3 ðrB3 rA3 Þ as appearing in Eqs. (3) and (4) is plotted versus the ratio of ionic radii rB =rA . The slope of the straight lines is used for the derivative on the right hand side of Eq. (4).
sodium glasses and K ¼ 55 GPa for lithium glasses. These values were calculated for similar compositions from YoungÕs modulus and PoissonÕs ratio given in [19]. With these data the following result is obtained: dQm B dðrB =rA Þ 104 kJ=mol ¼ 1:08 eV for Na-glasses; ¼ 80 kJ=mol ¼ 0:83 eV for Li-glasses: ð5Þ Straight lines with these slopes are drawn in Fig. 1 indicating good agreement with the behavior of the experimental results. 3. Width of energy distribution for ion exchange Smaller cations are assumed to sit in the sites of lowest energy and, therefore, the width of the site energy distribution wx has to be known. The result obtained for wx in [9,10] has to be modified, in order to take the double charge of the alkaline earth ions into account. In [10] the width wx of the energy distribution for an exchange of two different alkali ions was calculated by considering electrostatic contributions only. In the present context a double charged alkaline earth has to be replaced by two alkali ions, in order to maintain charge neutrality. For the sake of simplicity and the
160
R. Kirchheim / Journal of Non-Crystalline Solids 328 (2003) 157–163
The exchange energy is given by DE ¼ Eb Ea " 2 X 1 2e2 2e2 ¼ 4pe0 e k¼1 rB þ ra þ db rk rB þ ra þ da rk # 4 X e2 e2 þ : r A þ r a þ db r i r A þ r a þ da r i i¼1 ð8Þ
Fig. 3. Hypothetical configurations of two non-bridging oxygen atoms with two alkali ions or one alkaline earth ion respectively. The various separations of the surfaces of ions, i.e. dri stems from other atoms which are not shown and which hinder a closer approach of the ions.
advantage of getting a closed solution, the interaction of the cations with two non-bridging oxygen ions is considered only (cf. Fig. 3). The distances between the centers of ions are rB þ ra þ drk (k ¼ 1; 2) for the cation B2þ , where ra is the radius of the non-bridging oxygen anion. The parameter drk > 0 takes care of configurations where the cation due to blocking by other atoms is not allowed to approach the center of the negative charge as close as possible. For the two alkali ions the corresponding distances are rA þ ra þ dri (i ¼ 1, 2, 3, 4). Before the exchange of B2þ and 2Aþ the set of distance variations may be db ri or db rk respectively. Then the energy for the interaction between anions and cations is " # 2 4 X X 1 2e2 e2 b þ : E ¼ 4pe0 e k¼1 rB þ ra þ db rk r þ ra þ db ri i¼1 A ð6Þ After the ion exchange the distance variations shall be da rk and the Coulomb interaction energy becomes " # 2 4 X X 1 2e2 e2 a þ : E ¼ 4pe0 e k¼1 rB þ ra þ da rk r þ ra þ da ri i¼1 A ð7Þ
With the assumption that all dr are much smaller than rB Eq. (8) can be simplified to " 2 X 1 2e2 ðda rk db rk Þ DE ¼ 4pe0 e k¼1 ðrB þ ra Þ2 # 4 X e2 ðda ri db ri Þ þ : ð9Þ ðrA þ ra Þ2 i¼1 If for a given configuration of ions the smaller alkaline earth ion gets closer to the anions after the exchange and, therefore, the average hðda rk db rk Þi will be negative. Then the exchanged alkali ions move further apart from the adjacent anions and on the average the distance changes hðda ri db ri Þi become positive. The tacit assumption for this reasoning is a stiff amorphous network allowing elastic expansions only. The whole situation of alkaline earth ions coming closer to centers of the negative charge is approximated by an average distance change dr ¼ hda ri db ri i ¼ hda rk db rk i > 0 and an average exchange energy # " 1 4e2 dr 4e2 dr hDEi ¼ þ : 4pe0 e ðrB þ ra Þ2 ðrA þ ra Þ2
ð10Þ
ð11Þ
Because of rA > rB this average energy is negative. The reverse situation that the alkali ions come closer to the anions after the exchange, i.e. dr < 0 leads to a positive average exchange energy and, therefore, it has a much lower probability to occur. Thus in the stiff amorphous network below the glass transition temperature the smaller ions occupy sites with a smaller distance to the center of negative charge. To remove them from these sites during diffusion requires a higher activation energy when compared with the one of the larger cations sitting further apart from the anions. This
R. Kirchheim / Journal of Non-Crystalline Solids 328 (2003) 157–163
120 100 wx, kJ/Mol
b Qm A ¼ QA þ wxð1 yÞ
D(y,400C) D(y,450C) Q(y)
80
40 20 0
25
50
75
100
difference of radii (rR-rA), pm
b Qm B ¼ QB þ
Fig. 4. Width wx of the site energy distribution as a function of the difference of cation radii (rR rA ). The straight line are least square fits to the experimental points. Data obtained from diffusion coefficients do not lie on a straight line through the origin because they contain a temperature dependent (i.e. entropic) contribution (cf. [10]).
difference in activation energies of diffusion has been quantitatively described by assuming a box type distribution of exchange energies with a width wx ¼ hDEi in [10]. In accordance with this former treatment the ionic radii rA and rB in Eq. (11) will be replaced by an average one as defined by rBA ¼ ðrB þ rA Þ=2. For Dr ¼ rA rB rBA Eq. (11) becomes wx ¼ hDEi ¼
2e2 dr Dr pe0 eðrAB þ ra Þ
3
:
ð12Þ
Because of the double charge of alkaline earth ions and the interaction with two oxygen ions the width is by a factor of 4 larger than the one calculated for an alkali/alkali exchange. By plotting wx values obtained from diffusivity and conductivity data of mixed alkali glasses [10] versus the difference of radii Dr straight lines were obtained as shown in Fig. 4. Their slope yields a value of e2 dr 2pee0 ðrAB þ ra Þ
3
¼ 1:7 1012
kJ : m mol
2KViA ðViR ViA Þxð1 yÞ; VO ð14Þ
where wx is the width of the energy distribution for site exchanges between the alkalis A and R. Note that in the present context of rB =rA < 1 A in Eq. (14) has to be replaced by B and R by A and again the activation volume will be replaced by the ionic volume. Then we obtain for the mixed alkaline earth alkali glasses of this study
60
0
161
ð13Þ
4. Smaller alkaline earth ions For smaller alkali ions A in mixed alkali glasses the activation energy for diffusion was derived as
wx KViB ðViA ViB Þ: 2 3VO
ð15Þ
The value of x ¼ 1=3 was not inserted in the term wx because this term was anticipated in [9] to be proportional to x. However such a dependency cannot be proved at present. In addition, it will not affect the following reasoning. By using Eq. (12) we obtain b Qm B ¼ QB þ
e2 dr Dr pe0 eðrAB þ ra Þ
3
KViB ðViA ViB Þ: 3VO ð16Þ
Only changes of Qm B as a function of rB =rA are considered, in order to allow a comparison with the data in Fig. 1. Thus the following derivative is calculated: dQm e2 dr dðrA rB Þ B ¼ dðrB =rA Þ pe0 eðrAB þ ra Þ3 dðrB =rA Þ
16p2 KL2 d½rB3 ðrB3 rA3 Þ : dðrB =rA Þ 27VO
ð17Þ
Reasonable values for the derivatives on the right hand side of Eq. (17) are obtained from the slopes of the straight lines in Figs. 5 and 6, where rA rB and rB3 ðrB3 rA3 Þ values for relevant combinations of alkaline earth and alkali ions are plotted versus rB =rA . The corresponding slopes are )1.6 · 1010 m and )1.2 · 1059 m6 . The latter value is less reliable because of the pronounced scatter of data points in Fig. 6. This is of minor importance only, because the magnitude of the second term on the right hand side of Eq. (17) is much smaller than the first one. This is proved by using Eq. (13) and as before
162
R. Kirchheim / Journal of Non-Crystalline Solids 328 (2003) 157–163
5. Conclusions
0.1
rA-rB [nm]
0.08 0.06 0.04 0.02 0 0.5
0.6
0.7
0.8 rB/rA
0.9
1
Fig. 5. The difference rB rA as appearing in Eq. (17) is plotted versus the ratio of ionic radii rB =rA . The slope of the straight lines is used for the derivative on the right hand side of Eq. (17).
rB3(rB3-rA3), [10-7nm6]
60 50 40 30 20 10 0 0.5
0.6
0.7
0.8 rB/rA
0.9
1
Fig. 6. The polynomial rB3 ðrB3 rA3 Þ as appearing in Eq. (17) is plotted versus the ratio of ionic radii rB =rA . The slope of the straight line which is obtained from a least square fit to the data points is used for the derivative on the right hand side of Eq. (17).
K ¼ 45–55 GPa 50 GPa, VO ¼ 1:36 105 m3 / mol: dQm B ¼ 272 kJ=mol þ 93 kJ=mol dðrB =rA Þ ¼ 179 kJ=mol ¼ 1:85 eV:
The choice of the variable rB =rA for the presentation of activation energies in Fig. 1 was more or less arbitrary and not based on a physical model. In addition the error bars at the experimental data are large enough to allow other dependencies of Qm B on rB and rA to be fitted equally well as the ones developed in this study. Nevertheless the asymmetry around rB ¼ rA remains as a challenging feature to be answered qualitatively at least. This has been done in chapter 2 for rB =rA > 1 by assuming that a decreasing size of alkali ions give rise to a decreasing mesh size of the silicon/oxygen network. The larger the alkaline earth ions are the more their mobility is reduced by the narrowing of the network. For rB =rA < 1 the smaller alkaline earth ions occupy low energy configurations in the glass, because they can approach the centers of the negative charge more closely than the larger alkali ions. The larger the size difference is between these two types of ions the larger the energy difference becomes and the larger the activation energy for diffusion is. The larger changes of Qm B for rB =rA < 1 in comparison with rB =rA > 1 can be rationalized, because electrostatic interaction forces are usually stronger than elastic ones. This is also supported by the quantitative modeling in chapters 3 and 4. Of course the quantitative treatment in this study is not rigorous because of the complexity of the structure and the variety of interactions between the atoms. Major objections are the simplicity of the treatment of electrostatic interactions considering nearest neighbors only and the assumption of QbB (the activation energy of diffusion of B-ions in a binary glass) does not depend on rB =rA . Nevertheless, the agreement of the calculated slopes in Fig. 1 with the experimental data supports the model of transport in mixed glasses, the more as the agreement was achieved without adjusting a parameter.
ð18Þ Acknowledgement
A straight line with a slope given by Eq. (18) is included in Fig. 1 yielding good agreement with experimental data.
The author is grateful for financial support by the Deutsche Forschungsgemeinschaft (SFB 602).
R. Kirchheim / Journal of Non-Crystalline Solids 328 (2003) 157–163
References [1] K. Hughes, J.O. Isard, in: J. Hladik (Ed.), Physics of Electrolytes, vol. 1, Academic Press, London, 1972, p. 387. [2] G.H. Frischat, Ionic Diffusion in Oxide Glasses, TransTech, Aedermansdorf, 1975, p. 161. [3] M.D. Ingram, in: R.W. Cahn, P. Haasen, E.J. Kramer (Eds.), Materials Science and Technology, vol. 9, VCH, New York, 1991, p. 715. [4] R.L. Myuller, Zr. Tek. Fiz. 25 (1955) 236. [5] J.R. Hendrickson, P.J. Bray, Phys. Chem. Glasses 13 (1972) 43. [6] K. Funke, Prog. Solid State Chem. 22 (1993) 111; J. Non-Cryst. Solids 172–174 (1994) 1215. [7] P. Maass, A. Bunde, M.D. Ingram, Phys. Rev. Lett. 68 (1992) 3064; A. Bunde, M.D. Ingram, P. Maass, J. Non-Cryst. Solids 172–174 (1994) 1222.
163
[8] J.E. Davidson, M.D. Ingram, A. Bunde, K. Funke, J. NonCryst. Solids 203 (1996) 246. [9] R. Kirchheim, J. Non-Cryst. Solids 272 (2000) 85. [10] R. Kirchheim, D. Paulmann, J. Non-Cryst. Solids 286 (2001) 210. [11] G.N. Greaves, J. Non-Cryst. Solids 71 (1985) 203. [12] M.D. Ingram, Philos. Mag. B 60 (1989) 729. [13] P. Jund, W. Kob, R. Jullien, Philos. Mag 82 (2002) 597. [14] T. M€ utschele, R. Kirchheim, Scripta Metall. 21 (1987) 135; R. Kirchheim, Phys. Scripta T 94 (2001) 58. [15] R. Kirchheim, U. Stolz, J. Non-Cryst. Solids 70 (1985) 323. [16] F. Natrup, H. Bracht, C. Martiny, S. Murugavel, B. Roling, Phys. Chem. Chem. Phys 4 (2002) 3225. [17] P.W.S.K. Bandaranayake, C.T. Imirie, M.D. Ingram, Phys. Chem. Chem. Phys 4 (2002) 3209. [18] R. Kirchheim, Glass Sci. Technol. 75 (2002) 294. [19] O.V. Mazurin, M.V. Steltsina, T.P. Shvaiko-Shvaikovskaya, in: Handbook of Glass Data, vol. C, Elsevier, Amsterdam, 1991, p. 201ff.