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The relevance of site energy distribution for the mixed alkali effect
Journal of Non-Crystalline Solids 286 (2001) 210±223
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The relevance of site energy distribution for the mixed alkali eect Reiner Kirchheim *, Dirk Paulmann Institut f ur Materialphysik, Der Georg-August-Universitat Gottingen, Hospitalstrasse 3-7, D-37073 Gottingen, Germany Received 14 September 2000
Abstract The mixed alkali eect (MAE) can be explained as an interplay between network modi®cations induced by alkali ions of dierent size and an energy distribution of sites with a preferential occupancy of the low energy sites by the smaller cations. The underlying physical assumptions of this concept are discussed in the present paper in more detail and they are compared with results of Monte Carlo simulations of diusion and the Anderson±Stuart model. It will be shown that the new concept is actually hidden in the Anderson±Stuart model. Applying the new concept to experimental data has been done so far with a box-type distribution of site energies. Besides the oversimpli®cation, a major drawback of this distribution is that the diusion coecient of the smaller and minority cation is underestimated. It will be shown in the present study that by a dierent choice of distribution the concentration dependence of the diusivity of the minority cations can be understood. In addition some results in quaternary glasses will be explained by a more general distribution function. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Diusion of alkali cations in oxide glasses is complicated because both the arrangement of all atoms and the arrangement of the ions are disordered. Thus both the local disorder of neutral matrix atoms around a cation and the arrangement of anions and other cations in the neighborhood determine the potential energy of a cation. In the following, the ®rst part will be called structural disorder and the second one charge disorder. Although the anions as non-bridging oxygen atoms in silicate glasses are ®xed below the
* Corresponding author. Tel.: +49-551 39 5001; fax: +1-49 551 39 5012. E-mail address: [email protected] (R. Kirchheim).
glass transition temperature the cations can move and, therefore, their con®guration is changing as a function of time. These eects can be studied best by Monte Carlo simulations for the ideal case of pure charge disorder, i.e., anions distributed randomly on a regular lattice and cations moving among them. Results of these studies [1±3] are compiled and discussed below in Section 3. The eect structural disorder has on diusion can be elucidated best for hydrogen atoms in metallic glasses and/or small molecules in glassy polymers [4,5]. At low solute concentrations the structure of these glasses is not changed much and the small particles are incorporated into the interatomic space. Because of the structural disorder the potential energy depends on position. Minima in this energy landscape are called sites and the number of sites in an energy interval is called site
0022-3093/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 1 ) 0 0 5 2 4 - 5
R. Kirchheim, D. Paulmann / Journal of Non-Crystalline Solids 286 (2001) 210±223
energy distribution. If a site can be ®lled with one particle only, the thermal occupancy will be determined by Fermi±Dirac statistics. At low temperatures this corresponds to a ®lling of the site energy distribution from bottom up to the Fermi energy or chemical potential l. The signi®cance of this ®lling on diusion can be studied best for the simpli®ed case of constant saddle point energies. Thus correlation eects are avoided and by averaging over all site energies and corresponding jump frequencies the diusion coecient can be calculated as [4] D D0 exp
Gs l ; kB T
1
where Gs is the saddle point energy, kB is Boltzmann's constant and D0 is a prefactor depending on average jump distance and attempt frequency. Then the activation energy is the dierence between saddle point energy and the chemical potential, i.e., the highest energy of the ®lled sites. Increasing concentration leads to occupation of higher energy sites and, therefore, l increases and because of the resulting lower activation energy D increases. In [5] it was shown that thermally activated hopping over an energy barrier is not a necessary condition for the validity of Eq. (1). It can be derived as well by starting with the ergodic hypothesis. Then ratios of the mean time of residence particles spend in sites are equal to ratios of the corresponding occupancy. Thus particles within the sites of the highest energy (those at the Fermi level l) have the shortest time of residence and contribute the most to long range diusion. Increasing l requires occupancy of sites with higher energy and leads to shorter times of residence and, therefore, to a higher mobility. Independent of the atomistic mechanisms like tunneling through or hopping over barriers Eq. (1) remains valid. Despite the attempt in [4,6] the scenario described for diusion in structurally disordered materials is not applicable to oxide glasses, although the diusion coecient of alkali cations increases with increasing concentration, too. The two major objections are: (i) structural disorder changes remarkably as a function of the alkali
211
content and (ii) ionic disorder has to be taken into account as well. Nevertheless, it was tempting to use the concept of an energy landscape in the past [4,7]. However, for a ®xed alkali content as used in most studies on the mixed alkali eect (MAE) in ternary glasses, the number of non-bridging oxygen atoms and, therefore, the ionic disorder on average does not change by substituting one cation by another one. This simpli®es the discussion of cation diusion and provided the basis for an explanation of the MAE [8,9]. The MAE as described in many review papers and books [10±12] gives rise to a decrease of DC-conductivity by orders of magnitude if two binary glasses are mixed. In a ®rst modeling of the MAE by one of the authors [8,9] a box type distribution of site energies was used. It will be shown in the present paper that a dierent choice of the distribution function leads to a better description of the diusion of minority cations and that some eects in quaternary silicate glasses could be explained. In addition the crucial assumption that smaller cations occupy low energy sites preferentially is discussed in more detail. In order to provide the basis for understanding the re®ned treatment of the MAE, the relevant ideas and assumptions of the previous work are repeated ®rst. 2. Mixed alkali eect with a box-type distribution function In the following a mixed glass shall have the composition xyA2 O 1 yR2 O 1 xSiO2 , where A is a small and R a large alkali cation. Thus y describes the cation fraction of the smaller one. Measured densities of mixed glasses reveal that for y increasing from 0 to 1 the packing density increases which corresponds to a decrease of the mesh size of the network [8,9]. With the aid of the schematic picture in Fig. 1 the MAE will be ®rst explained qualitatively as a combined result of network modi®cations and site energy distribution. If R is a minority cation (upper part of Fig. 1) its mobility is reduced because it has to migrate through a lattice with a mesh size formed by the smaller cation A. In the lower part of Fig. 1 A is the minority cation but it is not able to pro®t much
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describe oxygen packing density by the partial molar volume of oxygen VO . Then it can be shown that for a constant value of the total alkali content x this partial molar volume, VO , varies linearly as a function of y. In agreement with expectation VO increases as the smaller cation is replaced by the larger one where the magnitude of the increase is proportional to the dierence of oxide volumes VR VA . There are a variety of dierent eects discussed in [8] which provide evidence that the cation mobility is reduced if VO decreases. Measurements of the ionic conductivity as a function of external hydrostatic pressure where VO is changed in a de®ned way allow a quantitative treatment. Thus changes of the activation energy of diusion Q as a function of VO are possible using measured quantities only. The result is expressed by the following equations [8]: oQA oy
kB T
o ln DA KVA VR oy VO
VA x
2
kB T
o ln DR KVR VR oy VO
VA x;
3
and
Fig. 1. Schematic presentation of network modi®cations and site occupancy in a mixed alkali glass. As examples of a smaller cation and a larger cation Na and Cs were used. Substituting with the larger cation enlarges the mesh size of the network which increases ion mobility of both species. This increase is overcompensated for the smaller cation by a preferred occupancy of low energy sites. If both cations are present with about the same fraction, their diusivity is reduced because the larger cation has to migrate through a lattice which is shrunken by the presence of the smaller cation and the smaller cation is sitting in low energy sites with a large time of residence.
from the large mesh size set up by the majority of large R ions because the eect of sitting in deep energy sites reduces mobility. In the middle with y 0:5 the diusivitiy of both ions is reduced with respect to the binaries because R experiences the reduced mesh size and A sits in low energy sites. In the following we will outline the quantitative treatment of this scenario. In order to get a measure of the network density or the mesh size, respectively, the particle density of the largest atom oxygen was calculated from measured densities of mass [8]. It is convenient to
oQR oy
where DA and DR are the diusion coecients of cations A and R, QA and QR are the corresponding activation energies, VA and VR are the corresponding activation volumes, VA and VR are the partial molar volumes of the alkali oxides A2 O and R2 O in the binary glasses (y 1 and y 0) and K is the bulk modulus. Comparing with available experimental data it can be shown that the activation volume V is equal to the ionic volume Vi [8] and from an evaluation of mass density data it could be shown that the molar volumes VA and VR of the oxides A2 O and R2 O are equal to two times the ionic volumes, i.e., 2VAi and 2VRi . Including this dependence Eqs. (2) and (3) yield after integration QnR QbR QnA QbA
2KViR ViR VO 2KViA ViR VO
ViA xy
and 4
ViA x 1
y:
Then the activation energy Qn in the network of the mixed glass is increased with respect to the
R. Kirchheim, D. Paulmann / Journal of Non-Crystalline Solids 286 (2001) 210±223
value in the binary glasses Qb for R and decreased for A. This is in accordance with R(A) moving through a network shrunk by A (expanded by R). With a rectangular site energy distribution and the assumption of a preferred occupancy of low energy sites by A leads via Eq. (1) to an additional change of the activation energy for A expressed by the following equation [8]: QwA
QbA
w 1
yx;
5
where w is the width of the box distribution for x 1 (cf. Fig. 1). Although not relevant for the present model of the MAE it has been assumed that the width of the box depends in a simple way on the total alkali content (cf. Appendix A). Both eects of network mesh size and preference for low energy sites are ®nally combined to give the activation energies of diusion in a mixed glass as QmA QbA wx 1
y
2KViA ViR VO
ViA x 1
y 6
213
modulus deviate from measured values by a factor of about 0.5±2. The DC-conductivity of a mixed glass rm is obtained by adding the partial contribution from A and R according to the following equation [8]: rm yrmA 1 yrmR DQmA yrbA exp 1 kB T DQmR exp ; kB T
yrbR 10
where DQ Qm Qb is the change of activation energy from the binary to the mixed case given by Eqs. (6) and (7) and rbA or rbR are the DC-conductivities of the binary glasses for y 1 or y 0. The last equation describes the dependence of rm on T and y quite well as shown in Fig. 2. However, the experimental results on the diffusion coecient of A reveal remarkable deviations from the predicted behavior for low y values. This is shown for Na- and Cs-diusion coecients in Fig. 3. These deviations will not show up in
and QmR QmR
2KViR ViR VO
ViA xy
7
or in terms of the diusion coecients assuming no change of the prefactors m DA kB T ln DmA 2KViA wx 1 y ViR ViA x 1 y 8 VO and kB T ln
DmR DbR
2KViR ViR VO
ViA xy:
9
The last four equations contain one unknown parameter only which is the width w of the box distribution or the parameter wx, respectively. Nevertheless, the bulk modulus was used in [8] as a free parameter, too, in order to obtain a better agreement between experimental results and the model. Values obtained this way for the bulk
Fig. 2. DC-conductivity of a mixed Na±Cs silicate glass [13] with 25 mol% alkali oxide at 200°C, 300°C and 400°C. The lines are calculated according to Eq. (10) with the two parameters K 17:5 GPa and wx 89:8 kJ/mol.
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Fig. 3. Diusion coecient of Na (open circles) and Cs (solid circles) at 397°C in a 25 mol% alkali oxide silicate glass [13]. The straight lines are ®ts to the linear behavior for A ions at y ! 1 and R ions at y ! 0, i.e., in regions where the corresponding ions are the majority component.
Fig. 2 because they belong to the minority cations. Therefore, the linear dependence of ln DA on y as predicted by Eq. (8) is not obeyed. Another example is presented in Fig. 4 for a ternary Na±Rb silicate glass. It will be shown in the following that this de®ciency with respect to the diusion of the smaller cations at low y values can be overcome by choosing a dierent site energy distribution. As the concept of a site energy distribution is seldom used in the science of oxide glasses we repeat some of the fundamental ideas and their relation to the popular Anderson±Stuart model [15].
Fig. 4. Diusion coecient of the smaller Na ion in a mixed Na/Rb silicate glass with 75 mol% SiO2 at dierent temperatures [14]. The diusion coecients of the larger Rb-ion do not show similar systematic deviations from straight lines as Na does.
3. Physical reasons for a distribution of site energies
Coulomb energy is the sum of all contributions from the other cations and anions. For temperatures far below the glass transition temperature the anions have ®xed positions. However, the cations are mobile and, therefore, the electrostatic potential depends on time. In order to avoid this, we assume a constant mean ®eld contribution of the remaining cations on the electrostatic energy of our cation under consideration. For the elastic part in site energies the Eshelby theory [16] yields for a spherical site of volume Vm " # 2ls Vi Vm 2 el Em ; 11 3 Vm
For the sake of simplicity we start with a discussion of the binary alkali silicate glass. The potential energy of a given cation in a silicate glass or its site energy, respectively, contains two contributions an elastic and an electrostatic one. The former arises, if the cation does not ®t into an empty site of the unoccupied volume of the network forming atoms. The electrostatic or
where ls is the shear modulus of the matrix and Vi is the volume of the cation. In the Anderson± Stuart model [15] the elastic contribution to the site energy is based on an expression proposed by Frenkel [17] which is derived in a less rigorous treatment than the one by Eshelby. Then a distribution of site energies arises, if a distribution of site volume exists.
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The electrostatic part of the Anderson±Stuart model [15] can be written as EmCou
e2 1 ; 4pe0 e ri rma
12
where ri is the radius of the cation and rma is the smallest distance a cation surface can approach the center of negative charge. A distribution of site energies arises, if other oxygen atoms hinder a cation to approach a non-bridging oxygen as close as possible (cf. Fig. 5(a)). Dierent to the variation of rma a distribution of site energies may also arise from dierent con®gurations of anions and cations. Namely the anions are closest to cations and, therefore, strongly aect the electrostatic energies. In Monte Carlo simulations of cation diusion in a regular lattice [1±3] the anions were randomly distributed among lattice positions and the same number of cations could move through the interstitial lattice. The dierence of the electrostatic energy between a site occupied by a jumping cation and the site left was monitored during the simulation and the
215
results are shown in Fig. 6. For low concentrations c ( fraction of sites occupied) and low temperatures the distribution is bimodal. The ®rst peak in Fig. 6 at E 0 stems from cations jumping around anions without leaving the nearest neighborhood or from cations moving far away from the electric ®eld of the anion. The second peak around E 0:5 eV is due to cations leaving a site next to an anion (dissociative step). At low temperatures cations spend most of their time in the neighborhood of the anion and the situation may be best described by the weak electrolyte model [18]. In terms of the site energy model this situation was treated in [4]. For high ion concentrations the distribution of electrostatic energies becomes broader because in the neighborhood of a cation several other anions and to a lesser extent cations interfere and, therefore, the energy depends on the various con®gurations of other ions (cf. Fig. 5(b)). In the following sections we will be concerned mainly with higher alkali contents and, therefore, the broader and not the bimodal distribution is relevant. In addition we will model ternary glasses. Thus the question
Fig. 5. (a) A con®guration corresponding to dierent distances rma between an anion (shaded circle) and the surface of a smaller and a larger cation (two circles with solid lines). The radii are chosen according to the rules presented in [8], where the smaller cation corresponds to Na and the larger one to Rb . (b) At higher alkali contents the interaction with other non-bridging oxygen ions (shaded circles) becomes important. As shown in the schematic structure of a possible con®guration it is comprehensive that smaller ions are less hindered to approach the minimum energy position than the larger ones. Thus anion con®gurations of lowest energies are preferably occupied by the smaller cations. As low energy con®gurations correspond to a long time of residence small ions become more immobilized this way.
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Fig. 6. Distribution of the dierence of Coulomb energies DEcoul associated with a jump from one to another site for two dierent temperatures and concentrations according to [1]. Concentrations c 0:01 and 0.1 are given in terms of number of cations per lattice sites.
arises how two cations occupy the various sites and especially how they compete for the low energy sites. 4. Occupancy of sites within a distribution of site energies Let us consider the exchange of an A- or a R-ion sitting in site m within the ternary glass xyA2 O 1 yR2 O 1 xSiO2 by a corresponding ion from a reservoir were the chemical potentials lA of A and lR of R are ®xed. This exchange can be expressed by the following equations: Am () Ar
and Rm () Rr :
13
The site m belongs to a subsystem of sites having the same volume Vm and the same value for rma . The number of m-sites Nmo is given from the distribution function of Vm and rma . It can be shown [4,19±21] that the number NA of A-cations and NR of R-cations obeys the following relations: X NAm NA m
X
Nmo kAm and 1 kAm kRm m X X Nmo kRm NR NRm ; 1 kAm kRm m m
14
where NAm is the number of A-cations in m-sites and NRm is the number of R-cations in m-sites. The thermodynamic activities kim are de®ned by the following equations: lAm EAm kAm exp and kB T 15 lRm ERm kRm exp ; kB T where EAm or ERm are the site energies and lAm or lRm are the chemical potentials for A or R cations in m-sites. Changes of site energy caused by changes of the total alkali content shall be included in E. As the sum of NA and NB corresponds to the total alkali content x it has to be constant, i.e., the two chemical potential lA and lR are related to each other and the corresponding relation has to be maintained in the reservoir. This way an cation exchange between a ternary glass and a melt of corresponding alkali salt mixtures changes the concentrations of A and R in the glass until the ratio of the chemical potentials is the same in the glass and in the melt. Experimental studies reveal that it is very dicult to replace the rest of smaller ions in a glass by larger ones [22]. Despite the diculty of straining the network by the incorporation of a larger cation this behavior is in agreement with the preferential occupancy of low energy sites by smaller cations.
R. Kirchheim, D. Paulmann / Journal of Non-Crystalline Solids 286 (2001) 210±223
217
In equilibrium the chemical potentials of the species in reaction (13) have to be equal yielding lAr lAm EAm kB T ln
Nmo
NAm NAm
NRm
Nmo
NRm NAm
NRm
16
and lRr lRm ERm kB T ln
:
17
This corresponds to a chemical potential with a contribution from con®gurational entropy according to a distribution of A and R cations among the available Nmo sites without direct interaction between A and R cations. It is not the equation corresponding to an ideal solution because EAm and ERm depend on the total alkali content. However, for a ternary glass with ®xed x and varying y Eqs. (14)±(17) are useful to discuss the distribution of A and R among the various sites and its eect on activation energies of diusion.From Eqs. (14) and (15) the following ratio is obtained: NAm lAr lRr ERm EAm exp NRm kB T Dl DEm exp : 18 kB T
Fig. 7. Box distribution n E of site energy dierences DE (dashed line) and occupancy ym n DE (according to Eq. (19)) by either A or R ions depending on whether DE is negative (A wins in competing for low energy sites) or positive. The shaded area is equivalent to y, i.e., the total amount of A ions for the ®rst case. Following the assumption made in this work they shall succeed in occupying sites of lowest energy.
with all the available A ions is applicable and Dl is directly obtained from y. This is true for other broad distribution as well as shown in Fig. 8. By adding a small fraction dy of A on the expense of R a useful expression can be derived (as trivial by looking at Fig. 8) dy n Dl dDl:
20
Depending on the sign of Dl DEm site m is preferentially occupied by A or R. The cation fraction ym of a given site m can be obtained from Eq. (18) as ym
NAm 1 : NAm NRm 1 exp DEm Dl kB T
19
This function has the form of the Fermi±Dirac function being about 1 for DEm < Dl and 0 vice versa. For a box-like distribution of DEm it is plotted in Fig. 7. For small temperatures the Fermi±Dirac function has a step like behavior, i.e., ym 1 for DE < Dl and ym 0 for DE > Dl. Increasing temperature leads to a smearing out of the occupancy around Dl over an energy range of about 3kT (cf. Fig. 7). Thus for a width wx kT the simpli®ed model of ®lling the sites of lowest energy
Fig. 8. A general broad distribution is ®lled with A ions of content y (equal to the gray area) up to Dl according to the discussion in the text. The black area corresponds to a substitution of R-ions by A-ions if the chemical potential dierence is raised by Dl. Approximating the black area by a rectangular one gives dy n Dl dDl.
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In order to describe diusion and DC-conductivity for a distribution of site energies two approaches have been used: (i) calculating an average jump rate for thermally activated hopping [4] and (ii) averaging over the times of residence [5]. In both cases the activation energy of diusion contains a term which is the negative of the chemical potential or the negative energy of the uppermost ®lled level (cf. Eq. (1)). For thermally activated hopping over the barrier the activation energy is simply the dierence between the average saddle point energy and the chemical potential. With respect to the situation in Fig. 5 the latter corresponds to Dl for A ions and to the uppermost level of the box for B ions. Then changes of the activation energy are related to changes of y via the relation dQ
dDl
dy : n Dl
21
Thus we have shown that the simple concept of ®lling a site energy distribution is an appropriate approximation for ternary glasses. In the following section we will discuss how DE depends upon the size of A and R.
5. Dierence of site energies for A and R cations First we consider the elastic contribution to DE by using Eq. (11) " # 2 2 2l V V V V Ai m Ri m DEmel s : 22 3 Vm Vm By our convention for the ionic volumes VRi > VAi the energy dierence is always negative and increases in magnitude with decreasing Vm . Therefore, larger sites are preferentially occupied by the larger ions R. There is also a distribution of site energies ERm and EAm because of dierent Coulomb interaction with non-bridging oxygen ions. The sites in the broad distribution obtained from Monte Carlo simulations and shown in Fig. 6 are occupied by both A and R with the same probabilities, if the cations are approximated by point charges. However, for realistic sizes of the cations and network
atoms there will be arrangements which do not allow a closest approach of cations as shown in Fig. 5. Then the corresponding variation of the smallest distance rma a cation surface can approach the center of negative charge gives rise to an electrostatic contribution to DE according to the Anderson±Stuart model [15] this can be written as e2 1 1 Coul DEm : 23 4pe0 e rA rma rR rma Because of rR > rA this energy dierence is always positive and increases in magnitude with decreasing rma and, therefore, those sites are preferentially occupied by the smaller ions A which allow a closer approach of the anion. In the following we assume that the direct Coulomb interaction is larger than the elastic one and, therefore, the smaller A ions occupy the low energy sites because of their larger ionic strength. As discussed in Appendix A this assumptions gives rise to a dependence of the site energy distribution wx on the cation radii which is in agreement with experimental ®ndings. One may argue that the larger R ion is pushing matrix atoms away in order to approach the anion as close as possible until it is repelled by the Pauli repulsion. Although this corresponds to no distribution of rma -values, it leads to a distribution of DE because the distortion energy is larger for R compared to A and the distortion energy will depend on the varying site volumes.
6. Mixed alkali eect with a general distribution functions So far a box-type distribution of site energies has been used which in connection with a step-like behavior of the occupancy gave rise to a linear dependence of ln D or the activation energy Q of ion A with respect to y (cation fraction of A). Deviations from this prediction are obvious for small values of y for A Na in a Na/Cs glass in Fig. 3 as well as for a Na/Rb glass in Fig. 4. These discrepancies can be overcome by using a site energy distribution n E which is decreasing with increasing E (cf. Fig. 9). Then changes of QA are expressed by (cf. Eq. (21))
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dQmA dy QmA QR A
1 n Dl Z y 0
or d~ y ; nDl ~ y
24
where QR A is the activation energy of A-diusion in a binary R-glass. The physical signi®cance of the last equation for small and large values of y is exempli®ed in Fig. 9. Because of high density of sites at low y-values any increase in y causes a small energy change of the top-most occupied energy level only (cf. Fig. 9(a)). The corresponding change of activation energy QmA or log DA , respectively is small, too. For a larger fraction of the
219
cation A sites with high energy Dl but low density n Dl are occupied and a corresponding increase of y causes a much larger change of the top-most occupied energy level (cf. Fig. 9(b)). Adding the eect of network modi®cations as described by Eq. (6) there might be even a relative maximum in QA or a relative minimum in log DA , respectively (see Fig. 9), because the decrease of the Q-value caused by the site energy distribution could be overcompensated by the increase caused by the shrinking network. At high values of y the in¯uence of the site energy distribution dominates as n Dl decreases. Thus deviation from a linear dependence of QA on y can be explained in a natural way. In
Fig. 9. A decreasing site energy distribution function n DE and its occupancy for low (a) and high y-values (b) is shown. The gray area is a measure of y, i.e., the fraction of A-cations. The black area corresponds to the same increment by raising the top-most energy level more (b) or less (a) pronounced. This raise is accompanied by a small or large decrease of activation energy dQmA (cf. Eq. (24)). In (c) n DE and its reciprocal function are shown. The eect n DE has on the activation energy of diusion QmA is described by the integral in Eq. (24) yielding the dashed curve in (d) labeled s.d. The counteracting eect of the decreasing mesh size with increasing y (cf. Eq. (6)) is shown by the lower dashed line labeled m.s. and the sum of the two is presented by a solid line. Thus depending on the magnitude of the two contributions QmA may go through a maximum with increasing y. This corresponds to a minimum in ln Dma vs y diagrams.
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addition the deviation should be the larger the larger the size dierence is between A and R, because this dierence determines the slope of the dashed line labeled m.s. (cf. Eq. (2)). A larger size mis®t leads to a larger slope of the line m.s. Then the negative contribution of the mesh size (m.s.) becomes more pronounced in comparison with the eect of the site energy distribution (s.d.) and the sum of both, i.e., the activation energy described by the solid line, is lowered. This conclusion is in agreement with the experimental data presented in Figs. 3 and 4, where the Na/Cs glass has the larger size dierence. In addition the large values obtained for the width w of a rectangular distribution [8] can be understood for a distribution as the one shown in Fig. 9. We have to consider the case of large yvalues shown in Fig. 9(b)) because the ®tting parameter w was obtained in [8] from data with large y-values. A rectangular distribution has the same eect on the y-dependence of the activation energy, if its heights are the same as the small n Dlvalue for the top-most ®lled level and, in order to get the same area as presented by the gray area in Fig. 9(b)), the width w of the rectangular distribution has to become very large. A rigorous quantitative treatment requires an assumption about the function n Dl to be used in Eq. (24). This is avoided because not much is known about n Dl and the mathematical complexity increases without a concomitant increase of physical understanding. In order to use the simple concept of a rectangular distribution further on, we may de®ne the width as a function of y. For the following this is used in a qualitative way only with w being small for y ! 0 and large for y ! 1. Whereas for the Na-ions in Fig. 3 deviations from the straight line behavior are observed but do not lead to a relative minimum of log D, the corresponding relative maximum of Q has been measured for Na tracer diusion in a mixed Li±K glass [23]. As an ion of intermediate size with respect to Li and K sodium is expected to occupy sites on the energy scale in between Li and K and, therefore, its activation energy changes in a similar way as the one for Li with respect to the energy scale (cf. Fig. 10). As the migrating species are Na ions their activation volume of diusion has to be
Fig. 10. Occupancy of sites in a distribution n E for a quaternary glass containing three alkaline ions A, M and R. The smallest ion A is occupying the sites of lowest energy (dark grey) and the largest ion R the highest ones (light grey). The ion M of intermediate size shall be present as a tracer only and, therefore, occupies a small fraction of the sites in between A and R on the energy scale (black). The corresponding energy levels for M are equal to the top-most levels of A. Then changes of the activation energies caused by changes of y due to the energy distribution are the same for A and M.
used in Eq. (6) whereas the change of mesh size is caused by the contents of Li and K ions and, therefore, their dierence of partial molar volumes VR VA 2 ViR ViR ) has to be included. Thus Eq. (6) is modi®ed for this case to QmM QA M wx 1
y
2KViM ViR VO
ViA x 1
y 25
or QmM QR M
ywx y
2KViM ViR VO
ViA x
;
26
where the index M refers to the tracer ion of intermediate size, e.g., Na for the case discussed R before and QA M QM is the activation energy of diusion of tracer M in the binary A±(R±)glass. For small values of y the width w was shown to be small and the third term on the right-hand side of Eq. (26) dominates and QmM increases with increasing y. For larger values of y the width w is enlarged and now the second term on the righthand side of Eq. (26) determines the dependence on y and, therefore, QmM decreases with further
R. Kirchheim, D. Paulmann / Journal of Non-Crystalline Solids 286 (2001) 210±223
Fig. 11. Activation energy of Na tracer diusion in two mixed Li±K silicate glasses [23] as a function of the Li-fraction. The changes can be understood qualitatively in the following way. Activation energy increases at low Li-contents because the mesh size of the network becomes narrower and the counteracting eect of the energy distribution is less pronounced as shown in Fig. 9(a)). This counteracting eect of occupying energy levels of higher energy becomes dominant at higher Li-contents as shown schematically in Fig. 9(b)). In agreement with Eq. (26) the changes are more pronounced for larger values of the total alkali content x (®lled circles).
increasing y. The corresponding experimental results are presented in Fig. 11. A similar behavior but without an extreme value occurs for Rb tracer diusion in a mixed Na±Cs glass. The experimental results [24] are presented in Fig. 12. In the framework of the model of this study it has to be interpreted by using a modi®ed version of Eq. (8) in accordance with the previous discussion of Na in a Li±K glass m DM kB T ln DbM 2KViM ViR ViA x 1 y: 27 wx 1 y VO Again the term describing the eect of the mesh size (second on the right) dominates for low y values because w is small for distribution of n DE like the one shown in Fig. 9. Increasing y gives rise to larger w values and a total compensation of the mesh size eect. For y > 0:6 the two terms on the right-hand side of Eq. (27) are equal. Then w can be calculated with the corresponding value of
221
Fig. 12. Ratio of tracer diusion coecients of Cs and Rb in a mixed Na±Cs silicate glass and a binary Na glass with 75 mol% SiO2 at 397°C [24]. Cs mobility in the mixed glass is reduced by the decreasing mesh size with increasing y. The same is true for the Rb mobility but the eect is ®nally compensated by the decrease of activation energy due to successively occupying sites of higher energy (cf. Figs. 9 and 10).
K 17:5 GPa [8] to be 51 kJ/mol which is about 40% smaller than the ®gure evaluated from the ternary case of Na±Cs glasses [8]. For the last two examples Rb-diusion occurred by an ion exchange in a ternary glass leading to a quaternary glass and, therefore, the size mis®t of cations may lead to a relaxation of the network as discussed in [8]. However, the amount of tracer ions is so small in comparison with the cations in the ternary glass that the corresponding changes of the network density are solely determined by the latter ions and not by the tracer.
7. Conclusions It could be shown that in a mixed glass other than a box-type distribution of site energies allows to explain the non-exponential increase of the diusion coecient with increasing concentration of the minority cations. Thus the new concept of diusion in mixed glasses stressing the interplay between network modi®cations and a distribution
222
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of site energies is capable of treating even secondorder eects in mixed alkali glasses. In addition it can be extended from ternary to quaternary glasses explaining some of the data on tracer diusion measured there. Although the new model is in accordance with the Anderson±Stuart model [15] it cannot provide a recipe of what site energy distribution is suitable for oxide glasses. Ionic diusion in oxide glasses is complicated by the fact that both Coulomb disorder and structural disorder seemed to aect the energy of an ion and, therefore, analytical solution may not be possible and computer simulations could be the appropriate choice. It should be mentioned that the concept of a distribution of site energies and its relevance for the MAE has been rediscovered [25±27] after its ®rst application in [9] based on a more general concept developed for interstitial diusion in amorphous solids [6,28]. The additional assumption about distinct environments around A and R ions used in [26] is not in contradiction to the present concept as larger R-ions displace the matrix atoms further compared to the smaller A-ions. However, the special ingredient of the treatment in [29,30] that a former and larger R-site being energetically less favorable for the smaller A-ion is dicult to comprehend as discussed in [9].
By assuming a distribution of the distances rma between anion center and cation surface (cf. Fig. 5) around an average value roa with a variation dra < roa leads to the following approximation for the variance of the site exchange energy or the width of the energy distribution wx, respectively oDE dra oroa e2 rR rA rR rA 2roa dra ; 4pe0 e rR roa 2 rA roa 2
wx
A:2
which is simpli®ed further on by using an average ionic radius rRA rR rA =2 and the approxi2 mation rR roa rA roa rRA roa wx
e2 2 rR rA dra : 4pe0 e rRA roa 3
A:3
The values of wx as obtained from curve ®tting in [8] are plotted in Fig. 13 vs the dierence of cation radii rR rA . In agreement with Eq. (A.3) there is a linear relation between wx and rR rA , although the straight line obtained from ®tting diffusion data does not go through the origin. This
Acknowledgements The authors are grateful for ®nancial support provided by the Deutsche Forschungsgemeinschaft (SFB 345). Appendix A. Width of the site energy distribution and its dependence on ionic radii In Section 5 it was argued that the major contribution to the site exchange energy DE stems from the electrostatic interaction between the small and large cations of radius rA and rR , respectively, according to Eq. (23) e2 1 1 DE : A:1 4pe0 e rA rma rR rma
Fig. 13. Width wx of the site energy distribution as a function of the dierence of cation radii rR rA . The data are obtained from ®tting diusion coecients according to Eq. (8) (from [8, Table 2] at 400°C (closed circles) and at 450°C (open circles)) and activation energies according to Eq. (6) (from [8, Table 3(b)]). The straight line are least-square ®ts to the experimental points. Data obtained from diusion coecients do not lie on a straight line through the origin because they contain a temperature dependent (i.e., entropic) contribution.
R. Kirchheim, D. Paulmann / Journal of Non-Crystalline Solids 286 (2001) 210±223
may be caused by an entropic contribution to wx as discussed in [8]. However, the slope of the various straight line is nearly equal corresponding to about 1.7 kJ/Mol/pm. Using e 3 for the dielectric constant of a silicate class and rRA roa 200 pm as the average distance between an oxygen anion and a middle sized alkali cation comparison of the slope with Eq. (A.3) yields dra 150 pm. This value is higher than expected although the order of magnitude is about right. Again this may be attributed to the crude approximation of the site energy distribution by a rectangular one. A distribution as shown in Fig. 9 would lead to smaller variances of DE and, therefore, to smaller values of dra . References [1] D. Paulmann, PhD thesis, University of G ottingen, 1998. [2] D. Kn odler, W. Dietrich, J. Petersen, Solid State Ionics 53± 56 (1992) 1135. [3] D. Kn odler, PhD thesis, University of Konstanz, 1994. [4] R. Kirchheim, Progr. Mater. Sci. 32 (1988) 262. [5] R. Kirchheim, Def. Dius. Forum 143 (1997) 911. [6] R. Kirchheim, J. Non-Cryst. Solids 55 (1983) 243. [7] J.M. Stevels, in: S. Fl ugge (Ed.), Handbuch der Physik, vol. 20/2, Springer, Berlin, 1957, p. 350. [8] R. Kirchheim, J. Non-Cryst. Solids 272 (2000) 85. [9] R. Kirchheim, in: H. Jain, D. Gupta (Eds.), Diusion in Amorphous Materials, TMS, Warrendale, PA, 1994, p. 43. [10] M.D. Ingram, in: R.W. Cahn, P. Haasen, E.J. Kramer (Eds.), Materials Science and Technology, vol. 9, VCH, Weinheim, p. 715.
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[11] G.H. Frischat, Ionic Diusion in Oxide Glasses, TransTech, Aedermansdorf, 1975. [12] K. Hughes, J.O. Isard, in: J. Hladik (Ed.), Physics of Electrolytes, vol. 1, Academic Press, London, 1972, p. 387. [13] H. Jain, N.L. Peterson, H.L. Downing, J. Non-Cryst. Solids 55 (1983) 283. [14] O.V. Mazurin, M.V. Steltsina, T.P. Shaiko-Shaikovskaya, Handbook of Glass Data Part A-E, Elsevier, Amsterdam, New York, part C, p. 85. [15] O.L. Anderson, D.A. Stuart, J. Am. Ceram. Soc. 37 (1954) 573. [16] J.D. Eshelby, in: F. Seitz, D. Turnbull (Eds.), Solid State Physics, Academic Press, New York, 1956. [17] J. Frenkel, Kinetic Theory of Liquids, Oxford University, London, 1949, p. 10. [18] D. Ravaine, J.L. Soquet, Phys. Chem. Glasses 18 (1977) 27. [19] R. Kirchheim, in: D. Wolf, S. Yip (Eds.), Materials Interfaces, Atomic Level Structure and Properties, Chapman and Hall, London, 1992, p. 481. [20] P. Pekarski, R. Kirchheim, J. Memb. Sci. 152 (1999) 251. [21] A.A. Gusev, U.W. Sutter, Phys. Rev. B 27 (1991) 6488. [22] Ref. [11] p. 78. [23] V.K. Pavlovski, Stekloobr. Sostoy. 5 (1970) 148, and Ref. [14], part C, p. 84. [24] H. Jain, N.L. Peterson, J. Am. Ceram. Soc. 66 (1983) 174, and Ref. [14], part C, p. 89. [25] A. Hunt, J. Non-Cryst. Solids 160 (1993) R183. [26] P. Mass, J. Non-Cryst. Solids 255 (1999) 35. [27] S.D. Baranowskii, H. Cordes, J. Chem. Phys. 111 (1999) 7546. [28] R. Kirchheim, Acta Metall. 30 (1982) 1069. [29] P. Maass, A. Bunde, M.D. Ingram, Phys. Rev. Lett. 68 (1992) 3064. [30] A. Bunde, M.D. Ingram, P. Maass, J. Non-Cryst. Solids 172±174 (1994) 1222.
www.elsevier.com/locate/jnoncrysol
The relevance of site energy distribution for the mixed alkali eect Reiner Kirchheim *, Dirk Paulmann Institut f ur Materialphysik, Der Georg-August-Universitat Gottingen, Hospitalstrasse 3-7, D-37073 Gottingen, Germany Received 14 September 2000
Abstract The mixed alkali eect (MAE) can be explained as an interplay between network modi®cations induced by alkali ions of dierent size and an energy distribution of sites with a preferential occupancy of the low energy sites by the smaller cations. The underlying physical assumptions of this concept are discussed in the present paper in more detail and they are compared with results of Monte Carlo simulations of diusion and the Anderson±Stuart model. It will be shown that the new concept is actually hidden in the Anderson±Stuart model. Applying the new concept to experimental data has been done so far with a box-type distribution of site energies. Besides the oversimpli®cation, a major drawback of this distribution is that the diusion coecient of the smaller and minority cation is underestimated. It will be shown in the present study that by a dierent choice of distribution the concentration dependence of the diusivity of the minority cations can be understood. In addition some results in quaternary glasses will be explained by a more general distribution function. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Diusion of alkali cations in oxide glasses is complicated because both the arrangement of all atoms and the arrangement of the ions are disordered. Thus both the local disorder of neutral matrix atoms around a cation and the arrangement of anions and other cations in the neighborhood determine the potential energy of a cation. In the following, the ®rst part will be called structural disorder and the second one charge disorder. Although the anions as non-bridging oxygen atoms in silicate glasses are ®xed below the
* Corresponding author. Tel.: +49-551 39 5001; fax: +1-49 551 39 5012. E-mail address: [email protected] (R. Kirchheim).
glass transition temperature the cations can move and, therefore, their con®guration is changing as a function of time. These eects can be studied best by Monte Carlo simulations for the ideal case of pure charge disorder, i.e., anions distributed randomly on a regular lattice and cations moving among them. Results of these studies [1±3] are compiled and discussed below in Section 3. The eect structural disorder has on diusion can be elucidated best for hydrogen atoms in metallic glasses and/or small molecules in glassy polymers [4,5]. At low solute concentrations the structure of these glasses is not changed much and the small particles are incorporated into the interatomic space. Because of the structural disorder the potential energy depends on position. Minima in this energy landscape are called sites and the number of sites in an energy interval is called site
0022-3093/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 1 ) 0 0 5 2 4 - 5
R. Kirchheim, D. Paulmann / Journal of Non-Crystalline Solids 286 (2001) 210±223
energy distribution. If a site can be ®lled with one particle only, the thermal occupancy will be determined by Fermi±Dirac statistics. At low temperatures this corresponds to a ®lling of the site energy distribution from bottom up to the Fermi energy or chemical potential l. The signi®cance of this ®lling on diusion can be studied best for the simpli®ed case of constant saddle point energies. Thus correlation eects are avoided and by averaging over all site energies and corresponding jump frequencies the diusion coecient can be calculated as [4] D D0 exp
Gs l ; kB T
1
where Gs is the saddle point energy, kB is Boltzmann's constant and D0 is a prefactor depending on average jump distance and attempt frequency. Then the activation energy is the dierence between saddle point energy and the chemical potential, i.e., the highest energy of the ®lled sites. Increasing concentration leads to occupation of higher energy sites and, therefore, l increases and because of the resulting lower activation energy D increases. In [5] it was shown that thermally activated hopping over an energy barrier is not a necessary condition for the validity of Eq. (1). It can be derived as well by starting with the ergodic hypothesis. Then ratios of the mean time of residence particles spend in sites are equal to ratios of the corresponding occupancy. Thus particles within the sites of the highest energy (those at the Fermi level l) have the shortest time of residence and contribute the most to long range diusion. Increasing l requires occupancy of sites with higher energy and leads to shorter times of residence and, therefore, to a higher mobility. Independent of the atomistic mechanisms like tunneling through or hopping over barriers Eq. (1) remains valid. Despite the attempt in [4,6] the scenario described for diusion in structurally disordered materials is not applicable to oxide glasses, although the diusion coecient of alkali cations increases with increasing concentration, too. The two major objections are: (i) structural disorder changes remarkably as a function of the alkali
211
content and (ii) ionic disorder has to be taken into account as well. Nevertheless, it was tempting to use the concept of an energy landscape in the past [4,7]. However, for a ®xed alkali content as used in most studies on the mixed alkali eect (MAE) in ternary glasses, the number of non-bridging oxygen atoms and, therefore, the ionic disorder on average does not change by substituting one cation by another one. This simpli®es the discussion of cation diusion and provided the basis for an explanation of the MAE [8,9]. The MAE as described in many review papers and books [10±12] gives rise to a decrease of DC-conductivity by orders of magnitude if two binary glasses are mixed. In a ®rst modeling of the MAE by one of the authors [8,9] a box type distribution of site energies was used. It will be shown in the present paper that a dierent choice of the distribution function leads to a better description of the diusion of minority cations and that some eects in quaternary silicate glasses could be explained. In addition the crucial assumption that smaller cations occupy low energy sites preferentially is discussed in more detail. In order to provide the basis for understanding the re®ned treatment of the MAE, the relevant ideas and assumptions of the previous work are repeated ®rst. 2. Mixed alkali eect with a box-type distribution function In the following a mixed glass shall have the composition xyA2 O 1 yR2 O 1 xSiO2 , where A is a small and R a large alkali cation. Thus y describes the cation fraction of the smaller one. Measured densities of mixed glasses reveal that for y increasing from 0 to 1 the packing density increases which corresponds to a decrease of the mesh size of the network [8,9]. With the aid of the schematic picture in Fig. 1 the MAE will be ®rst explained qualitatively as a combined result of network modi®cations and site energy distribution. If R is a minority cation (upper part of Fig. 1) its mobility is reduced because it has to migrate through a lattice with a mesh size formed by the smaller cation A. In the lower part of Fig. 1 A is the minority cation but it is not able to pro®t much
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describe oxygen packing density by the partial molar volume of oxygen VO . Then it can be shown that for a constant value of the total alkali content x this partial molar volume, VO , varies linearly as a function of y. In agreement with expectation VO increases as the smaller cation is replaced by the larger one where the magnitude of the increase is proportional to the dierence of oxide volumes VR VA . There are a variety of dierent eects discussed in [8] which provide evidence that the cation mobility is reduced if VO decreases. Measurements of the ionic conductivity as a function of external hydrostatic pressure where VO is changed in a de®ned way allow a quantitative treatment. Thus changes of the activation energy of diusion Q as a function of VO are possible using measured quantities only. The result is expressed by the following equations [8]: oQA oy
kB T
o ln DA KVA VR oy VO
VA x
2
kB T
o ln DR KVR VR oy VO
VA x;
3
and
Fig. 1. Schematic presentation of network modi®cations and site occupancy in a mixed alkali glass. As examples of a smaller cation and a larger cation Na and Cs were used. Substituting with the larger cation enlarges the mesh size of the network which increases ion mobility of both species. This increase is overcompensated for the smaller cation by a preferred occupancy of low energy sites. If both cations are present with about the same fraction, their diusivity is reduced because the larger cation has to migrate through a lattice which is shrunken by the presence of the smaller cation and the smaller cation is sitting in low energy sites with a large time of residence.
from the large mesh size set up by the majority of large R ions because the eect of sitting in deep energy sites reduces mobility. In the middle with y 0:5 the diusivitiy of both ions is reduced with respect to the binaries because R experiences the reduced mesh size and A sits in low energy sites. In the following we will outline the quantitative treatment of this scenario. In order to get a measure of the network density or the mesh size, respectively, the particle density of the largest atom oxygen was calculated from measured densities of mass [8]. It is convenient to
oQR oy
where DA and DR are the diusion coecients of cations A and R, QA and QR are the corresponding activation energies, VA and VR are the corresponding activation volumes, VA and VR are the partial molar volumes of the alkali oxides A2 O and R2 O in the binary glasses (y 1 and y 0) and K is the bulk modulus. Comparing with available experimental data it can be shown that the activation volume V is equal to the ionic volume Vi [8] and from an evaluation of mass density data it could be shown that the molar volumes VA and VR of the oxides A2 O and R2 O are equal to two times the ionic volumes, i.e., 2VAi and 2VRi . Including this dependence Eqs. (2) and (3) yield after integration QnR QbR QnA QbA
2KViR ViR VO 2KViA ViR VO
ViA xy
and 4
ViA x 1
y:
Then the activation energy Qn in the network of the mixed glass is increased with respect to the
R. Kirchheim, D. Paulmann / Journal of Non-Crystalline Solids 286 (2001) 210±223
value in the binary glasses Qb for R and decreased for A. This is in accordance with R(A) moving through a network shrunk by A (expanded by R). With a rectangular site energy distribution and the assumption of a preferred occupancy of low energy sites by A leads via Eq. (1) to an additional change of the activation energy for A expressed by the following equation [8]: QwA
QbA
w 1
yx;
5
where w is the width of the box distribution for x 1 (cf. Fig. 1). Although not relevant for the present model of the MAE it has been assumed that the width of the box depends in a simple way on the total alkali content (cf. Appendix A). Both eects of network mesh size and preference for low energy sites are ®nally combined to give the activation energies of diusion in a mixed glass as QmA QbA wx 1
y
2KViA ViR VO
ViA x 1
y 6
213
modulus deviate from measured values by a factor of about 0.5±2. The DC-conductivity of a mixed glass rm is obtained by adding the partial contribution from A and R according to the following equation [8]: rm yrmA 1 yrmR DQmA yrbA exp 1 kB T DQmR exp ; kB T
yrbR 10
where DQ Qm Qb is the change of activation energy from the binary to the mixed case given by Eqs. (6) and (7) and rbA or rbR are the DC-conductivities of the binary glasses for y 1 or y 0. The last equation describes the dependence of rm on T and y quite well as shown in Fig. 2. However, the experimental results on the diffusion coecient of A reveal remarkable deviations from the predicted behavior for low y values. This is shown for Na- and Cs-diusion coecients in Fig. 3. These deviations will not show up in
and QmR QmR
2KViR ViR VO
ViA xy
7
or in terms of the diusion coecients assuming no change of the prefactors m DA kB T ln DmA 2KViA wx 1 y ViR ViA x 1 y 8 VO and kB T ln
DmR DbR
2KViR ViR VO
ViA xy:
9
The last four equations contain one unknown parameter only which is the width w of the box distribution or the parameter wx, respectively. Nevertheless, the bulk modulus was used in [8] as a free parameter, too, in order to obtain a better agreement between experimental results and the model. Values obtained this way for the bulk
Fig. 2. DC-conductivity of a mixed Na±Cs silicate glass [13] with 25 mol% alkali oxide at 200°C, 300°C and 400°C. The lines are calculated according to Eq. (10) with the two parameters K 17:5 GPa and wx 89:8 kJ/mol.
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Fig. 3. Diusion coecient of Na (open circles) and Cs (solid circles) at 397°C in a 25 mol% alkali oxide silicate glass [13]. The straight lines are ®ts to the linear behavior for A ions at y ! 1 and R ions at y ! 0, i.e., in regions where the corresponding ions are the majority component.
Fig. 2 because they belong to the minority cations. Therefore, the linear dependence of ln DA on y as predicted by Eq. (8) is not obeyed. Another example is presented in Fig. 4 for a ternary Na±Rb silicate glass. It will be shown in the following that this de®ciency with respect to the diusion of the smaller cations at low y values can be overcome by choosing a dierent site energy distribution. As the concept of a site energy distribution is seldom used in the science of oxide glasses we repeat some of the fundamental ideas and their relation to the popular Anderson±Stuart model [15].
Fig. 4. Diusion coecient of the smaller Na ion in a mixed Na/Rb silicate glass with 75 mol% SiO2 at dierent temperatures [14]. The diusion coecients of the larger Rb-ion do not show similar systematic deviations from straight lines as Na does.
3. Physical reasons for a distribution of site energies
Coulomb energy is the sum of all contributions from the other cations and anions. For temperatures far below the glass transition temperature the anions have ®xed positions. However, the cations are mobile and, therefore, the electrostatic potential depends on time. In order to avoid this, we assume a constant mean ®eld contribution of the remaining cations on the electrostatic energy of our cation under consideration. For the elastic part in site energies the Eshelby theory [16] yields for a spherical site of volume Vm " # 2ls Vi Vm 2 el Em ; 11 3 Vm
For the sake of simplicity we start with a discussion of the binary alkali silicate glass. The potential energy of a given cation in a silicate glass or its site energy, respectively, contains two contributions an elastic and an electrostatic one. The former arises, if the cation does not ®t into an empty site of the unoccupied volume of the network forming atoms. The electrostatic or
where ls is the shear modulus of the matrix and Vi is the volume of the cation. In the Anderson± Stuart model [15] the elastic contribution to the site energy is based on an expression proposed by Frenkel [17] which is derived in a less rigorous treatment than the one by Eshelby. Then a distribution of site energies arises, if a distribution of site volume exists.
R. Kirchheim, D. Paulmann / Journal of Non-Crystalline Solids 286 (2001) 210±223
The electrostatic part of the Anderson±Stuart model [15] can be written as EmCou
e2 1 ; 4pe0 e ri rma
12
where ri is the radius of the cation and rma is the smallest distance a cation surface can approach the center of negative charge. A distribution of site energies arises, if other oxygen atoms hinder a cation to approach a non-bridging oxygen as close as possible (cf. Fig. 5(a)). Dierent to the variation of rma a distribution of site energies may also arise from dierent con®gurations of anions and cations. Namely the anions are closest to cations and, therefore, strongly aect the electrostatic energies. In Monte Carlo simulations of cation diusion in a regular lattice [1±3] the anions were randomly distributed among lattice positions and the same number of cations could move through the interstitial lattice. The dierence of the electrostatic energy between a site occupied by a jumping cation and the site left was monitored during the simulation and the
215
results are shown in Fig. 6. For low concentrations c ( fraction of sites occupied) and low temperatures the distribution is bimodal. The ®rst peak in Fig. 6 at E 0 stems from cations jumping around anions without leaving the nearest neighborhood or from cations moving far away from the electric ®eld of the anion. The second peak around E 0:5 eV is due to cations leaving a site next to an anion (dissociative step). At low temperatures cations spend most of their time in the neighborhood of the anion and the situation may be best described by the weak electrolyte model [18]. In terms of the site energy model this situation was treated in [4]. For high ion concentrations the distribution of electrostatic energies becomes broader because in the neighborhood of a cation several other anions and to a lesser extent cations interfere and, therefore, the energy depends on the various con®gurations of other ions (cf. Fig. 5(b)). In the following sections we will be concerned mainly with higher alkali contents and, therefore, the broader and not the bimodal distribution is relevant. In addition we will model ternary glasses. Thus the question
Fig. 5. (a) A con®guration corresponding to dierent distances rma between an anion (shaded circle) and the surface of a smaller and a larger cation (two circles with solid lines). The radii are chosen according to the rules presented in [8], where the smaller cation corresponds to Na and the larger one to Rb . (b) At higher alkali contents the interaction with other non-bridging oxygen ions (shaded circles) becomes important. As shown in the schematic structure of a possible con®guration it is comprehensive that smaller ions are less hindered to approach the minimum energy position than the larger ones. Thus anion con®gurations of lowest energies are preferably occupied by the smaller cations. As low energy con®gurations correspond to a long time of residence small ions become more immobilized this way.
216
R. Kirchheim, D. Paulmann / Journal of Non-Crystalline Solids 286 (2001) 210±223
Fig. 6. Distribution of the dierence of Coulomb energies DEcoul associated with a jump from one to another site for two dierent temperatures and concentrations according to [1]. Concentrations c 0:01 and 0.1 are given in terms of number of cations per lattice sites.
arises how two cations occupy the various sites and especially how they compete for the low energy sites. 4. Occupancy of sites within a distribution of site energies Let us consider the exchange of an A- or a R-ion sitting in site m within the ternary glass xyA2 O 1 yR2 O 1 xSiO2 by a corresponding ion from a reservoir were the chemical potentials lA of A and lR of R are ®xed. This exchange can be expressed by the following equations: Am () Ar
and Rm () Rr :
13
The site m belongs to a subsystem of sites having the same volume Vm and the same value for rma . The number of m-sites Nmo is given from the distribution function of Vm and rma . It can be shown [4,19±21] that the number NA of A-cations and NR of R-cations obeys the following relations: X NAm NA m
X
Nmo kAm and 1 kAm kRm m X X Nmo kRm NR NRm ; 1 kAm kRm m m
14
where NAm is the number of A-cations in m-sites and NRm is the number of R-cations in m-sites. The thermodynamic activities kim are de®ned by the following equations: lAm EAm kAm exp and kB T 15 lRm ERm kRm exp ; kB T where EAm or ERm are the site energies and lAm or lRm are the chemical potentials for A or R cations in m-sites. Changes of site energy caused by changes of the total alkali content shall be included in E. As the sum of NA and NB corresponds to the total alkali content x it has to be constant, i.e., the two chemical potential lA and lR are related to each other and the corresponding relation has to be maintained in the reservoir. This way an cation exchange between a ternary glass and a melt of corresponding alkali salt mixtures changes the concentrations of A and R in the glass until the ratio of the chemical potentials is the same in the glass and in the melt. Experimental studies reveal that it is very dicult to replace the rest of smaller ions in a glass by larger ones [22]. Despite the diculty of straining the network by the incorporation of a larger cation this behavior is in agreement with the preferential occupancy of low energy sites by smaller cations.
R. Kirchheim, D. Paulmann / Journal of Non-Crystalline Solids 286 (2001) 210±223
217
In equilibrium the chemical potentials of the species in reaction (13) have to be equal yielding lAr lAm EAm kB T ln
Nmo
NAm NAm
NRm
Nmo
NRm NAm
NRm
16
and lRr lRm ERm kB T ln
:
17
This corresponds to a chemical potential with a contribution from con®gurational entropy according to a distribution of A and R cations among the available Nmo sites without direct interaction between A and R cations. It is not the equation corresponding to an ideal solution because EAm and ERm depend on the total alkali content. However, for a ternary glass with ®xed x and varying y Eqs. (14)±(17) are useful to discuss the distribution of A and R among the various sites and its eect on activation energies of diusion.From Eqs. (14) and (15) the following ratio is obtained: NAm lAr lRr ERm EAm exp NRm kB T Dl DEm exp : 18 kB T
Fig. 7. Box distribution n E of site energy dierences DE (dashed line) and occupancy ym n DE (according to Eq. (19)) by either A or R ions depending on whether DE is negative (A wins in competing for low energy sites) or positive. The shaded area is equivalent to y, i.e., the total amount of A ions for the ®rst case. Following the assumption made in this work they shall succeed in occupying sites of lowest energy.
with all the available A ions is applicable and Dl is directly obtained from y. This is true for other broad distribution as well as shown in Fig. 8. By adding a small fraction dy of A on the expense of R a useful expression can be derived (as trivial by looking at Fig. 8) dy n Dl dDl:
20
Depending on the sign of Dl DEm site m is preferentially occupied by A or R. The cation fraction ym of a given site m can be obtained from Eq. (18) as ym
NAm 1 : NAm NRm 1 exp DEm Dl kB T
19
This function has the form of the Fermi±Dirac function being about 1 for DEm < Dl and 0 vice versa. For a box-like distribution of DEm it is plotted in Fig. 7. For small temperatures the Fermi±Dirac function has a step like behavior, i.e., ym 1 for DE < Dl and ym 0 for DE > Dl. Increasing temperature leads to a smearing out of the occupancy around Dl over an energy range of about 3kT (cf. Fig. 7). Thus for a width wx kT the simpli®ed model of ®lling the sites of lowest energy
Fig. 8. A general broad distribution is ®lled with A ions of content y (equal to the gray area) up to Dl according to the discussion in the text. The black area corresponds to a substitution of R-ions by A-ions if the chemical potential dierence is raised by Dl. Approximating the black area by a rectangular one gives dy n Dl dDl.
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R. Kirchheim, D. Paulmann / Journal of Non-Crystalline Solids 286 (2001) 210±223
In order to describe diusion and DC-conductivity for a distribution of site energies two approaches have been used: (i) calculating an average jump rate for thermally activated hopping [4] and (ii) averaging over the times of residence [5]. In both cases the activation energy of diusion contains a term which is the negative of the chemical potential or the negative energy of the uppermost ®lled level (cf. Eq. (1)). For thermally activated hopping over the barrier the activation energy is simply the dierence between the average saddle point energy and the chemical potential. With respect to the situation in Fig. 5 the latter corresponds to Dl for A ions and to the uppermost level of the box for B ions. Then changes of the activation energy are related to changes of y via the relation dQ
dDl
dy : n Dl
21
Thus we have shown that the simple concept of ®lling a site energy distribution is an appropriate approximation for ternary glasses. In the following section we will discuss how DE depends upon the size of A and R.
5. Dierence of site energies for A and R cations First we consider the elastic contribution to DE by using Eq. (11) " # 2 2 2l V V V V Ai m Ri m DEmel s : 22 3 Vm Vm By our convention for the ionic volumes VRi > VAi the energy dierence is always negative and increases in magnitude with decreasing Vm . Therefore, larger sites are preferentially occupied by the larger ions R. There is also a distribution of site energies ERm and EAm because of dierent Coulomb interaction with non-bridging oxygen ions. The sites in the broad distribution obtained from Monte Carlo simulations and shown in Fig. 6 are occupied by both A and R with the same probabilities, if the cations are approximated by point charges. However, for realistic sizes of the cations and network
atoms there will be arrangements which do not allow a closest approach of cations as shown in Fig. 5. Then the corresponding variation of the smallest distance rma a cation surface can approach the center of negative charge gives rise to an electrostatic contribution to DE according to the Anderson±Stuart model [15] this can be written as e2 1 1 Coul DEm : 23 4pe0 e rA rma rR rma Because of rR > rA this energy dierence is always positive and increases in magnitude with decreasing rma and, therefore, those sites are preferentially occupied by the smaller ions A which allow a closer approach of the anion. In the following we assume that the direct Coulomb interaction is larger than the elastic one and, therefore, the smaller A ions occupy the low energy sites because of their larger ionic strength. As discussed in Appendix A this assumptions gives rise to a dependence of the site energy distribution wx on the cation radii which is in agreement with experimental ®ndings. One may argue that the larger R ion is pushing matrix atoms away in order to approach the anion as close as possible until it is repelled by the Pauli repulsion. Although this corresponds to no distribution of rma -values, it leads to a distribution of DE because the distortion energy is larger for R compared to A and the distortion energy will depend on the varying site volumes.
6. Mixed alkali eect with a general distribution functions So far a box-type distribution of site energies has been used which in connection with a step-like behavior of the occupancy gave rise to a linear dependence of ln D or the activation energy Q of ion A with respect to y (cation fraction of A). Deviations from this prediction are obvious for small values of y for A Na in a Na/Cs glass in Fig. 3 as well as for a Na/Rb glass in Fig. 4. These discrepancies can be overcome by using a site energy distribution n E which is decreasing with increasing E (cf. Fig. 9). Then changes of QA are expressed by (cf. Eq. (21))
R. Kirchheim, D. Paulmann / Journal of Non-Crystalline Solids 286 (2001) 210±223
dQmA dy QmA QR A
1 n Dl Z y 0
or d~ y ; nDl ~ y
24
where QR A is the activation energy of A-diusion in a binary R-glass. The physical signi®cance of the last equation for small and large values of y is exempli®ed in Fig. 9. Because of high density of sites at low y-values any increase in y causes a small energy change of the top-most occupied energy level only (cf. Fig. 9(a)). The corresponding change of activation energy QmA or log DA , respectively is small, too. For a larger fraction of the
219
cation A sites with high energy Dl but low density n Dl are occupied and a corresponding increase of y causes a much larger change of the top-most occupied energy level (cf. Fig. 9(b)). Adding the eect of network modi®cations as described by Eq. (6) there might be even a relative maximum in QA or a relative minimum in log DA , respectively (see Fig. 9), because the decrease of the Q-value caused by the site energy distribution could be overcompensated by the increase caused by the shrinking network. At high values of y the in¯uence of the site energy distribution dominates as n Dl decreases. Thus deviation from a linear dependence of QA on y can be explained in a natural way. In
Fig. 9. A decreasing site energy distribution function n DE and its occupancy for low (a) and high y-values (b) is shown. The gray area is a measure of y, i.e., the fraction of A-cations. The black area corresponds to the same increment by raising the top-most energy level more (b) or less (a) pronounced. This raise is accompanied by a small or large decrease of activation energy dQmA (cf. Eq. (24)). In (c) n DE and its reciprocal function are shown. The eect n DE has on the activation energy of diusion QmA is described by the integral in Eq. (24) yielding the dashed curve in (d) labeled s.d. The counteracting eect of the decreasing mesh size with increasing y (cf. Eq. (6)) is shown by the lower dashed line labeled m.s. and the sum of the two is presented by a solid line. Thus depending on the magnitude of the two contributions QmA may go through a maximum with increasing y. This corresponds to a minimum in ln Dma vs y diagrams.
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R. Kirchheim, D. Paulmann / Journal of Non-Crystalline Solids 286 (2001) 210±223
addition the deviation should be the larger the larger the size dierence is between A and R, because this dierence determines the slope of the dashed line labeled m.s. (cf. Eq. (2)). A larger size mis®t leads to a larger slope of the line m.s. Then the negative contribution of the mesh size (m.s.) becomes more pronounced in comparison with the eect of the site energy distribution (s.d.) and the sum of both, i.e., the activation energy described by the solid line, is lowered. This conclusion is in agreement with the experimental data presented in Figs. 3 and 4, where the Na/Cs glass has the larger size dierence. In addition the large values obtained for the width w of a rectangular distribution [8] can be understood for a distribution as the one shown in Fig. 9. We have to consider the case of large yvalues shown in Fig. 9(b)) because the ®tting parameter w was obtained in [8] from data with large y-values. A rectangular distribution has the same eect on the y-dependence of the activation energy, if its heights are the same as the small n Dlvalue for the top-most ®lled level and, in order to get the same area as presented by the gray area in Fig. 9(b)), the width w of the rectangular distribution has to become very large. A rigorous quantitative treatment requires an assumption about the function n Dl to be used in Eq. (24). This is avoided because not much is known about n Dl and the mathematical complexity increases without a concomitant increase of physical understanding. In order to use the simple concept of a rectangular distribution further on, we may de®ne the width as a function of y. For the following this is used in a qualitative way only with w being small for y ! 0 and large for y ! 1. Whereas for the Na-ions in Fig. 3 deviations from the straight line behavior are observed but do not lead to a relative minimum of log D, the corresponding relative maximum of Q has been measured for Na tracer diusion in a mixed Li±K glass [23]. As an ion of intermediate size with respect to Li and K sodium is expected to occupy sites on the energy scale in between Li and K and, therefore, its activation energy changes in a similar way as the one for Li with respect to the energy scale (cf. Fig. 10). As the migrating species are Na ions their activation volume of diusion has to be
Fig. 10. Occupancy of sites in a distribution n E for a quaternary glass containing three alkaline ions A, M and R. The smallest ion A is occupying the sites of lowest energy (dark grey) and the largest ion R the highest ones (light grey). The ion M of intermediate size shall be present as a tracer only and, therefore, occupies a small fraction of the sites in between A and R on the energy scale (black). The corresponding energy levels for M are equal to the top-most levels of A. Then changes of the activation energies caused by changes of y due to the energy distribution are the same for A and M.
used in Eq. (6) whereas the change of mesh size is caused by the contents of Li and K ions and, therefore, their dierence of partial molar volumes VR VA 2 ViR ViR ) has to be included. Thus Eq. (6) is modi®ed for this case to QmM QA M wx 1
y
2KViM ViR VO
ViA x 1
y 25
or QmM QR M
ywx y
2KViM ViR VO
ViA x
;
26
where the index M refers to the tracer ion of intermediate size, e.g., Na for the case discussed R before and QA M QM is the activation energy of diusion of tracer M in the binary A±(R±)glass. For small values of y the width w was shown to be small and the third term on the right-hand side of Eq. (26) dominates and QmM increases with increasing y. For larger values of y the width w is enlarged and now the second term on the righthand side of Eq. (26) determines the dependence on y and, therefore, QmM decreases with further
R. Kirchheim, D. Paulmann / Journal of Non-Crystalline Solids 286 (2001) 210±223
Fig. 11. Activation energy of Na tracer diusion in two mixed Li±K silicate glasses [23] as a function of the Li-fraction. The changes can be understood qualitatively in the following way. Activation energy increases at low Li-contents because the mesh size of the network becomes narrower and the counteracting eect of the energy distribution is less pronounced as shown in Fig. 9(a)). This counteracting eect of occupying energy levels of higher energy becomes dominant at higher Li-contents as shown schematically in Fig. 9(b)). In agreement with Eq. (26) the changes are more pronounced for larger values of the total alkali content x (®lled circles).
increasing y. The corresponding experimental results are presented in Fig. 11. A similar behavior but without an extreme value occurs for Rb tracer diusion in a mixed Na±Cs glass. The experimental results [24] are presented in Fig. 12. In the framework of the model of this study it has to be interpreted by using a modi®ed version of Eq. (8) in accordance with the previous discussion of Na in a Li±K glass m DM kB T ln DbM 2KViM ViR ViA x 1 y: 27 wx 1 y VO Again the term describing the eect of the mesh size (second on the right) dominates for low y values because w is small for distribution of n DE like the one shown in Fig. 9. Increasing y gives rise to larger w values and a total compensation of the mesh size eect. For y > 0:6 the two terms on the right-hand side of Eq. (27) are equal. Then w can be calculated with the corresponding value of
221
Fig. 12. Ratio of tracer diusion coecients of Cs and Rb in a mixed Na±Cs silicate glass and a binary Na glass with 75 mol% SiO2 at 397°C [24]. Cs mobility in the mixed glass is reduced by the decreasing mesh size with increasing y. The same is true for the Rb mobility but the eect is ®nally compensated by the decrease of activation energy due to successively occupying sites of higher energy (cf. Figs. 9 and 10).
K 17:5 GPa [8] to be 51 kJ/mol which is about 40% smaller than the ®gure evaluated from the ternary case of Na±Cs glasses [8]. For the last two examples Rb-diusion occurred by an ion exchange in a ternary glass leading to a quaternary glass and, therefore, the size mis®t of cations may lead to a relaxation of the network as discussed in [8]. However, the amount of tracer ions is so small in comparison with the cations in the ternary glass that the corresponding changes of the network density are solely determined by the latter ions and not by the tracer.
7. Conclusions It could be shown that in a mixed glass other than a box-type distribution of site energies allows to explain the non-exponential increase of the diusion coecient with increasing concentration of the minority cations. Thus the new concept of diusion in mixed glasses stressing the interplay between network modi®cations and a distribution
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of site energies is capable of treating even secondorder eects in mixed alkali glasses. In addition it can be extended from ternary to quaternary glasses explaining some of the data on tracer diusion measured there. Although the new model is in accordance with the Anderson±Stuart model [15] it cannot provide a recipe of what site energy distribution is suitable for oxide glasses. Ionic diusion in oxide glasses is complicated by the fact that both Coulomb disorder and structural disorder seemed to aect the energy of an ion and, therefore, analytical solution may not be possible and computer simulations could be the appropriate choice. It should be mentioned that the concept of a distribution of site energies and its relevance for the MAE has been rediscovered [25±27] after its ®rst application in [9] based on a more general concept developed for interstitial diusion in amorphous solids [6,28]. The additional assumption about distinct environments around A and R ions used in [26] is not in contradiction to the present concept as larger R-ions displace the matrix atoms further compared to the smaller A-ions. However, the special ingredient of the treatment in [29,30] that a former and larger R-site being energetically less favorable for the smaller A-ion is dicult to comprehend as discussed in [9].
By assuming a distribution of the distances rma between anion center and cation surface (cf. Fig. 5) around an average value roa with a variation dra < roa leads to the following approximation for the variance of the site exchange energy or the width of the energy distribution wx, respectively oDE dra oroa e2 rR rA rR rA 2roa dra ; 4pe0 e rR roa 2 rA roa 2
wx
A:2
which is simpli®ed further on by using an average ionic radius rRA rR rA =2 and the approxi2 mation rR roa rA roa rRA roa wx
e2 2 rR rA dra : 4pe0 e rRA roa 3
A:3
The values of wx as obtained from curve ®tting in [8] are plotted in Fig. 13 vs the dierence of cation radii rR rA . In agreement with Eq. (A.3) there is a linear relation between wx and rR rA , although the straight line obtained from ®tting diffusion data does not go through the origin. This
Acknowledgements The authors are grateful for ®nancial support provided by the Deutsche Forschungsgemeinschaft (SFB 345). Appendix A. Width of the site energy distribution and its dependence on ionic radii In Section 5 it was argued that the major contribution to the site exchange energy DE stems from the electrostatic interaction between the small and large cations of radius rA and rR , respectively, according to Eq. (23) e2 1 1 DE : A:1 4pe0 e rA rma rR rma
Fig. 13. Width wx of the site energy distribution as a function of the dierence of cation radii rR rA . The data are obtained from ®tting diusion coecients according to Eq. (8) (from [8, Table 2] at 400°C (closed circles) and at 450°C (open circles)) and activation energies according to Eq. (6) (from [8, Table 3(b)]). The straight line are least-square ®ts to the experimental points. Data obtained from diusion coecients do not lie on a straight line through the origin because they contain a temperature dependent (i.e., entropic) contribution.
R. Kirchheim, D. Paulmann / Journal of Non-Crystalline Solids 286 (2001) 210±223
may be caused by an entropic contribution to wx as discussed in [8]. However, the slope of the various straight line is nearly equal corresponding to about 1.7 kJ/Mol/pm. Using e 3 for the dielectric constant of a silicate class and rRA roa 200 pm as the average distance between an oxygen anion and a middle sized alkali cation comparison of the slope with Eq. (A.3) yields dra 150 pm. This value is higher than expected although the order of magnitude is about right. Again this may be attributed to the crude approximation of the site energy distribution by a rectangular one. A distribution as shown in Fig. 9 would lead to smaller variances of DE and, therefore, to smaller values of dra . References [1] D. Paulmann, PhD thesis, University of G ottingen, 1998. [2] D. Kn odler, W. Dietrich, J. Petersen, Solid State Ionics 53± 56 (1992) 1135. [3] D. Kn odler, PhD thesis, University of Konstanz, 1994. [4] R. Kirchheim, Progr. Mater. Sci. 32 (1988) 262. [5] R. Kirchheim, Def. Dius. Forum 143 (1997) 911. [6] R. Kirchheim, J. Non-Cryst. Solids 55 (1983) 243. [7] J.M. Stevels, in: S. Fl ugge (Ed.), Handbuch der Physik, vol. 20/2, Springer, Berlin, 1957, p. 350. [8] R. Kirchheim, J. Non-Cryst. Solids 272 (2000) 85. [9] R. Kirchheim, in: H. Jain, D. Gupta (Eds.), Diusion in Amorphous Materials, TMS, Warrendale, PA, 1994, p. 43. [10] M.D. Ingram, in: R.W. Cahn, P. Haasen, E.J. Kramer (Eds.), Materials Science and Technology, vol. 9, VCH, Weinheim, p. 715.
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[11] G.H. Frischat, Ionic Diusion in Oxide Glasses, TransTech, Aedermansdorf, 1975. [12] K. Hughes, J.O. Isard, in: J. Hladik (Ed.), Physics of Electrolytes, vol. 1, Academic Press, London, 1972, p. 387. [13] H. Jain, N.L. Peterson, H.L. Downing, J. Non-Cryst. Solids 55 (1983) 283. [14] O.V. Mazurin, M.V. Steltsina, T.P. Shaiko-Shaikovskaya, Handbook of Glass Data Part A-E, Elsevier, Amsterdam, New York, part C, p. 85. [15] O.L. Anderson, D.A. Stuart, J. Am. Ceram. Soc. 37 (1954) 573. [16] J.D. Eshelby, in: F. Seitz, D. Turnbull (Eds.), Solid State Physics, Academic Press, New York, 1956. [17] J. Frenkel, Kinetic Theory of Liquids, Oxford University, London, 1949, p. 10. [18] D. Ravaine, J.L. Soquet, Phys. Chem. Glasses 18 (1977) 27. [19] R. Kirchheim, in: D. Wolf, S. Yip (Eds.), Materials Interfaces, Atomic Level Structure and Properties, Chapman and Hall, London, 1992, p. 481. [20] P. Pekarski, R. Kirchheim, J. Memb. Sci. 152 (1999) 251. [21] A.A. Gusev, U.W. Sutter, Phys. Rev. B 27 (1991) 6488. [22] Ref. [11] p. 78. [23] V.K. Pavlovski, Stekloobr. Sostoy. 5 (1970) 148, and Ref. [14], part C, p. 84. [24] H. Jain, N.L. Peterson, J. Am. Ceram. Soc. 66 (1983) 174, and Ref. [14], part C, p. 89. [25] A. Hunt, J. Non-Cryst. Solids 160 (1993) R183. [26] P. Mass, J. Non-Cryst. Solids 255 (1999) 35. [27] S.D. Baranowskii, H. Cordes, J. Chem. Phys. 111 (1999) 7546. [28] R. Kirchheim, Acta Metall. 30 (1982) 1069. [29] P. Maass, A. Bunde, M.D. Ingram, Phys. Rev. Lett. 68 (1992) 3064. [30] A. Bunde, M.D. Ingram, P. Maass, J. Non-Cryst. Solids 172±174 (1994) 1222.