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Towards a universal behaviour of ion dynamics in Na- and Rb-oxide glasses
Solid State Ionics 176 (2005) 1383 – 1391 www.elsevier.com/locate/ssi
Towards a universal behaviour of ion dynamics in Na- and Rb-oxide glasses ´ rpa´d W. Imrea, Helmut Mehrera, John N. Mundyb Stephan Vossa,1, Sergiy V. Divinskia,*, A a
Institut fu¨r Materialphysik, Universita¨t Mu¨nster, Wilhelm-Klemm-Str. 10, D-48149 Mu¨nster, Germany and Sonderforschungsbereich 458, D-48149 Mu¨nster, Germany b The Villages, Florida, 32159-9490 USA Received 3 March 2005; received in revised form 17 March 2005; accepted 17 March 2005
Abstract Activation enthalpies of ionic conductivities in Na – Rb alumino-germanate and borate glasses are reviewed. Correlations between the activation enthalpy and the ratio of average distances between like alkali ions, bd ion, to the average distances between network – former atoms, bd network, are elucidated for single- and mixed-alkali glasses. The Haven ratio is shown to decrease with decreasing bd ion/bd network. Interstitial-like and substitutional-like subnetworks of ion sites are suggested. The experimentally observed dependence of the Haven ratio on bd ion/bd network is consistently reproduced by a Monte Carlo simulation of ion dynamics on this random network including single and collective ion jumps. D 2005 Elsevier B.V. All rights reserved. PACS: 66.30Hs; 66.10.Ed Keywords: Ionic glasses; Ionic conductivity; Tracer diffusion; Haven ratio; Monte Carlo simulation
1. Introduction Although physical properties of different oxide glasses can vary significantly, it is equally true that many features of ionic motion in single- and mixed-alkali glasses are universal for all systems that have been examined. Such a universality of ion dynamics strongly suggests that it is largely determined by the interaction of the ions. The particular structure of a glass system simply serves as a suitable Fskeleton_ providing the space for the alkalis to move in. Although local features of a particular glass network are certainly different, the long-range ion dynamics (ionic conductivity and diffusion) reveal a universal character. Then a particular Fskeleton_ may be replaced by some Funiversal glass network_.
* Corresponding author. E-mail address: [email protected] (S.V. Divinski). 1 Now at Infineon Technologies, Mu¨nchen, Germany. 0167-2738/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ssi.2005.03.007
In this paper, we compile experimental data connected with the ion dynamics in Na –Rb alumino-germanate and in Na – Rb borate glasses. The structures of alkali-borate and of alkali-alumino-germanate glasses are quite different, thus providing a good playground to investigate which properties associated with the ion dynamics exhibit a universal behavior. We expect that such universal behavior is reflected by the fact that the ion dynamics are largely determined by the average ion – ion distance, bd ion, and the average separation between the network – former atoms, bd network. In this context, we will elucidate how the following quantities depend on their ratio, bd ion/bd network: & The activation enthalpy, DH r , associated with the temperature dependence of ionic conduction. & The strength of the mixed-alkali effect, which we define by the relative increase in the activation enthalpy for intermediate Na – Rb glass compositions as compared to a linear interpolation of DH r
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S. Voss et al. / Solid State Ionics 176 (2005) 1383 – 1391
2. 0
between the end-member compositions of a mixedalkali glass. & The Haven ratio of single-alkali Na-oxide glasses.
1. 5
∆Hσ (eV)
A concept of interstitial-like and substitutional-like ion sites is proposed and a Monte Carlo simulation of the Haven ratio is performed including single and collective ion jumps.
In Eq. (1) Y denotes the total alkali content and X the Na/ (Na + Rb) ratio. Three glass series (A, B, and C) were produced with different composition parameters Z which give the relative amount of Al2O3 content. The following ratios were chosen: Z = 0 (A-series), Z/Y = 1/3 (B-series) and Z/Y = 1 (C-series). The variables X, Y, and Z adopt values between 0 and 1. Furthermore, the ionic transport was studied in 17 single and mixed Na –Rb borate glasses also as a function of X and Y. The compositions of the Na – Rb borate glasses are described by ð2Þ
Two mixed-alkali systems were studied, with Y = 0.2 and Y = 0.3. The relevant data of all types of glass system have been published for Na – Rb alumino-germanate glasses in [1– 5] and for Na – Rb borate glasses in [6 –11]. Glass and sample preparation as well as the experimental procedures are described in detail in these papers.
3. Results and discussion 3.1. Activation enthalpy of ionic conduction in single-alkali glasses In single-alkali glasses, it is often observed that the activation enthalpy of the electrical conductivity, DH r , decreases with increasing number density, N ion, of the ions (see Ref. [12] for a review). Assuming a random distribution, we can estimate an average distance of the ions, bd ion, from their number density according to bdion ¼
1 1=3
Nion
I
ð3Þ
1. 5
2. 0
2. 5
/ 2. 0
1. 5
∆Hσ (eV)
ð1Þ
a)
1. 0
Ionic transport measurements of more than fifty Na –Rb alumino-germanate glasses have allowed a systematic examination of the changes of the ionic conductivity with the Na/(Na + Rb) and Al/Ge ratios. These glasses are described by the general formula
Y ½ X Na2 Oð1 X ÞRb2 O I ð1 Y ÞB2 O3 :
1. 0
0. 5
2. Glass compositions
Y ½ X Na2 O I ð1 X ÞRb2 O I ZAl2 O3 ð1 Y Z ÞGeO2 :
borates A-series B-series C-series
1. 0
0. 5 1. 0
borates A-series B-series C-series
b) 1. 5
2. 0
2. 5
/ Fig. 1. Activation enthalpy of the electrical conductivity, DH r , vs. average ion distance in Na- (a) and Rb- (b) borate- and alumino-germanate singlealkali glasses. For direct comparison of different glasses, the alkali – alkali interionic distances bd ion are normalised to the average distance bd network of the corresponding network – former atoms.
Thus we expect that DH r increases with increasing ion distance. We introduce a dimensionless parameter by dividing the average ion – ion distance by an average distance between the network –former atoms. The average distance of the central network atoms, bd network, can be calculated from the number density of the network atoms in a way analogous to Eq. (3). In our case, these are boron atoms on the one hand and Ge/Al atoms on the other hand. In both glass systems, the borate and the alumino-germanate glasses, these atoms serve as central atoms of the structure units of the network. For example, in borate glasses these are BO3 and BO4 units. In Fig. 1 the activation enthalpy DH r is plotted as a function of the ratio bd ion/bd network. Since bd ion and bd network are defined via the corresponding number densities of the alkali ions or network atoms in a way given by Eq. (3), respectively, the ratio bd Na/bd network is then 1 Y 1=3 bdNa =bdnetwork ¼ ð4Þ Y for the Na-borate glasses and 1 Y 1=3 bdNa =bdnetwork ¼ 2Y
ð5Þ
S. Voss et al. / Solid State Ionics 176 (2005) 1383 – 1391
for the Na-germanate glasses. A comparison of Fig. 1a and b shows that the increase in activation enthalpy with increasing ion separation is different for Na and Rb glasses. However, almost the same behaviour is observed for the glasses with the same alkali ion. In Na-based single-alkali glasses the activation enthalpy of the ionic conductivity, DH r , increases linearly with increasing values of the ratio bd ion/bd network, Fig. 1a. Practically all data (measured independently in different laboratories) fall onto a dashed line. Only the activation enthalpy for the C-series (largest Al2O3 content) of Na-alumino-germanate glasses deviates from the general trend. The activation enthalpies of the ionic conductivity of Rbbased single-alkali glasses reveal generally the same dependence on the ratio bd ion/bd network, Fig. 1b. The overall trend in alkali borate- and alumino-germanate glasses is the same: DH r varies roughly linearly as a function of the ratio bd ion/bd network as indicated by the dashed line in Fig. 1b. The slopes of the dashed lines in Figs. 1a and b are different for Na and Rb glasses. The activation enthalpy increase is more pronounced for the ions with larger ionic radius, i.e., for the Rb glasses. The ratio bd ion/bd network does not completely determine the activation enthalpy. However, for a given alkali ion, the A and B series of alumino-germanate glasses and the pure borate glasses show very similar behavior. Only glasses of the C series deviate from the general trend. In the following, we consider the influence of the network –former atoms on the activation enthalpy DH r . In Fig. 2 we have plotted DH r as a function of the average
alumino-germanates
borates
∆Hσ (eV)
1. 5
distance between the network atoms, bd network. Figs. 1 and 2 show the following salient features: (i) The network atoms are closer to each other than the alkali ions, which is reflected by values bd ion/ bd network > 1. (ii) Relatively small changes of bd network in Fig. 2 entail large changes in the activation enthalpy DH r as compared to those for bd ion variations. This is true for both glass systems. (iii) In each single-alkali glass, the activation enthalpy DH r decreases with increasing separation of the network atoms. This is in accordance with the Anderson –Stuart model [13]. A larger separation of the network atoms reduces the strain energy, which is required to open up doorways for ion jumps. Some ‘‘extra space’’ is created when the network expands. This expansion makes the jump process of an alkali ion easier. (iv) The same value of the activation enthalpy as for a Rbalumino-germanate glass is found for a Rb-borate glass at a lower value of bd network. DH r of Rb-borate and of Rb-alumino-germanate glasses as a function of bd network behave similarly, if one allows for a nearly constant shift of about 0.05 nm. This value roughly coincides with the difference between the covalent radii between B (0.082 nm) and Ge (0.122 nm) or Al (0.118 nm), respectively. A similar behavior for DH r as a function of bd network can be observed for Naborate and for Na-alumino-germanate glasses. The similarity between the covalent radii of Ge and Al may explain why DH r does not depend significantly on the Al-content of the alumino-germanate glasses. (v) For mixed-alkali glasses, the activation enthalpy does not monotonically depend on the separation of the network atoms. This can be seen in Fig. 2 (full symbols) for two Na –Rb borate glass systems with total alkali contents of Y = 0.2 (triangles down) and 0.3 (triangles up) and for three Na – Rb-germanate glass systems with Y = 0.1 (triangles up), 0.2 (triangles down) and 0.3 (full circles). 3.2. Activation enthalpy of ionic conduction in mixed-alkali glasses
1. 0 Rb Na Na 0. 5 0.30
1385
0.34
0.38
Rb
0.42
< dnetwork > (nm) Fig. 2. Activation enthalpy of the electrical conductivity, DH r , vs. the average distance of the central network atoms, bd network, i.e., boron in borate glasses and aluminum/germanium in the alumino-germanate glasses. Open squares—Na-oxide glasses; open circles—Rb-oxide glasses, filled symbols represent five Na – Rb mixed-alkali glass systems (see text for the relation to X and Y).
In this section we address the mixed-alkali effect (MAE). As pointed out in the introduction, we focus on features which are common to different mixed-alkali systems. Interactions between unlike alkali ions manifest themselves in a maximum in the activation enthalpy of the electrical conductivity, DH r . The strength of these interactions is reflected by the relative increase in DH r . To quantify this rise, we define DMAE in a mixed-alkali glass by ;
DMAE ¼
DHr ðXmax Þ DHr ðXmax Þ ;
DHr ðXmax Þ
ð6Þ
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S. Voss et al. / Solid State Ionics 176 (2005) 1383 – 1391
1. 2
as a measure for the strength of the MAE. Here X max denotes the relative composition where the maximum in ; DH r is observed. DHr denotes the activation enthalpy one would expect for a linear interpolation between the activation enthalpies of the end-member compositions according to ; DHr ð X Þ
H
R
0. 8 0. 6 0. 4
¼ DHr ð0Þ þ X ðDHr ð1Þ DHr ð0ÞÞ:
ð7Þ
To obtain DMAE for a given mixed-alkali system, we first fitted a third-order polynomial to the measured values of DH r (X). From these; fits, we then derived the values for X max and DH r (X max). DHr was calculated from Eq. (7) by inserting the measured values for DH r (0) and DH r (1) if available, otherwise we used the values from the polynomial fits. Fig. 3 displays DMAE vs. bd ion/bd network for Na –Rb mixed-alkali borate and alumino-germanate glasses. We observe the following remarkable features: (i) The values of DMAE as a function of bd ion/bd network are almost independent of the glass system. (ii) For all systems DMAE decreases with increasing bd ion/ bd network and vanishes at a value of bd ion/bd networkå 1.8. This suggests also for the Na – Rb borate glasses that no mixed-alkali effect will be observed when bd ion/bd network exceeds a value of about 1.8. The similarity of the values for DMAE as a function of bd ion/bd network independent of the network indicates that the ions themselves play an important role for the mixedalkali effect. Some insight into the reasons of such a universal behaviour of the activation enthalpy DH r comes from the Haven ratio discussed below. 3.3. Haven ratio The Haven ratio H R quantifies the ratio between the tracer diffusion coefficient, D*, and the conductivity borates A-series B-series C-series
0. 6
∆MAE
1. 0
0. 4 0. 2
0 1. 0
1. 4
1. 8
2. 2
/ Fig. 3. Strength of the mixed-alkali effect, DMAE, Eq. (6), as a function of bd ion/bd network. Open symbols represent Na – Rb alumino-germanate glasses, whereas the two filled circles refer to Na – Rb borate glasses, respectively.
0. 2 0 0
2
4
6
8
/ Fig. 4. Haven ratio H R as a function of the ratio of the average Na – Na distance to the network distance, bd Na/bd network. Values for Na-borate glasses: —Voss et al. [10] (T=380 -C), >—Kelly et al. [14] (T=300 -C); Na-alumino-germanate glasses: r—this work (T=380 -C), q—Kelly et al. [14] (T=300 -C).
&
diffusion coefficient, D r . For single-alkali glasses D r can be calculated from the measured dc conductivity, adc, via the Nernst –Einstein equation. The Haven ratio is then defined as HR ¼
D4 : Dr
ð8Þ
D* and D r are not identical due to correlation effects and/or due to the collective nature of the jump process. We talk about collective processes when two or more ions move in one jump event. For collective jump events, the displacement of a tracer ion is smaller than the displacement of the charge itself. Kelly et al. [14] have given an overview of the Haven ratio as a function of total alkali content Y for alkali borate, germanate, and silicate glasses. In each system, they reported a similar decrease of H R with increasing Y. To reveal a universal behavior of the Haven ratio for different glass systems, we plotted H R in Fig. 4 as a function of bd Na/bd network for the Na-borate and Naalumino-germanate glasses. Furthermore, we include data compiled by Kelly et al. [14] for H R in Na-borate and Na-alumino-germanate glasses. From Figs. 2 and 3 of reference [14], we read out the H R ( Y) data electronically. Using Eqs. (4) and (5), these literature data on the Haven ratio can be replotted vs. bd ion/bd network Fig. 4 reveals the following features: – For values of bd Na/bd network larger than 4, the Haven ratio is close to unity. This value suggests an uncorrelated motion of single ions. For a crystalline ionconducting material, a Haven ratio of unity gives evidence for a direct interstitial diffusion mechanism. – It is an open question what the lowest value for the Haven ratio is. From Fig. 4 it may be anticipated that the Haven ratio tends to zero with decreasing bd Na/ bd network. However, values of bd Na/bd network close to zero are physically unreasonable. A Haven ratio of
S. Voss et al. / Solid State Ionics 176 (2005) 1383 – 1391
about 0.25 is reported for bd Na/bd networkå1 (Fig. 4). Small values of H R are expected for collective jump processes of the ions. – The majority of experimental studies have been performed for bd Na/bd network between 1 and 4. In this composition region H R increases approximately linearly with increasing bd Na/bd network. A linear fit to the experimental data in Fig. 4 results in the following dependence: HR , 0:23
bdNa þ 0:03: bdnetwork
ð9Þ
In a recent study, some of us (S.V., A.I. and H.M. [9]) measured the Haven ratio in a Rb-borate glass for various temperatures. The temperature dependence of H R can be described by the Arrhenius relation H R = 20.9 exp(20.8 kJmol1/RT) within the measured temperature range 505 to 650 K. H R decreases with decreasing temperature and at 505 K a value of H R = 0.15 is found. In contrast, no temperature dependence of H R is observed in the corresponding Na-borate glass with H Rå0.459. In what follows, we introduce the concept of subnetworks for the ions and a diffusion mechanism which can explain the experimental finding. This will be supported by Monte Carlo simulations.
& The incorporation of Na2O (or Rb2O) in pure borate glass leads to an alkali-borate glass in which some BO3 units are transformed into BO4 units. The associated positions for alkali ions are denoted as substitutional-like or S sites. In the composition interval under consideration ( Y < 0.25), the number of BO4 units almost coincides with the number of alkali ions for charge-neutrality reasons (glasses with Y < 0.25 are free of non-bridging oxygen). In a singlealkali glass only one type of S site is considered. In a mixed-alkali glass–not treated in detail in this paper– two types of site would have to be introduced. & An alkali ion normally resides on an S site. It can jump to a neighboring I site as a result of thermal activation, leaving the S site unoccupied. & The numbers of I or S sites are assumed to be proportional to the partial concentration of borate or alkali oxides, respectively. The ratio of the number of S sites to the total number of sites is related to the number density of alkali ions. A larger number of S sites corresponds to a higher number density of ions and thus to a smaller average ion –ion distance in the glass. & We assume that the I and S sites keep their character. Relaxation effects, i.e., site transformation from I to S or vice versa, are not taken into account. & Only jumps to neighbouring sites are considered: (i) single jumps of an ion from an S or I site to a neighboring unoccupied I or S site, Fig. 5a; (ii) collective jumps in which two or more ions move simultaneously, Fig. 5b. The following kinds of collective jumps are
4. Monte Carlo calculations of the Haven ratio The previous discussion has shown that the Haven ratio decreases with decreasing bd Na/bd network independently of the glass matrix. This suggests that the physical origin of this decrease can be largely attributed to the ion –ion and network separations. Different glasses have their own glass network, with local configurations depending on various parameters such as composition, thermal history, preparation method, etc. However, the universality of Fig. 4 suggests that long-range diffusion Fscreens-out_ these differences. In the following, we reproduce a decreasing Haven ratio with decreasing ion –ion distance by a simple model. We then compare the calculations with the experiments.
1387
_ BO4 Na+
a)
BO3
_ BO4
Na+
4.1. Description of the model Our model has been derived to describe the ion dynamics in borate glasses. It is based on the following assumptions: & Trigonal BO3 configurations are characteristic of the structure of pure borate glass B2O3. The packing of the BO3 units provides interstitial-like sites for alkali ions. We denote these positions as I sites. An alkali ion has a certain residence time at such a site.
b)
BO3
Fig. 5. Schematic representation of jump processes considered in the present model. The Na+ ions reside either on I sites (provided by BO3 units) or on S sites (provided by BO 4 units), shown by the small shaded circles and the large dashed circles, respectively. In panel (a), single jumps of the Na+ ion are illustrated. In panel (b), a collective jump event involving more than one Na+ ion is shown.
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S. Voss et al. / Solid State Ionics 176 (2005) 1383 – 1391
taken into account: an Na ion at an I site and an Na ion at a neighbouring S site perform a collective (simultaneous) jump. As a result, the first Na ion occupies the S site now and the second Na ion resides at a neighbouring site (of type I or S). If the second Na ion has been displaced to an I site, the jump chain is finished. However, if this Na ion has landed at a neighbouring S site, the third Na ion participates in the collective jump event and its new position will be chosen in a similar manner as for the second Na ion, and so on. This chain of displacements within the collective jump event is considered until the last Na ion lands on an I site. Then the collective jump event is finished. We have denoted the I sites as Finterstitial-like_ sites. Alkali ion migration on the I subnetwork is similar to interstitial diffusion in crystals. On the other hand, if an ion leaves an S (Fsubstitutional-like_) site, a vacancy-like configuration is left (the charged BO4 configuration). Then an alkali ion should occupy this Fvacancy_. The appearance of collective jumps may be understood as follows: if a sodium ion is located at an S site, the total charge of the configuration is zero. However, the alkali ion is not necessarily located at the geometrical center of the negative charge. This configuration corresponds to an electrical dipole Na+ – BO4. If another Na+ ion occupies a neighbouring I site, the first Na+ ion may be additionally pushed from the center of the BO4 configuration, Fig. 5b. The electrostatic interaction with the neighbouring Na+ ion decreases the energy barrier for dissociation of the Na+ – BO4 dipole and its Na+ ion may leave the BO4 unit with a simultaneous jump of the second Na+ ion into this vacant BO4 unit. As a result, the two ions moved simultaneously within a collective jump event. Alternatively, the first Na ion may not only jump into an adjacent I site, but it may jump to a neighbouring occupied S site and activate a simultaneous jump of other alkali ion from this S site (see the last point in the above description of the model). This increases the number of ions participating in the collective jump event. The effectivity of collective jump events depends crucially on the Na – Na distance. The smaller the Na –Na separation, the larger is the effect of Coulomb repulsion on diffusion barriers for the collective jump (Fig. 5b). However, it is not only the ion – ion distance which affects the barriers. A smaller average network distance bd network corresponds to a larger number of I sites between the S sites. This causes two effects: (i) the I sites weaken the electrostatic interaction of ions between S sites; and (ii) the I sites hinder ion jumps towards the S sites. Both effects increase the barriers for collective ion motion. These features of the model qualitatively agree with the experimental observations that the activation enthalpy, DH r , is directly proportional to the average ion – ion distance, bd ion, and inversely proportional to the average network distance, bd network. The decrease of the energy barriers for collective jumps (Fig. 5b) increases their probability relative to single-ion
jumps (Fig. 5a). Thus the Fdegree of collectivity_ of the ion dynamics increases with decreasing bd ion/bd network. As a result, the Haven ratio decreases with increasing ion density and decreasing ion – ion separation. This is the trend which is experimentally observed in Fig. 4. The present model has features similar to the one introduced by Habasaki and Hiwatari [15]. Our model is more specific for the borate glasses and is extended by considering jump events which include several ions (collective motion of even four or five ions were observed, see below). Moreover, the simulation in reference [16] was carried out on a cubic lattice, whereas a random 3D network for diffusion jumps is generated in the present study. This is indeed crucial for reproducing the concentration dependence of the Haven ratio observed experimentally (see below). The different types of sites and ion jumps are common features of the present approach and the dynamic structure model [16]. However, site relaxation effects needed not be included in the present model. 4.2. Construction of a network for ionic motion A random glass-like network is constructed as follows. A cubic simulation box of size 40 40 40 (in arbitrary units) is chosen with periodic boundary conditions. This box is randomly filled with hard spheres of radius 1 (in the same units). A packing density of about 33% is reached. These positions are marked as I sites and they represent the BO3 units. A connectivity graph is constructed from neighbours of each I site. A site I * is considered as a neighbour for site I, if the distance I * –I is smaller than a chosen maximum value, d max, and if I * is not strongly overshaded by other sites. The value d max should be chosen reasonably large in order to sample a complete neighbourhood of each site.2 A certain amount of sites is then randomly transformed into S sites corresponding to the given concentration of the alkali ions in the glass. Our aim is the calculation of the Haven ratio. Only the ratio between the tracer diffusivity, D*, and the conductivity diffusion coefficient, D r , is of importance, but not the individual absolute values. For simplicity, we assume that all attempt frequencies are equal. Then the difference of the energy barriers between collective and single-ion jumps, DE, is the relevant parameter in the present model. The ratio of jump rates of single-ion to collective jumps is given by exp(DE/kT). Here k is the Boltzmann constant and T the absolute temperature. The simulation has been performed for various values of DE: DE = 0 (single jumps are preferred); DE = 0 (both types of jumps are equally probable); and DE < 0 (collective jumps are more probable).
2 A variation of d max from 3 to 4 resulted in marginal changes of the connectivity graph obtained. Note that d max = 2 is the shortest distance between centers of hard spheres with unity radius.
S. Voss et al. / Solid State Ionics 176 (2005) 1383 – 1391
If we simply start a kinetic Monte Carlo simulation of the ion dynamics within the constructed network, the following process is very often observed: a reverse jump of an ion occurs after its initial jump from an S site. Only occasionally an ion may leave its S site and initiate a collective jump event. In order to enhance the productivity of the simulation procedure, a modified approach is applied. An additional ion is introduced in the simulation block at an I site and a standard Monte Carlo procedure with the residence time algorithm [17,18] is applied. This additional ion can either perform a jump between I sites (Fig. 5a) or initiate a collective jump sequence (Fig. 5b) with the relative probabilities determined by exp(DE/kT). In the latter case, this ion will finally reside at an S site and a new ion will occupy an I site. At any moment, only one noncompensated charge (free alkali ion) is available in the system. Jump rates m l , l = 1, . . .z i, for the particular ion at the ith I site are calculated for any possible jump direction l (z i is the number of neighbours for the site i). For simplicity, the jump rate from the I site to an S site (this jump initiates a collective jump event) is taken as unity. The jump rate between I sites is then given by exp(DE/kT). A random number is generated and the jump direction l (including possible collective jump modes) is chosen according to the i probabilities Pl ; Pl ¼ Pl ¼ ml =~Zj¼1 mj . After the completed jump event, the coordinates of participating ions are updated and the time variable is incremented according to the residence time algorithm by the value s: s¼
1 Z1 X
I
ð10Þ
ml
l¼1
After a fixed number of jump events (usually 105 to 106), the tracer diffusivity, D*, and the conductivity diffusion coefficient, D r , are calculated via D4 ¼
s þ1 1 NX R2 6t m¼1 m
ð11Þ
R2 : 6t
ð12Þ
and Dr ¼
In Eq. (11), the summation is performed over all ions m which have moved the distance R m during the simulation time t. Note that there are N S ions at the S sites and one extra ion at an I position. The conductivity diffusion coefficient, Eq. (12), is defined by the total displacement R of the non-compensated charge during the simulation time t. According to the present model, only one ion on the I site subnetwork is available during the whole simulation time, which produces the non-compensated charge.
1389
For each alkali ion concentration 102 to 103 different glassy configurations were generated. For a given glassy configuration, the random walk was repeated 105 to 106 times and the results were averaged over 103 to 104 different distributions of ions. 4.3. Simulation results Simulations were performed for the values DE/kT = 6, 0, 3, and 6. These values correspond to the following ratios between the probabilities of single and collective jumps: 1:0.003, 1:1, 0.05:1, and 0.003: 1, respectively. The results as a function of the parameter bd ion/bd network for a glassy network are presented in Fig. 6a. Comparison of experimental data (Fig. 4) and simulation (Fig. 6a) suggests that our model can describe the composition dependence of the Haven ratio in single-alkali glasses, if we assume that collective jumps dominate (DE < 0). The specific value of DE is a fitting parameter of the present model. The calculations carried out at DE/ kT = 3 result in a composition dependence of the Haven ratio (Fig. 6a), which is in good agreement with experiments shown in (Fig. 4). At small ion concentrations, corresponding to bd ion/ bd network > 4, single-ion jumps prevail and the Haven ratio is about unity. Each jump of an ion equally contributes to the conductivity and to the tracer long-range diffusion, and therefore D r åD*. As the alkali content increases and the average ion – ion separation decreases, more alkali ions appear at neighbouring sites of a given alkali ion. Then collective jumps become more probable. The chains of sequential ion displacements result in a large total displacement of charge, whereas the individual displacements of ions are smaller. As a consequence, the Haven ratio decreases. The larger the contribution of collective jumps, the smaller is the Haven ratio. The average number of ions participating in a jump event gradually increases from unity at bd ion/bd network > 4 to about 2.1 at bd ion/bd networkå2. In the simulation, sometimes jump events were observed with four and more participating ions. One of the most striking features of Fig. 6a is that for DE/ kT = 3, H R decreases almost linearly with decreasing bd ion/bd network. This is indicated by the dashed line in Fig. 6a at 1 < bd ion/bd network < 4. In order to elucidate the effect of a glassy (random) network on ion motion, we have performed an analogous calculation of the Haven ratio for a regular FCC lattice. The results are presented in Fig. 6b. Qualitatively a similar behaviour is observed for the Haven ratio in random and FCC lattices. In the case of an FCC lattice, however, a more moderate decrease of H R is seen as the value of bd ion/ bd network falls below 4. For the same energy barriers the Haven ratio for ion motion on a random lattice is consistently smaller than that for the FCC lattice. The
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S. Voss et al. / Solid State Ionics 176 (2005) 1383 – 1391
∆E/kT = 6 1 ∆E/kT = 0
HR
0.8
0.6
∆E/kT = −6 ∆E/kT = −3
0.4
0.2
a) 0 0
4
2
6
8
/ 1.2
∆E/kT = 6 1.0
∆E/kT = 0
HR
0.8
0.6
∆E/kT = −6
∆E/kT = −3
0.4
0.2
b) 0.0 0
4
2
6
8
/ Fig. 6. Calculated Haven ratio as a function of the ratio of bd ion/bd network. Results are presented for a glassy network (a) and for a regular FCC lattice (b). Monte Carlo calculations have been performed for different values of DE: for DE > 0 single jumps are preferred, for DE < 0 collective jumps are more probable.
difference becomes more pronounced when the ion – ion separation decreases.
5. Summary and conclusions Numerous experimental results on ionic conductivity and tracer diffusion in single and mixed Na – Rb aluminogermanate and borate glasses are analysed. Universal features of the ion dynamics in these glasses are elucidated. The activation enthalpy of conductivity is largely determined by the ratio of average distance between alkali ions, bd ion, and network – former atoms, bd network. The Haven ratio consistently decreases, if the ratio bd ion/bd network decreases.
A new model of ion diffusion in borate and aluminogermanate glasses is suggested with interstitial-like and substitutional-like ion sites. Single and collective diffusion mechanisms are considered. Monte Carlo calculations of the Haven ratio result in a good agreement with the experimental findings. The decrease of the Haven ratio with decreasing bd ion/bd network is attributed to an increase of collective jump events.
Acknowledgments One of us (J.N.M.) gratefully acknowledges the invitation of the Sonderforschungsbereich to spend several weeks in Munster as a visiting scientist. We are grateful to
S. Voss et al. / Solid State Ionics 176 (2005) 1383 – 1391
Prof. Dr. K. Funke and to Dr. C. Cramer for their helpful comments on the manuscript. Financial support of the DFG via Sonderforschungbereich 458 is acknowledged.
[8] [9] [10]
References [1] [2] [3] [4] [5] [6]
J.N. Mundy, G.L. Jin, Solid State Ionics 21 (1986) 305. J.N. Mundy, G.L. Jin, Solid State Ionics 24 (1987) 263. J.N. Mundy, G.L. Jin, Solid State Ionics 25 (1987) 71. G.L. Jin, Y. Liu, J.N. Mundy, J. Mater. Sci. 22 (1987) 3672. J.N. Mundy, G.L. Jin, Solid State Ionics 66 (1993) 69. A.W. Imre, S. Voss, H. Mehrer, Phys. Chem. Chem. Phys. 4 (2002) 3219. [7] H. Mehrer, A.W. Imre, S. Voss, in: P. Vincenzini, V. Buscaglia (Eds.), Mass and Charge Transport in Inorganic Materials: II. CIMTEC,
[11] [12] [13] [14] [15] [16] [17] [18]
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Advances in Science and Technology, vol. 37, 2002, (TECHNA Srl, 2003). A.W. Imre, S. Voss, H. Mehrer, J. Non-Cryst. Solids 333 (2004) 231. S. Voss, A.W. Imre, H. Mehrer, Phys. Chem. Chem. Phys 6 (2004) 3669. S. Voss, F. Berkemeier, A.W. Imre, H. Mehrer, Z. Phys. Chem. 218 (2004) 1353. F. Berkemeier, S. Voss, A.W. Imre, and H. Mehrer, Phys. Chem. Chem. Phys. (submitted for publication). M.D. Ingram, Phys. Chem. Glasses 28 (1987) 215. O.L. Anderson, D.A. Stuart, J. Am. Ceram. Soc. 37 (1954) 573. J.E. Kelly, J.F. Cordaro, M. Tomozawa, J. Non-Cryst. Solids 41 (1980) 47. J. Habasaki, Y. Hiwatari, Phys. Rev., E 62 (2000) 8790. A. Bunde, P. Maass, M.D. Ingram, J. Non-Cryst. Solids 172 – 174 (1994) 1222. G.E. Murch, Am. J. Phys. 47 (1979) 78. S.V. Divinski, M. Salamon, H. Mehrer, Philos. Mag. 84 (2004) 757.
Towards a universal behaviour of ion dynamics in Na- and Rb-oxide glasses ´ rpa´d W. Imrea, Helmut Mehrera, John N. Mundyb Stephan Vossa,1, Sergiy V. Divinskia,*, A a
Institut fu¨r Materialphysik, Universita¨t Mu¨nster, Wilhelm-Klemm-Str. 10, D-48149 Mu¨nster, Germany and Sonderforschungsbereich 458, D-48149 Mu¨nster, Germany b The Villages, Florida, 32159-9490 USA Received 3 March 2005; received in revised form 17 March 2005; accepted 17 March 2005
Abstract Activation enthalpies of ionic conductivities in Na – Rb alumino-germanate and borate glasses are reviewed. Correlations between the activation enthalpy and the ratio of average distances between like alkali ions, bd ion, to the average distances between network – former atoms, bd network, are elucidated for single- and mixed-alkali glasses. The Haven ratio is shown to decrease with decreasing bd ion/bd network. Interstitial-like and substitutional-like subnetworks of ion sites are suggested. The experimentally observed dependence of the Haven ratio on bd ion/bd network is consistently reproduced by a Monte Carlo simulation of ion dynamics on this random network including single and collective ion jumps. D 2005 Elsevier B.V. All rights reserved. PACS: 66.30Hs; 66.10.Ed Keywords: Ionic glasses; Ionic conductivity; Tracer diffusion; Haven ratio; Monte Carlo simulation
1. Introduction Although physical properties of different oxide glasses can vary significantly, it is equally true that many features of ionic motion in single- and mixed-alkali glasses are universal for all systems that have been examined. Such a universality of ion dynamics strongly suggests that it is largely determined by the interaction of the ions. The particular structure of a glass system simply serves as a suitable Fskeleton_ providing the space for the alkalis to move in. Although local features of a particular glass network are certainly different, the long-range ion dynamics (ionic conductivity and diffusion) reveal a universal character. Then a particular Fskeleton_ may be replaced by some Funiversal glass network_.
* Corresponding author. E-mail address: [email protected] (S.V. Divinski). 1 Now at Infineon Technologies, Mu¨nchen, Germany. 0167-2738/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ssi.2005.03.007
In this paper, we compile experimental data connected with the ion dynamics in Na –Rb alumino-germanate and in Na – Rb borate glasses. The structures of alkali-borate and of alkali-alumino-germanate glasses are quite different, thus providing a good playground to investigate which properties associated with the ion dynamics exhibit a universal behavior. We expect that such universal behavior is reflected by the fact that the ion dynamics are largely determined by the average ion – ion distance, bd ion, and the average separation between the network – former atoms, bd network. In this context, we will elucidate how the following quantities depend on their ratio, bd ion/bd network: & The activation enthalpy, DH r , associated with the temperature dependence of ionic conduction. & The strength of the mixed-alkali effect, which we define by the relative increase in the activation enthalpy for intermediate Na – Rb glass compositions as compared to a linear interpolation of DH r
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S. Voss et al. / Solid State Ionics 176 (2005) 1383 – 1391
2. 0
between the end-member compositions of a mixedalkali glass. & The Haven ratio of single-alkali Na-oxide glasses.
1. 5
∆Hσ (eV)
A concept of interstitial-like and substitutional-like ion sites is proposed and a Monte Carlo simulation of the Haven ratio is performed including single and collective ion jumps.
In Eq. (1) Y denotes the total alkali content and X the Na/ (Na + Rb) ratio. Three glass series (A, B, and C) were produced with different composition parameters Z which give the relative amount of Al2O3 content. The following ratios were chosen: Z = 0 (A-series), Z/Y = 1/3 (B-series) and Z/Y = 1 (C-series). The variables X, Y, and Z adopt values between 0 and 1. Furthermore, the ionic transport was studied in 17 single and mixed Na –Rb borate glasses also as a function of X and Y. The compositions of the Na – Rb borate glasses are described by ð2Þ
Two mixed-alkali systems were studied, with Y = 0.2 and Y = 0.3. The relevant data of all types of glass system have been published for Na – Rb alumino-germanate glasses in [1– 5] and for Na – Rb borate glasses in [6 –11]. Glass and sample preparation as well as the experimental procedures are described in detail in these papers.
3. Results and discussion 3.1. Activation enthalpy of ionic conduction in single-alkali glasses In single-alkali glasses, it is often observed that the activation enthalpy of the electrical conductivity, DH r , decreases with increasing number density, N ion, of the ions (see Ref. [12] for a review). Assuming a random distribution, we can estimate an average distance of the ions, bd ion, from their number density according to bdion ¼
1 1=3
Nion
I
ð3Þ
1. 5
2. 0
2. 5
1. 5
∆Hσ (eV)
ð1Þ
a)
1. 0
Ionic transport measurements of more than fifty Na –Rb alumino-germanate glasses have allowed a systematic examination of the changes of the ionic conductivity with the Na/(Na + Rb) and Al/Ge ratios. These glasses are described by the general formula
Y ½ X Na2 Oð1 X ÞRb2 O I ð1 Y ÞB2 O3 :
1. 0
0. 5
2. Glass compositions
Y ½ X Na2 O I ð1 X ÞRb2 O I ZAl2 O3 ð1 Y Z ÞGeO2 :
borates A-series B-series C-series
1. 0
0. 5 1. 0
borates A-series B-series C-series
b) 1. 5
2. 0
2. 5
Thus we expect that DH r increases with increasing ion distance. We introduce a dimensionless parameter by dividing the average ion – ion distance by an average distance between the network –former atoms. The average distance of the central network atoms, bd network, can be calculated from the number density of the network atoms in a way analogous to Eq. (3). In our case, these are boron atoms on the one hand and Ge/Al atoms on the other hand. In both glass systems, the borate and the alumino-germanate glasses, these atoms serve as central atoms of the structure units of the network. For example, in borate glasses these are BO3 and BO4 units. In Fig. 1 the activation enthalpy DH r is plotted as a function of the ratio bd ion/bd network. Since bd ion and bd network are defined via the corresponding number densities of the alkali ions or network atoms in a way given by Eq. (3), respectively, the ratio bd Na/bd network is then 1 Y 1=3 bdNa =bdnetwork ¼ ð4Þ Y for the Na-borate glasses and 1 Y 1=3 bdNa =bdnetwork ¼ 2Y
ð5Þ
S. Voss et al. / Solid State Ionics 176 (2005) 1383 – 1391
for the Na-germanate glasses. A comparison of Fig. 1a and b shows that the increase in activation enthalpy with increasing ion separation is different for Na and Rb glasses. However, almost the same behaviour is observed for the glasses with the same alkali ion. In Na-based single-alkali glasses the activation enthalpy of the ionic conductivity, DH r , increases linearly with increasing values of the ratio bd ion/bd network, Fig. 1a. Practically all data (measured independently in different laboratories) fall onto a dashed line. Only the activation enthalpy for the C-series (largest Al2O3 content) of Na-alumino-germanate glasses deviates from the general trend. The activation enthalpies of the ionic conductivity of Rbbased single-alkali glasses reveal generally the same dependence on the ratio bd ion/bd network, Fig. 1b. The overall trend in alkali borate- and alumino-germanate glasses is the same: DH r varies roughly linearly as a function of the ratio bd ion/bd network as indicated by the dashed line in Fig. 1b. The slopes of the dashed lines in Figs. 1a and b are different for Na and Rb glasses. The activation enthalpy increase is more pronounced for the ions with larger ionic radius, i.e., for the Rb glasses. The ratio bd ion/bd network does not completely determine the activation enthalpy. However, for a given alkali ion, the A and B series of alumino-germanate glasses and the pure borate glasses show very similar behavior. Only glasses of the C series deviate from the general trend. In the following, we consider the influence of the network –former atoms on the activation enthalpy DH r . In Fig. 2 we have plotted DH r as a function of the average
alumino-germanates
borates
∆Hσ (eV)
1. 5
distance between the network atoms, bd network. Figs. 1 and 2 show the following salient features: (i) The network atoms are closer to each other than the alkali ions, which is reflected by values bd ion/ bd network > 1. (ii) Relatively small changes of bd network in Fig. 2 entail large changes in the activation enthalpy DH r as compared to those for bd ion variations. This is true for both glass systems. (iii) In each single-alkali glass, the activation enthalpy DH r decreases with increasing separation of the network atoms. This is in accordance with the Anderson –Stuart model [13]. A larger separation of the network atoms reduces the strain energy, which is required to open up doorways for ion jumps. Some ‘‘extra space’’ is created when the network expands. This expansion makes the jump process of an alkali ion easier. (iv) The same value of the activation enthalpy as for a Rbalumino-germanate glass is found for a Rb-borate glass at a lower value of bd network. DH r of Rb-borate and of Rb-alumino-germanate glasses as a function of bd network behave similarly, if one allows for a nearly constant shift of about 0.05 nm. This value roughly coincides with the difference between the covalent radii between B (0.082 nm) and Ge (0.122 nm) or Al (0.118 nm), respectively. A similar behavior for DH r as a function of bd network can be observed for Naborate and for Na-alumino-germanate glasses. The similarity between the covalent radii of Ge and Al may explain why DH r does not depend significantly on the Al-content of the alumino-germanate glasses. (v) For mixed-alkali glasses, the activation enthalpy does not monotonically depend on the separation of the network atoms. This can be seen in Fig. 2 (full symbols) for two Na –Rb borate glass systems with total alkali contents of Y = 0.2 (triangles down) and 0.3 (triangles up) and for three Na – Rb-germanate glass systems with Y = 0.1 (triangles up), 0.2 (triangles down) and 0.3 (full circles). 3.2. Activation enthalpy of ionic conduction in mixed-alkali glasses
1. 0 Rb Na Na 0. 5 0.30
1385
0.34
0.38
Rb
0.42
< dnetwork > (nm) Fig. 2. Activation enthalpy of the electrical conductivity, DH r , vs. the average distance of the central network atoms, bd network, i.e., boron in borate glasses and aluminum/germanium in the alumino-germanate glasses. Open squares—Na-oxide glasses; open circles—Rb-oxide glasses, filled symbols represent five Na – Rb mixed-alkali glass systems (see text for the relation to X and Y).
In this section we address the mixed-alkali effect (MAE). As pointed out in the introduction, we focus on features which are common to different mixed-alkali systems. Interactions between unlike alkali ions manifest themselves in a maximum in the activation enthalpy of the electrical conductivity, DH r . The strength of these interactions is reflected by the relative increase in DH r . To quantify this rise, we define DMAE in a mixed-alkali glass by ;
DMAE ¼
DHr ðXmax Þ DHr ðXmax Þ ;
DHr ðXmax Þ
ð6Þ
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S. Voss et al. / Solid State Ionics 176 (2005) 1383 – 1391
1. 2
as a measure for the strength of the MAE. Here X max denotes the relative composition where the maximum in ; DH r is observed. DHr denotes the activation enthalpy one would expect for a linear interpolation between the activation enthalpies of the end-member compositions according to ; DHr ð X Þ
H
R
0. 8 0. 6 0. 4
¼ DHr ð0Þ þ X ðDHr ð1Þ DHr ð0ÞÞ:
ð7Þ
To obtain DMAE for a given mixed-alkali system, we first fitted a third-order polynomial to the measured values of DH r (X). From these; fits, we then derived the values for X max and DH r (X max). DHr was calculated from Eq. (7) by inserting the measured values for DH r (0) and DH r (1) if available, otherwise we used the values from the polynomial fits. Fig. 3 displays DMAE vs. bd ion/bd network for Na –Rb mixed-alkali borate and alumino-germanate glasses. We observe the following remarkable features: (i) The values of DMAE as a function of bd ion/bd network are almost independent of the glass system. (ii) For all systems DMAE decreases with increasing bd ion/ bd network and vanishes at a value of bd ion/bd networkå 1.8. This suggests also for the Na – Rb borate glasses that no mixed-alkali effect will be observed when bd ion/bd network exceeds a value of about 1.8. The similarity of the values for DMAE as a function of bd ion/bd network independent of the network indicates that the ions themselves play an important role for the mixedalkali effect. Some insight into the reasons of such a universal behaviour of the activation enthalpy DH r comes from the Haven ratio discussed below. 3.3. Haven ratio The Haven ratio H R quantifies the ratio between the tracer diffusion coefficient, D*, and the conductivity borates A-series B-series C-series
0. 6
∆MAE
1. 0
0. 4 0. 2
0 1. 0
1. 4
1. 8
2. 2
0. 2 0 0
2
4
6
8
&
diffusion coefficient, D r . For single-alkali glasses D r can be calculated from the measured dc conductivity, adc, via the Nernst –Einstein equation. The Haven ratio is then defined as HR ¼
D4 : Dr
ð8Þ
D* and D r are not identical due to correlation effects and/or due to the collective nature of the jump process. We talk about collective processes when two or more ions move in one jump event. For collective jump events, the displacement of a tracer ion is smaller than the displacement of the charge itself. Kelly et al. [14] have given an overview of the Haven ratio as a function of total alkali content Y for alkali borate, germanate, and silicate glasses. In each system, they reported a similar decrease of H R with increasing Y. To reveal a universal behavior of the Haven ratio for different glass systems, we plotted H R in Fig. 4 as a function of bd Na/bd network for the Na-borate and Naalumino-germanate glasses. Furthermore, we include data compiled by Kelly et al. [14] for H R in Na-borate and Na-alumino-germanate glasses. From Figs. 2 and 3 of reference [14], we read out the H R ( Y) data electronically. Using Eqs. (4) and (5), these literature data on the Haven ratio can be replotted vs. bd ion/bd network Fig. 4 reveals the following features: – For values of bd Na/bd network larger than 4, the Haven ratio is close to unity. This value suggests an uncorrelated motion of single ions. For a crystalline ionconducting material, a Haven ratio of unity gives evidence for a direct interstitial diffusion mechanism. – It is an open question what the lowest value for the Haven ratio is. From Fig. 4 it may be anticipated that the Haven ratio tends to zero with decreasing bd Na/ bd network. However, values of bd Na/bd network close to zero are physically unreasonable. A Haven ratio of
S. Voss et al. / Solid State Ionics 176 (2005) 1383 – 1391
about 0.25 is reported for bd Na/bd networkå1 (Fig. 4). Small values of H R are expected for collective jump processes of the ions. – The majority of experimental studies have been performed for bd Na/bd network between 1 and 4. In this composition region H R increases approximately linearly with increasing bd Na/bd network. A linear fit to the experimental data in Fig. 4 results in the following dependence: HR , 0:23
bdNa þ 0:03: bdnetwork
ð9Þ
In a recent study, some of us (S.V., A.I. and H.M. [9]) measured the Haven ratio in a Rb-borate glass for various temperatures. The temperature dependence of H R can be described by the Arrhenius relation H R = 20.9 exp(20.8 kJmol1/RT) within the measured temperature range 505 to 650 K. H R decreases with decreasing temperature and at 505 K a value of H R = 0.15 is found. In contrast, no temperature dependence of H R is observed in the corresponding Na-borate glass with H Rå0.459. In what follows, we introduce the concept of subnetworks for the ions and a diffusion mechanism which can explain the experimental finding. This will be supported by Monte Carlo simulations.
& The incorporation of Na2O (or Rb2O) in pure borate glass leads to an alkali-borate glass in which some BO3 units are transformed into BO4 units. The associated positions for alkali ions are denoted as substitutional-like or S sites. In the composition interval under consideration ( Y < 0.25), the number of BO4 units almost coincides with the number of alkali ions for charge-neutrality reasons (glasses with Y < 0.25 are free of non-bridging oxygen). In a singlealkali glass only one type of S site is considered. In a mixed-alkali glass–not treated in detail in this paper– two types of site would have to be introduced. & An alkali ion normally resides on an S site. It can jump to a neighboring I site as a result of thermal activation, leaving the S site unoccupied. & The numbers of I or S sites are assumed to be proportional to the partial concentration of borate or alkali oxides, respectively. The ratio of the number of S sites to the total number of sites is related to the number density of alkali ions. A larger number of S sites corresponds to a higher number density of ions and thus to a smaller average ion –ion distance in the glass. & We assume that the I and S sites keep their character. Relaxation effects, i.e., site transformation from I to S or vice versa, are not taken into account. & Only jumps to neighbouring sites are considered: (i) single jumps of an ion from an S or I site to a neighboring unoccupied I or S site, Fig. 5a; (ii) collective jumps in which two or more ions move simultaneously, Fig. 5b. The following kinds of collective jumps are
4. Monte Carlo calculations of the Haven ratio The previous discussion has shown that the Haven ratio decreases with decreasing bd Na/bd network independently of the glass matrix. This suggests that the physical origin of this decrease can be largely attributed to the ion –ion and network separations. Different glasses have their own glass network, with local configurations depending on various parameters such as composition, thermal history, preparation method, etc. However, the universality of Fig. 4 suggests that long-range diffusion Fscreens-out_ these differences. In the following, we reproduce a decreasing Haven ratio with decreasing ion –ion distance by a simple model. We then compare the calculations with the experiments.
1387
_ BO4 Na+
a)
BO3
_ BO4
Na+
4.1. Description of the model Our model has been derived to describe the ion dynamics in borate glasses. It is based on the following assumptions: & Trigonal BO3 configurations are characteristic of the structure of pure borate glass B2O3. The packing of the BO3 units provides interstitial-like sites for alkali ions. We denote these positions as I sites. An alkali ion has a certain residence time at such a site.
b)
BO3
Fig. 5. Schematic representation of jump processes considered in the present model. The Na+ ions reside either on I sites (provided by BO3 units) or on S sites (provided by BO 4 units), shown by the small shaded circles and the large dashed circles, respectively. In panel (a), single jumps of the Na+ ion are illustrated. In panel (b), a collective jump event involving more than one Na+ ion is shown.
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taken into account: an Na ion at an I site and an Na ion at a neighbouring S site perform a collective (simultaneous) jump. As a result, the first Na ion occupies the S site now and the second Na ion resides at a neighbouring site (of type I or S). If the second Na ion has been displaced to an I site, the jump chain is finished. However, if this Na ion has landed at a neighbouring S site, the third Na ion participates in the collective jump event and its new position will be chosen in a similar manner as for the second Na ion, and so on. This chain of displacements within the collective jump event is considered until the last Na ion lands on an I site. Then the collective jump event is finished. We have denoted the I sites as Finterstitial-like_ sites. Alkali ion migration on the I subnetwork is similar to interstitial diffusion in crystals. On the other hand, if an ion leaves an S (Fsubstitutional-like_) site, a vacancy-like configuration is left (the charged BO4 configuration). Then an alkali ion should occupy this Fvacancy_. The appearance of collective jumps may be understood as follows: if a sodium ion is located at an S site, the total charge of the configuration is zero. However, the alkali ion is not necessarily located at the geometrical center of the negative charge. This configuration corresponds to an electrical dipole Na+ – BO4. If another Na+ ion occupies a neighbouring I site, the first Na+ ion may be additionally pushed from the center of the BO4 configuration, Fig. 5b. The electrostatic interaction with the neighbouring Na+ ion decreases the energy barrier for dissociation of the Na+ – BO4 dipole and its Na+ ion may leave the BO4 unit with a simultaneous jump of the second Na+ ion into this vacant BO4 unit. As a result, the two ions moved simultaneously within a collective jump event. Alternatively, the first Na ion may not only jump into an adjacent I site, but it may jump to a neighbouring occupied S site and activate a simultaneous jump of other alkali ion from this S site (see the last point in the above description of the model). This increases the number of ions participating in the collective jump event. The effectivity of collective jump events depends crucially on the Na – Na distance. The smaller the Na –Na separation, the larger is the effect of Coulomb repulsion on diffusion barriers for the collective jump (Fig. 5b). However, it is not only the ion – ion distance which affects the barriers. A smaller average network distance bd network corresponds to a larger number of I sites between the S sites. This causes two effects: (i) the I sites weaken the electrostatic interaction of ions between S sites; and (ii) the I sites hinder ion jumps towards the S sites. Both effects increase the barriers for collective ion motion. These features of the model qualitatively agree with the experimental observations that the activation enthalpy, DH r , is directly proportional to the average ion – ion distance, bd ion, and inversely proportional to the average network distance, bd network. The decrease of the energy barriers for collective jumps (Fig. 5b) increases their probability relative to single-ion
jumps (Fig. 5a). Thus the Fdegree of collectivity_ of the ion dynamics increases with decreasing bd ion/bd network. As a result, the Haven ratio decreases with increasing ion density and decreasing ion – ion separation. This is the trend which is experimentally observed in Fig. 4. The present model has features similar to the one introduced by Habasaki and Hiwatari [15]. Our model is more specific for the borate glasses and is extended by considering jump events which include several ions (collective motion of even four or five ions were observed, see below). Moreover, the simulation in reference [16] was carried out on a cubic lattice, whereas a random 3D network for diffusion jumps is generated in the present study. This is indeed crucial for reproducing the concentration dependence of the Haven ratio observed experimentally (see below). The different types of sites and ion jumps are common features of the present approach and the dynamic structure model [16]. However, site relaxation effects needed not be included in the present model. 4.2. Construction of a network for ionic motion A random glass-like network is constructed as follows. A cubic simulation box of size 40 40 40 (in arbitrary units) is chosen with periodic boundary conditions. This box is randomly filled with hard spheres of radius 1 (in the same units). A packing density of about 33% is reached. These positions are marked as I sites and they represent the BO3 units. A connectivity graph is constructed from neighbours of each I site. A site I * is considered as a neighbour for site I, if the distance I * –I is smaller than a chosen maximum value, d max, and if I * is not strongly overshaded by other sites. The value d max should be chosen reasonably large in order to sample a complete neighbourhood of each site.2 A certain amount of sites is then randomly transformed into S sites corresponding to the given concentration of the alkali ions in the glass. Our aim is the calculation of the Haven ratio. Only the ratio between the tracer diffusivity, D*, and the conductivity diffusion coefficient, D r , is of importance, but not the individual absolute values. For simplicity, we assume that all attempt frequencies are equal. Then the difference of the energy barriers between collective and single-ion jumps, DE, is the relevant parameter in the present model. The ratio of jump rates of single-ion to collective jumps is given by exp(DE/kT). Here k is the Boltzmann constant and T the absolute temperature. The simulation has been performed for various values of DE: DE = 0 (single jumps are preferred); DE = 0 (both types of jumps are equally probable); and DE < 0 (collective jumps are more probable).
2 A variation of d max from 3 to 4 resulted in marginal changes of the connectivity graph obtained. Note that d max = 2 is the shortest distance between centers of hard spheres with unity radius.
S. Voss et al. / Solid State Ionics 176 (2005) 1383 – 1391
If we simply start a kinetic Monte Carlo simulation of the ion dynamics within the constructed network, the following process is very often observed: a reverse jump of an ion occurs after its initial jump from an S site. Only occasionally an ion may leave its S site and initiate a collective jump event. In order to enhance the productivity of the simulation procedure, a modified approach is applied. An additional ion is introduced in the simulation block at an I site and a standard Monte Carlo procedure with the residence time algorithm [17,18] is applied. This additional ion can either perform a jump between I sites (Fig. 5a) or initiate a collective jump sequence (Fig. 5b) with the relative probabilities determined by exp(DE/kT). In the latter case, this ion will finally reside at an S site and a new ion will occupy an I site. At any moment, only one noncompensated charge (free alkali ion) is available in the system. Jump rates m l , l = 1, . . .z i, for the particular ion at the ith I site are calculated for any possible jump direction l (z i is the number of neighbours for the site i). For simplicity, the jump rate from the I site to an S site (this jump initiates a collective jump event) is taken as unity. The jump rate between I sites is then given by exp(DE/kT). A random number is generated and the jump direction l (including possible collective jump modes) is chosen according to the i probabilities Pl ; Pl ¼ Pl ¼ ml =~Zj¼1 mj . After the completed jump event, the coordinates of participating ions are updated and the time variable is incremented according to the residence time algorithm by the value s: s¼
1 Z1 X
I
ð10Þ
ml
l¼1
After a fixed number of jump events (usually 105 to 106), the tracer diffusivity, D*, and the conductivity diffusion coefficient, D r , are calculated via D4 ¼
s þ1 1 NX R2 6t m¼1 m
ð11Þ
R2 : 6t
ð12Þ
and Dr ¼
In Eq. (11), the summation is performed over all ions m which have moved the distance R m during the simulation time t. Note that there are N S ions at the S sites and one extra ion at an I position. The conductivity diffusion coefficient, Eq. (12), is defined by the total displacement R of the non-compensated charge during the simulation time t. According to the present model, only one ion on the I site subnetwork is available during the whole simulation time, which produces the non-compensated charge.
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For each alkali ion concentration 102 to 103 different glassy configurations were generated. For a given glassy configuration, the random walk was repeated 105 to 106 times and the results were averaged over 103 to 104 different distributions of ions. 4.3. Simulation results Simulations were performed for the values DE/kT = 6, 0, 3, and 6. These values correspond to the following ratios between the probabilities of single and collective jumps: 1:0.003, 1:1, 0.05:1, and 0.003: 1, respectively. The results as a function of the parameter bd ion/bd network for a glassy network are presented in Fig. 6a. Comparison of experimental data (Fig. 4) and simulation (Fig. 6a) suggests that our model can describe the composition dependence of the Haven ratio in single-alkali glasses, if we assume that collective jumps dominate (DE < 0). The specific value of DE is a fitting parameter of the present model. The calculations carried out at DE/ kT = 3 result in a composition dependence of the Haven ratio (Fig. 6a), which is in good agreement with experiments shown in (Fig. 4). At small ion concentrations, corresponding to bd ion/ bd network > 4, single-ion jumps prevail and the Haven ratio is about unity. Each jump of an ion equally contributes to the conductivity and to the tracer long-range diffusion, and therefore D r åD*. As the alkali content increases and the average ion – ion separation decreases, more alkali ions appear at neighbouring sites of a given alkali ion. Then collective jumps become more probable. The chains of sequential ion displacements result in a large total displacement of charge, whereas the individual displacements of ions are smaller. As a consequence, the Haven ratio decreases. The larger the contribution of collective jumps, the smaller is the Haven ratio. The average number of ions participating in a jump event gradually increases from unity at bd ion/bd network > 4 to about 2.1 at bd ion/bd networkå2. In the simulation, sometimes jump events were observed with four and more participating ions. One of the most striking features of Fig. 6a is that for DE/ kT = 3, H R decreases almost linearly with decreasing bd ion/bd network. This is indicated by the dashed line in Fig. 6a at 1 < bd ion/bd network < 4. In order to elucidate the effect of a glassy (random) network on ion motion, we have performed an analogous calculation of the Haven ratio for a regular FCC lattice. The results are presented in Fig. 6b. Qualitatively a similar behaviour is observed for the Haven ratio in random and FCC lattices. In the case of an FCC lattice, however, a more moderate decrease of H R is seen as the value of bd ion/ bd network falls below 4. For the same energy barriers the Haven ratio for ion motion on a random lattice is consistently smaller than that for the FCC lattice. The
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∆E/kT = 6 1 ∆E/kT = 0
HR
0.8
0.6
∆E/kT = −6 ∆E/kT = −3
0.4
0.2
a) 0 0
4
2
6
8
∆E/kT = 6 1.0
∆E/kT = 0
HR
0.8
0.6
∆E/kT = −6
∆E/kT = −3
0.4
0.2
b) 0.0 0
4
2
6
8
difference becomes more pronounced when the ion – ion separation decreases.
5. Summary and conclusions Numerous experimental results on ionic conductivity and tracer diffusion in single and mixed Na – Rb aluminogermanate and borate glasses are analysed. Universal features of the ion dynamics in these glasses are elucidated. The activation enthalpy of conductivity is largely determined by the ratio of average distance between alkali ions, bd ion, and network – former atoms, bd network. The Haven ratio consistently decreases, if the ratio bd ion/bd network decreases.
A new model of ion diffusion in borate and aluminogermanate glasses is suggested with interstitial-like and substitutional-like ion sites. Single and collective diffusion mechanisms are considered. Monte Carlo calculations of the Haven ratio result in a good agreement with the experimental findings. The decrease of the Haven ratio with decreasing bd ion/bd network is attributed to an increase of collective jump events.
Acknowledgments One of us (J.N.M.) gratefully acknowledges the invitation of the Sonderforschungsbereich to spend several weeks in Munster as a visiting scientist. We are grateful to
S. Voss et al. / Solid State Ionics 176 (2005) 1383 – 1391
Prof. Dr. K. Funke and to Dr. C. Cramer for their helpful comments on the manuscript. Financial support of the DFG via Sonderforschungbereich 458 is acknowledged.
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