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A cooperative model for the mixed mobile ion effect in covalent glasses
Journal of Non-Crystalline Solids 241 (1998) 149±165
A cooperative model for the mixed mobile ion eect in covalent glasses Beatrix M. Schulz *, Manfred Dubiel, Michael Schulz Fachbereich Physik, Martin±Luther±Universit at Halle, 06099 Halle (Saale), Germany Received 2 October 1997; received in revised form 17 June 1998
Abstract A thermodynamical model of the long±standing problem of the mixed mobile ion eect (MMIE) or the mixed alkali eect, respectively, in glasses is presented. The basic idea of the model is an internal dynamic interaction between the cations and the dynamics of the cooperative regions in glasses. Two types of cation traps due to structural disorder in the glassy network are assumed, in which the deeper traps mainly determine the glass relaxation properties and the cations with a reduced binding energy are responsible for the movement of ions. The modi®cation of the usual diusion coecient of cations has been calculated under these considerations. That leads to a variation of dc-conductivity in dependence on the composition ratio, f , of the dierent cations introduced into a covalent network, i.e. the occurrence of the MMIE. The main result is a relation between the dc conductivity and temperature dependent relaxation time that is controlled by f . Furthermore, the conductivity properties of ion exchanged glasses can be predicted quantitatively if the exchange of ions take place either above or below the glass transition temperature. Ó 1998 Elsevier Science B.V. All rights reserved. PACS: 66.30.Dn; 72.80.Ng; 81.05 Kf
1. Introduction A large number of liquids have been observed to develop slower dynamics without the evolution of any observable long range correlated or ordered structure. This glass transition process shows a cooperativity [1] and can be described as a typical nonlocal process. On the other hand, the structure of glasses at the atomic level is relatively complex. It can be expected that both, local structure and nonlocal cooperative dynamics determine the ionic conduction in glasses and supercooled liquids. For example, the silicate glass system being the overwhelming application of glass products consists of randomly arranged SiO4 tetrahedrons. The network forms an in®nite large cluster of chemical bonds between these structural elements. Below the glass transition temperature, Tg , this random structure is almost ®xed, i.e. most of the chemical bonds are stable over the total observation time. The glass dynamics consist in the destruction of bonds, cooperative motion of the silicate network and the formation of new bonds.
*
Corresponding author. Tel.: +49-345 5527159; e-mail: [email protected].
0022-3093/98/$ ± see front matter Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 8 ) 0 0 7 6 1 - 3
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The introduction of alkali ions into a pure SiO2 glass leads to an increased electrical conductivity. Additionally, if the glass contains two types of cations simultaneously, e.g. Na and K , the conductivity has a non-linear dependence on the composition ratio of the cations [2,3]. This eect is denoted as mixed alkali eect, MAE, or more generally as mixed mobile ion eect, (MMIE). The decrease of conductivity is of the order about of 103 ±104 [4]. This decrease is in contradiction to the electrical conductivity expected from superposition of the single conductivities. The MMIE represents a property of ion-conducting disordered systems such as glasses. To explain this unusual eect in glasses a variety of models was developed. For a review see Day [2] or Ingram [3,5]. Till now, however, there exists no theoretical framework considering a nonlocal dynamical interaction between the alkali cations via cooperative rearrangements of the glass matrix, to our knowledge. Furthermore, the consideration of ion-exchanged glasses in the discussion of MMIE involves additional complications. Some measurements of the ionic conductivity or the diusion coecients in ion-exchanged silicate glasses indicate the presence of a reduced mobility of cations as measured for glasses produced by melting procedures, whereas other experiments do not show such eects (see for example Ref. [6]). Recently, the uni®ed site relaxation model (USRM) has been proposed by Bunde, Funke and Ingram [6,7]. The authors combined the relaxation of the Coulomb ®eld and the site relaxation during ion hopping processes to explain the frequency dependence of the ionic conductivity as well as eects such as the MMIE. According to this model the ions move to neighboring sites, if appropriate target sites exist. On the other hand, successful ion jumps require localized structural relaxations to match the environment for the speci®c cation. With respect to the ion-exchanged glasses, the authors predict the MMIE in a homogeneous mixture of dierent cations, whereas the eect disappears in a completely inhomogeneous mixture. In general, the cation mobility is reduced whenever two dierent cations are able to create their speci®c environment by local processes. A speci®c environment, the so-called site, is characterized by typical bond lengths and coordination numbers of surrounding oxygen atoms. These assumptions suggest a remarkable importance of glass dynamics below as well as above the glass transition temperature. The aim of the present work is to show a new phenomenological explanation of the mobility in mixed cation glasses by considering cooperative rearrangements of the introduced cations owing to their interactions with the covalent glassy network. The USRM is a typical local theory based on structural relaxations. In comparison to this model, our new theory is based on an eective dynamical interaction of cations situated in a volume of the order of magnitude of a cooperative region. This interaction will be calculated under the assumption of the existence of strongly and weakly trapped cations, i.e. the existence of two types of cation binding energies resulting from the disordered structure of glasses. The strongly trapped or relatively immobile cations should modify the relaxation process of the glass matrix, i.e. this type of cations has an aect on the mobility of all cations inside the glass matrix. This eect is described in terms of a conductivity relaxation time by which the change of the cation mobility and of the electrical conductivity can be explained. The results are discussed with respect to the aect of cation composition on the conductivity over a range cation concentrations in the temperature range below and above the glass transition temperature, Tg . Additionally, the introduction of cations into the glass network by an ion exchange will be considered. 2. Two-site model of cation bonds We restrict our investigations to glasses which form a network of covalent bonds. Typical materials are SiO2 , B2 O3 and GeO2 . Some polymers and polymer networks are also candidates. If such glasses are modi®ed by addition of salts or oxides, then the following scenario can be observed. Since all anions or oxygen atoms are tied to the network of covalent bonds, only the cations, i.e. alkali ions as Na or K ; remain as most mobile species. In this network the cations behave like ions in a potential landscape
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including traps of various depths. A large spectrum of local potential minima exist in the glass where cations can be trapped. We assume that there are only two types of potential minima: deep traps and shallow traps. Of course, this is a very simpli®ed assumption. In reality one has to consider a smooth distribution function with respect to the potential depths. The assumption of deep and shallow traps, however, is an approximation of the real distribution in disordered materials by a suciently simple discrete distribution. Because of the Coulombic attraction cation sites in the very neighborhood of negatively charged anions, e.g. of nonbridging oxygens, are thought to be deep traps in covalent networks. If this interaction is reduced, e.g. by neighboring cations or defects within the glass network, weaker traps are formed. For the following discussion we assume that strongly trapped cations determine the dynamics of the glass matrix. Obviously, a glass with two kinds of cation bonds has concentrations, cA and cB , of A and B components, respectively, or by the total concentration c cA cB and the composition ratio of cations f cA = cA cB . We introduce further quantities, such as the concentration of deep traps, Ud , the total concentration of strongly trapped cations, cH (with cH 6 Ud ), and the composition ratio, f H , of the strongly trapped cations. The concentration of deep traps depends on the structure of the glassy network which is mainly determined by the distribution of defects, e.g. of nonbridging oxygen atoms and other anions. Thus Ud implicitly becomes a function of the total cation concentration c and partially of the composition ratio f , i.e. Ud Ud c; f . In general, there is a monotonic relation between the total concentration of cations, c, and the concentration of the deep traps, Ud , i.e. the number of deep traps increases with increasing cation content. On the other hand, we assume that the dependence on the composition ratio, f , is weaker. Therefore, in the following discussion we will use the sucient assumption Ud Ud c, with the extension to the general case Ud Ud c; f being always possible. In general, the binding energy of a trapped cation depends also on the type of the cation, i.e. there are four energy levels: EA and EB ; the energies of the weakly trapped cations, and EAH and EBH , the energy levels of strongly trapped cations. Consequently, the concentration of the relatively immobile cations in deep traps is a function of the temperature, T , of the concentration of deep traps, Ud , and both cation contents, hence cH cH T ; Ud ; cA ; cB cH T ; Ud c; cf ; c 1 ÿ f or cH cH T ; c; f . The composition ratio f H of the cations in the deep traps is controlled by this parameters too, i.e. f H f H T ; c; f . An interesting case is given if the binding energies do not depend on the cation type EA EB and EAH EBH : Mostly, this speci®c borderline case is only an approximation for real glasses, but it will be very helpful for ®rst calculations. With that, we obtaine cH cH T ; Ud ; c cH T ; c and f H f . We will use this approximation in the further discussion (see below). Finally, we introduce the concentration of the weakly trapped cations, cmob c ÿ cH , and the corresponding composition ratio, fmob , with fmob f for the above mentioned speci®c borderline case. Considering the bonding of the two kinds of cations de®ned above, it is reasonable to identify the weakly trapped ions with the most mobile species because of the their smaller activation energies. On the other hand, the deep traps are assumed to be relatively immobile. 3. Calculation of relaxations time, diusion coecient and conductivity The fraction of cations occupying deep or shallow traps becomes a well de®ned quantity in the thermodynamical limit, i.e. thermodynamical ¯uctuations of cation concentrations are asymptotically irrelevant observables. Each cation occupies a trap only for a randomly distributed ®nite time and it moves to another trap during a shorter hopping time, i.e. the fraction of cations outside the traps may be neglected. The probability of a jump and, consequently the trapping time, is determined by the depth of the trap and the local mobility of the surrounding glass matrix. During a sucient time, however, each ion visits both kinds of traps. Thus, the average individual diusion coecients of a given kind of cation get equivalent
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values for sucient observation times, i.e. these coecients are weighted averages over the corresponding hopping rates for deep and weak traps [8]. On the other hand, conductivity is a collective property of all cations. In the thermodynamical equilibrium (principle of detailed balance) the number of cations escaping from deep traps is equivalent to the number of cations occupying such traps. Although the individual energetic situation of each cation is permanently changed, the collective state always remains conserved. Each cation leaving a fraction (i.e. the fraction of deeply trapped cations or weakly trapped cation, respectively) is replaced by another cation entering this fraction. Thus, the whole system can be described by well de®ned fractions of always strongly trapped cations and fractions of always weakly trapped cations, respectively. From a statistical point of view this consideration corresponds to the principle of exchangeable particles. Therefore the conductivity can be described by separate fractions of relatively mobile cations and of relatively immobile cations, respectively. Because of the energy dierence DEc EcH ÿ Ec between cations in deep and in shallow traps, whereas the index, c A; B, denotes the kind of cations, there is the relation Dc Dcmob Dimmob eDEc =kB T ;
1
between the diusion coecients of the mobile and immobile fractions of each type of cations. The conductivity of each component is given by the well known Nernst±Einstein relation composed of the contributions of the mobile and the immobile fractions q2 Dc q2c c c c mob c H ÿDEc =kB T Dmob cmob Dimmob cc cc cH ÿ 1 with c A; B 2 rc c e kB T kB T in which qc is the electrical charge of the cations. The Haven ratio is assumed to be unity to simplify the calculations. Usually, this ratio decreases with increasing concentration of cation from 1:0 to 0.2. The diusion coecients depend on the hopping ions and the mobility of the covalent network. Therefore, the diusion coecients, Dcmob , can be written as a product of two parts Dcmob Lc sÿ1 :
3
c
The ®rst factor, L , contains individual properties of the cation type c A, B, e.g. the mass of the cations, trapping energies, Ec , and possible interactions with the local environment of the cation. The second factor, sÿ1 , is mainly determined by relaxation of the glass matrix and depends on temperature and alkali content. This factor de®nes the true time scale, s, of the diusion. Therefore, s can be interpreted as the averaged conductivity relaxation time. As assumed above, the incorporation of relatively immobile cations into the glass matrix leads to a modi®cation of the covalent network. This modi®cation must be considered for calculating the averaged relaxation time, s. It can be expected that s depends on cH and f H . Obviously, a jump of a cation from one trap into another is hindered by the surrounding network acting like a cage. The averaged time in which an ion leaves its cage depends on the size of the corresponding cooperative region. In other words, a number, N , of atoms of the covalent network must move over a small distance, before the elementary hopping process of the captured cation can be realized. The modi®cation of the covalent network by the incorporation of the strongly trapped cations leads to a relation N N T ; cH ; f H . Following the arguments of Adam and Gibbs [1] (see also [9]) the conductivity relaxation time can be written as ÿ sc sc;1 exp ac N : 4 Here, a is an undetermined factor which is ®nally used as a ®t parameter. sc;1 is de®ned by a natural time scale which is given by sc;1 sc T ! 1. Using Eqs. (3) and (4) the conductivity of glass including cations is determined by ÿ 2 ÿ ÿ exp ÿ ac N H H ÿDEA =kB T 2 H H ÿDEB =kB T : 5 ÿ 1 qB LB c c 1 ÿ f e ÿ 1 r qA LA cf c f e sc;1 kB T
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Here LA and LB are temperature dependent quantities which consider in particular the trapping energies, EA and EB , respectively. For a further discussion of Eq. (5) we need an explicit relation N N T ; cH ; f H . The size of a cooperative region of a pure covalent network depends on temperature [1]. At temperatures suciently above the Vogel temperature, T1 [10], the size of cooperative regions can be described by [9] N 0 T
h0 : T ÿ T1
6
Obviously, this is an empirical representation which fails for T ! T1 . Near T1 other divergenceless approaches, e.g. ln s N 0 T T ÿ2 [11], should be used. Furthermore, it can be expected that close to and below T1 a hopping regime remains eective [12] which leads to an Arrhenius-like dependence of the relaxations time with a larger activation energy, e.g. ln s N 0 T T ÿ1 . A large part of ionically conducting glasses (based for example on SiO2 or GeO2 ) showing the MMIE belong to the strong glasses [13], i.e. the corresponding Vogel temperature is small or vanishes T1 ! 0). Therefore, the size of the cooperatively rearranging regions of strong glasses is approximately N 0 T ' T ÿ1 . Thus, Eq. (6) at temperatures suciently above T1 (fragile glasses) or over the whole temperature range (strong glasses, T1 ! 0) can be used as a reasonable approach. The detailed relation between N 0 and the temperature, however, is irrelevant for the following investigations. The modi®cation of the network by an incorporation of two kinds of cations (A,B) changes the size of the cooperative regions. We separate the volume of the glass into subvolumes, indicated by the index, I, of size, NI , of a cooperative region. Thus, one has the condition X NI Ntotal ; 7 I
where Ntotal is the number of all atoms of the glass. If the subvolume, I, contains NIA immobile cations of type A and NIB cations of type B, the size of the cooperative region is less changed as compared to a region without immobile cations. The consideration of only linear corrections leads to NI N 0 T g
MI X ÿ ÿ ni hMI N 0 T g NIA ÿ NIB h NIA NIB :
8
i1
From the present point of view g and h are empirical factors which should depend on the the underlying covalent network. In particular, the dierence, NIA ÿ NIB , denotes the eect of mass dierences and speci®c volume dierences on the mobility of the modi®ed glass. On the other hand, the cations inside a cooperative region can be assumed to be randomly distributed. The actual relaxation time of a cooperative region I is given by sc;I sc;1 exp ac NI : The time scale sc follows from the average over all possible distributions of A and B ÿ sc sc;1 exp ac N hsI i sc;1 h exp ac NI i:
9 10
Therefore, the averaged number of cations per cooperative region is given by N
1 ln h exp ac NI i: ac
11
This quantity can be obtained by a simple algebraic calculation, see Appendix B. We obtain ÿ H ac H U d ÿ c H 2 ÿ N H 1 g 2f ÿ 1 h c c g 2f H ÿ 1 h 2ac g2 cH f H 1 ÿ f H 2 Ud N 0 T 2 cH2 g 2f H ÿ1 h :
12
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The present relation allows the calculation of the relaxation time sc . For a better representation we determine the averaged number of particles of a cooperative region for only one component. The case of a pure component B f H 0) leads to ac cH Ud ÿ cH 2 2 H 0 H H2 13 N B c ; T N T 1 c h ÿ g h ÿ g c h ÿ g ; 2 Ud where the pure component A f H 1) corresponds to ac cH Ud ÿ cH 2 2 H 0 H H2 N A c ; T N T 1 c h g g h c g h : 2 Ud
14
Hence, the averaged number of ions per cooperative region for an available composition ratio becomes ÿ ÿ ac N N B cH ; T 1 ÿ f H N A cH ; T f H 2N 0 T ÿ 2 cH2 g2 f H 1 ÿ f H : 15 Ud Finally we obtain the averaged relaxation time s H H2 H ÿ H 1ÿf H s0 T lc f 1ÿf fH H sc sA c ; T sB c ; T sc;1
16
with the coecient l 2 ac =Ud ÿ 2g2 and the relaxation times of one component glasses sc cH ; T c A,B) given by ÿ ÿ sA cH ; T sc;1 exp ac N A and sB cH ; T sc;1 exp ac N B : 17 The third relaxation time, s0 T , of Eq. (16) corresponds to the extrapolated case of a covalent network with the same structure and distribution of deep and shallow traps as the underlying glass without any immobile cations. Note that this virtual relaxation time can be extrapolated only from experimental data, since an empty covalent network with the same structure as a network containing a ®nite fraction of cations is always unstable and experimentally unapproachable. s0 T , however, is never identical with the relaxation time of the pure covalent glass, e.g. amorphous SiO2 . It should be remarked also that the material dependent factor, ac , is a speci®c quantity for the investigated relaxation process. Another process in the same glass, e.g. the structural relaxation, has the same average size of cooperative regions, N , but a dierent factor, as . Therefore, dierent processes in the same glass show dierent relaxation times, e.g. structural relaxations s s exp a N , whereas the conductivity relaxation time is given by are described by s;1 s s sc sc;1 exp ac N . This dependence corresponds to the general assumption that the underlying dynamics of the glass are determined by the size of the cooperatively rearranging regions and is approximately universal whereas physical observables, e.g. electrical conductivity, dynamical structure factor, dielectric moduli, are coupled to this dynamic process by dierent strengths, e.g. as , ac . Usually, one obtains as > ac . Thus the decoupling index [14], Rs T ln ss =sc ln ss =sc as ÿ ac N , increases with decreasing temperatures. Rs T is small or nearly zero at high temperatures, i.e. structural and conductivity relaxation seem to be coupled at the same process. On the other hand, at low temperatures T 6 Tg ) the decoupling index becomes larger and the conductivity relaxation is apparently decoupled from the structural relaxation. It follows that a change of mechanical properties occurs close to or below Tg , whereas the conductivity shows only a smaller rate of decrease with decreasing temperature. Additionally, it should be remarked that sometimes both ac and as show a dependence on temperature. The substitution of Eq. (16) into Eq. (5) leads to the electrical conductivity ÿ ÿ 1 q2A LA cf cH f H eÿDEA =kB T ÿ 1 q2B LB c 1 ÿ f cH 1 ÿ f H eÿDEB =kB T ÿ 1 : 18 r fH 1ÿf H lcH2 f H 1ÿf H kB T s0 T =sc;1 sA cH ; T sB cH ; T
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Two additional remarks are necessary: At ®rst, the expansion to the ®rst order of cumulants leads to the mean ®eld approximation, hsI i sc;1 h exp ac NI i sc;1 exp ac hNI i. This approximation neglects all ¯uctuations of the composition ratio and the concentration. Thus, the non-linear exponent, lcH2 f H 1 ÿ f H , in Eq. (16) results from the deviation of the cation number from their average inside a cooperative region. Secondly, the result (16) can be obtained also by a phenomenological expansion of the averaged number of particles per cooperative region N N T ; cH ; f H in powers of the composition ratio up to the second order, i.e. N N 0 1 # 1 f H # 2 f H2 . The substitution of N in Eq. (4) by this power expansion yields an analogous dependence as Eq. (16), only the dependence on the concentration is unde®ned. 4. Discussion 4.1. Distribution of cation bonds The assumption of mobile and immobile cations, i.e. the existence of two dierent cation bonds, is similar to the model of ionic transport in silicate glasses by Greaves and Ngai [15]. These authors interpret the decrease of the macroscopic activation energy as well as binding energy with increasing alkali content by means of network hopping and intra-channel hopping of cations. For dilute concentrations the binding energy becomes equal to the Coulombic binding energy of an isolated alkali. The Coulombic interaction between cations and counter ions is to be interpreted in terms of deep traps following our ideas. Since the fraction of deeply trapped ions is determined by binding energies and corresponding Boltzmann factors, this fraction concerns the largest fraction of the cations. On the other hand, at higher alkali concentrations non-bridging oxygens channels are formed and the migration of cations is mainly controlled by the intrachannel hopping mechanism [15]. The channel sites can be considered as shallow traps. Hopping processes between these shallow traps, i.e. the motion along the channels, give the main contribution to the conductivity. It should be remarked, however, that the similarity of our model with the model of Greaves and Ngai is limited. The present model needs only the existence of deep and shallow traps, and speci®c structural properties, e.g. channels due to microsegregation of alkali, are not required. Therefore, our model can be generally applied to ionically conducting glasses, and should explain also the conductivity of borate or germanate glasses containing dierent cations. For these glasses a decreasing activation energy of the dc-ionic conductivity always was found with increasing fraction of alkali [16,17]. This dependence on the cation concentration supports the present assumptions, although its interpretation diers from that applied to silicates [18]. These authors calculate the activation energies for conductivity of Rb2 O±GeO2 glasses from data of far-infrared spectroscopy measurements as well as on the basis of an `unoccupied volume'. The long-range alkali motion is correlated to alkali ions occupying high-frequency IR active sites. An additional low frequency band, which was also assigned to vibrations of metal ions, gives experimental evidence for the presence of immobile cations as proposed by our thermodynamical model. Moreover, the presence of two IR bands which can be assigned to cation vibrations could be observed for many other glass systems as, e.g., silicates and borates (see Ref. [19]). However, no direct relationship of IR vibration frequencies with the binding energy of corresponding alkali ions is given. Especially, the sources of the low frequency vibrations are not completely known. Therefore, further investigations should try to identify dierent cations species which are responsible for the dynamical properties to examine this model. 4.2. Dependence of conductivity on cation concentration The qualitative dependence of conductivity on the concentration shall be discussed by assuming a covalent network with only one type of cation (e.g. A), i.e. we choose f f H 1. The conductivity then becomes
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ÿ q2A LA c ÿ cH cH eÿDEA =kB T : r kB T sA cH ; T
19
For the further discussion of Eq. (19) we need an approximate relation, cH cH c; T . This relation is usually obtained from experimental measurements. Here we will use an approximation leading to an almost qualitative result. A more detailed analysis requires better knowledge of cH , if possible, determined by direct experimental measurements. We assume that the covalent network is insensitive against changes of the cation content, i.e. the concentrations of deep and shallow traps do not depend on the cation concentration. Obviously, this is a very strong restriction, since it is well known that the number of traps increases continuously with increasing alkali content. Each addition of alkali cations implies simultaneously the incorporation of anions, e.g. nonbridging oxygens, into the covalent network. Thus, the physical properties of the network vary because of the changed cation concentration and it can be expected that the quantities describing the network, e.g. Ud and DEA , also change with alkali content. For a qualitative discussion, however, this assumption should be sucient. According to this consideration the concentration of deeply trapped cations is determined by U d ÿ c Ud ÿ c exp c Utot ÿ Ud ÿ c c
ÿ
DEA kB T
20
(see Appendix A), where Utot is the total concentration of traps in the modi®ed glass. From Eq. (20) a saturation at suciently high concentration c follows, i.e. limc!1 c c; T Ud . The opposite case c ! 0), however, can be described by the relation limc!0 c c; T c= 1 exp fÿDEA =kB T g, i.e. the main part of cations will be strongly trapped at suciently low concentrations. Eq. (19) is non-linear for smaller cation concentrations c>Ud ) r
q2A LA eÿDEA =kB T c ; ÿDE =k T A B kB T 1 e sA c; T
21
whereas at much larger concentrations a more linear dependence dominates: ÿ 1 q2A LA c ÿ Ud Ud eÿDEA =kB T r kB T sA Ud ; T
:
22
Note that this linear dependence is a result of the approximation applied. For real glasses we expect a non-linear dependence of the conductivity at smaller concentrations whereas the dependence at larger concentrations is less. Fig. 1 shows the electrical conductivity versus the concentration for the case of large energy dierences, DEA . This dependence is observed qualitatively for various materials [16]. In particular, the variation of the conductivity or the corresponding eective diusion coecient is a property of a glass network at smaller cation concentrations. SiO2 glasses, for example, show a nonlinear dependence on alkali content for suciently low (in the ppm range) concentrations of Na2 O [20,16], whereas at increased concentrations above (5 mol% Na2 O) an approximately linear behavior is observed. It should be remarked again, however, that the discussion of the concentration dependence is only of qualitative nature. As mentioned above, a thorough analysis is possible if the relation c c c; T is known in detail. Nevertheless, the qualitative behavior discussed above should be eective also for real glasses. Furthermore, the dependence on the composition ratio at a ®xed concentration discussed in the following is not in¯uenced by this restriction.
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Fig. 1. Qualitative behavior of the reduced conductivity rR r c=r c0 of a glass matrix modi®ed by one type of cations versus the reduced concentration x c=c0 c0 is the maximum allowed concentration of cations in the glass matrix). The reduced diusion coecient DR is de®ned by DR rR =x. The example is calculated by using the parameters aN0 T 80; a=c0 0:5, g hc0 ÿ0:5; c1 =c0 0:5 and exp ÿDE=kB T 0:005.
4.3. Mixed alkali glasses 4.3.1. Concentration of cations < Ud Here we will analyze the conductivity dependence on the composition ratio, f, for a low concentration of cations, i.e. the predominant part of cations is localized in deep traps. Using the above introduced speci®c borderline case EAH EBH EH and EA EB EH ÿ DE), the concentration of relatively immobile cations is given approximately by cH c= 1 exp fÿDE=kB T g and the composition ratio is de®ned by f H f . The exponent of s0 T =sc;1 (see Eq. (18)) can be neglected on condition that the cation content is suciently low, i.e. lc2 ! 0. We obtain 2e2 c LA f LB 1 ÿ f r : 23 kB T 1 eDE=kB T sA c; T f sB c; T 1ÿf
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Note that we have used the simpli®cation qA qB e. The conductivity shows an extremum at the composition ratio, fextr: : ÿ1 ÿ1 LA sB c; T ÿ ln 24 fextr: 1 ÿ LB sA c; T with the restriction that 0 6 fextr: 6 1. If this condition is not ful®lled, the conductivity increases or decreases monotonoicly with decreasing composition ratio over whole the range fextr: 2 0; 1. Fig. 2 shows the characteristic dependence on temperature and composition ratio for suciently small concentrations of cations. At suciently high temperatures a simple linear relation occurs. Only the low temperature regime (near the Vogel temperature, T1 < T < 1.2T1 ) shows a deviation from this dependence. It is evident that a MMIE for glasses with lower concentrations of cations cannot be observed. This result is in agreement with various experimental observations [21]. 4.3.2. Concentration of cations > Ud a. High temperature regime: Another case is given for high alkali content. It is well known [2,3] that the conductivity has a non-linear dependence on the composition ratio, f . Again we use the reasonable assumption EAH EBH EH and EA EB EH ÿ DE, which now corresponds to cH Ud and f H f . This simpli®cation leads to the conductivity ÿ e2 c Ud eÿDE=kB T ÿ 1 LA f LB 1 ÿ f : 25 r 2 f kB T sA Ud ; T sB Ud ; T 1ÿf s0 T =sc;1 lUd f 1ÿf
Fig. 2. Reduced electrical dc±conductivity r T =rB T versus the composition ratio f for a low cation concentration c. The used parameters are D0A =D0B 10, h 0 and h0 cH g=T1 0:1. The reduced temperature T 0 is de®ned by T 0 T =T1 . The dotted line corresponds to the case T 0 ! 1.
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159
Fig. 3. Reduced electrical dc-conductivity r T =rB T versus the composition ratio f for a high cation concentrations c. The used parameters are D0A =D0B 1, h 0 and h0 cH g=T1 0:1 and h0 a=c1 ÿ 2cH2 g2 =T1 8. The reduced temperature T 0 is given by T 0 T =T1 .
There is a non-linear dependence on f with a minimum, as shown in Fig. 3. As an example we consider the dependence on the composition ratio of the dc-conductivity of the glass system (K2 O)f (Na2 O) 1±f (SiO2 )3 obtained by usual melting procedures. The conductivity data measured by Moynihan [4] et al. ®tted by Eq. (25) are shown in Fig. 4. Within the accuracy of the measuring method there is a good agreement of our model with the experimental results. The conductivity shows the predicted minimum near the point of equal concentration of the two cations, sodium and potassium. This non-linearity with respect to the composition is a characteristic property of the mixed alkali eect. Finally, the MMIE becomes relevant if the quantity lcH2 cannot be neglected. The criterion for the observation of larger MMIE (or mixed alkali eect) can be approximated by using Eqs. (13)±(15). We obtain lcH g: As mentioned above, a detailed analysis of the dependence on the cation concentration should be discussed more carefully. In spite of the simplistic assumptions used the dependence on f for a given c, however, is approximately correct. It is an obvious result that the total conductivity is not a simple linear superposition of the single conductivity of cations of both types having the concentration cA fc and cB 1 ÿ f c, respectively. b. Low temperature regime T < Tg ). Eq. (25) is also valid after cooling below the glass transition temperature, i.e. the general dependence, with a extremum at f ' 1=2, can be observed below and above the glass transition temperature. Thus the two regimes are not strictly separated by Tg , since a continuous crossover occurs between both regimes. An interesting eect is to be expected, however, if a partial exchange of cations, e.g. by a low-temperature ion-exchange, is made below Tg . Such an exchange leads to a dierent change of the composition
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Fig. 4. Reduced electrical dc-conductivity of (K2 O)f (Na2 O)1ÿf (Si2 O)3 versus the composition ratio f (cation type A: K , cation type B: Na ). The experimental data are obtained from Ref. [14]. The dotted curves correspond to the theoretical prediction with D0A D0B , lg sA c; T =sB c; T 13:7, and lc21 lgs0 T =s1 0:8425. The last two ratios are ®tted at the temperature T 373 K. The second curve follows immediately using the reasonable Vogel temperature T1 122 K.
ratios for strongly trapped cations and weakly trapped cations, respectively. The change of f H and partially also of cH for strongly trapped cations is essentially smaller than the corresponding change for the relatively mobile cations. The system takes on a non-equilibrium state which approaches equilibrium only after a sucient relaxation time. It can be assumed that far below Tg and immediately after the exchange procedure both cH and f H have approximately the same magnitudes as before the exchange. Since the total concentration of cations is constant during the exchange procedure, the total composition ratio will be changed, i.e. f ! f 0 . Thus the conductivity after the cation exchange is given by ÿ ÿ e2 LA cf 0 Ud f H eÿDEA =kB T ÿ 1 LB c 1 ÿ f 0 Ud 1 ÿ f H eÿDEB =kB T ÿ 1 : 26 r fH 1ÿf H lU2 f H 1ÿf H kB T s0 T =sc;1 d sA Ud ; T sB Ud ; T Now we use again the assumption EAH EBH EH and EA EB EH ÿ DE. Therefore, the conductivity 0 . On the after the exchange of cations shows a linear dependence with respect to the composition ratio, fmob other hand, if the exchange is realized above Tg the system approaches equilibrium between strongly and weakly trapped cations in a shorter time, i.e. shortly after the exchange f H f 0 . Thus, the MMIE is to be observed as usual. We obtain the important result that with respect to the MMIE cation-exchanged glasses have a dependence that is determined by an exchange above or below Tg . An exchange suciently above Tg is equivalent to a preparation starting from the melt and usually leads to glasses with a larger MMIE. An exchange well below Tg prevents at least partially a thermodynamical equilibrium between the strongly and weakly trapped cations, i.e. no MMIE or almost no MMIE can be observed. Ion-exchanged alkali glasses show contrary dependence with respect to the deviation from linearity in dynamic and transport properties. On the one hand, the absence of the MMIE was proved for Na/Li exchanged aluminosilicate glasses [22] as well as for K/Na exchanged soda±lime glasses [23]. Therefore,
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according to our model, we assume that the ion exchange was carried out suciently below the glass transition temperature. Actually, the K/Na-exchange by Tomandl and Schaeer [23] was performed at utz [24] found for a Rb/Na substitution in binary 440°C, well below Tg . On the other hand, Frischat and Sch sodium silicates and for a Na/Rb substitution in binary rubidium silicates the respective conductivity dependence according to the MMIE. These Na2 Oá3SiO2 and Rb2 Oá3SiO2 glasses were cation-exchanged at 530°C, i.e. the exchange was at temperatures well above Tg of both glass systems. In addition, the samples were heat treated at 530°C after the cation exchange to achieve a homogeneous distribution of cations. After this procedure the complete non-linear conductivity dependence was observed. Note that to interpret these results, Tomandl and Schaeer assumed structural dierences for a cation exchange below the glass transformation temperature and the mixed-alkali structure as a result of melting processes. According to the thermodynamical model introduced here, deviations from the expected MMIE are only possible for an ion exchange below Tg since a total thermodynamical equilibration for the cation distribution shortly after the cation exchange is possible only for T > Tg . Note that the characteristic control parameter for a cation exchange well below Tg is the ratio n cH =c (with n Ud =c for c > Ud ) which determines mainly the behavior shortly after the exchange procedure. There exists a ®nite fraction of cations cmob c 1 ÿ n which can be exchanged without a signi®cant in¯uence on the dynamics of the covalent network. Thus, a remarkable deviation from the MMIE below Tg should be expected if n is suciently smaller than unity. 4.4. Excess enthalpy and MMIE It should be remarked that a connection between the MMIE in dc-conductivity and the enthalpy of mixing can be observed [25]. This connection leads to the suggestion that such an enthalpy shows a negative excess if the MMIE occurs. An excess enthalpy is correlated with the formation of a non-random distribution of the cations in equilibrium. In a number of papers [25,26] a considerable excess enthalphy is assumed resulting in a inhomogeneous distribution of cations. Such an inhomogeneous cation distribution may also explain the non-linear dependence of conductivity on the composition ratio of cations. Continuous changes of cation±cation correlations as a function of the Na/(Na+Li)-ratio [26] have been demonstrated recently by multi-nuclear single and double resonance magic-angle spinning nuclear magnetic resonance in Li2 O/Na2 O±SiO2 mixed cation glasses. It should be noted, however, that there is no direct experimental evidence of a preferred interaction between unlike cations, to our knowledge. The interaction between cations should be smaller than interaction with network ions because of their chemical equivalence assuming moderate dilution in the covalent network, i.e. the density ¯uctuations have a relatively short correlation length. Since in the modi®ed glass the average size of a cooperative region (1±10 nm) [27] is expected to be larger than this correlation length of the nonrandom cation distribution on the scale of the cooperative region an apparently random distribution of the cations is obtained. Thus, the existence of an excess enthalpy of mixing and a random distribution on the scale of the cooperative region do not contradict. 5. Conclusions The present model con®rms essential experimental results, concerning the mobility-fractions in ionconducting glasses. In contrast to other current theories our model considers the dynamic interaction between the cation and covalent network. This interaction is realized by the integration of a part of the cations into the covalent network of the underlying glass. From this point of view the main result of the present paper is deduced. The dc-conductivity is not a simple linear superposition of the conductivities of the single cations. The strong interaction of the cation with the dynamics of the covalent network leads to a typical many body system with internal restrictions. The MMIE with an extremum in the conductivity of
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mixed cation glasses is a result of a characteristic non-linear superposition, which was analyzed in the present paper. Future studies are required to con®rm of the present theory. The evidence of two types of cation binding energies or a corresponding asymmetric distribution of a simple binding energy represent one of the crucial question. Furthermore, for the ion exchange of cations above and below the glass transition temperature further experiments in proof of the characteristic non-linear dependence on the composition ratio of the conductivity above Tg as well as of a linear dependence of the conductivity on the composition ratio for a cation exchange below Tg are needed. Acknowledgements This work has been supported by the Deutsche Forschungsgemeinschaft, SFB 418 and schu 934/3-1. Appendix A. Concentration of immobile cations We suppose that the glass matrix contains Nd deep traps and Nw shallow traps. Furthermore, the number of cations should be de®ned by v. Thus, there exist Nw Nd k vÿk possibilities for a con®guration with k cations in deep traps in and v ÿ k weakly trapped cations. The vÿk with statistical weight of each con®guration is given by pk 1 ÿ p eDE=kB T : 1 eDE=kB T Therefore, the probability that k cations occupy deep traps is given by Nw vÿk ÿ1 Nd pk 1 ÿ p : A:1 P k N k vÿk P N realizes the normalization condition k P k 1. The thermodynamical limit Nd ; Nw ! 1) leads to a sharp maximum of P k at km . The maximum follows from p
oP k 0 ok
or
1 oP k o ln P k 0: P k ok ok
Using Stirling's formula, one obtains the approximation R exp f R ln R ÿ m ln m ÿ R ÿ m ln R ÿ mg m and therefore o ln P k DE 0: ln Nd ÿ k ÿ lnk ln v ÿ k ÿ ln Nw ÿ v k ok kB T Hence Nd ÿ k v ÿ k eÿDE=kB T : k Nw ÿ v k
A:2
The total number of particles in the modi®ed glass may be Np . Thus, the total concentration of the traps is given by Utot Nd Nw =Np , the concentration of the deep traps is de®ned by Ud Nd =Np . Furthermore,
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the concentrations of relatively immobile (or strong trapped) cations is cH k=Np and the total concentration of cations is c v=Np . Finally, the concentration of the relatively immobile cations is given by the equation U d ÿ cH c ÿ cH eÿDE=kB T : cH Utot ÿ Ud ÿ c cH
A:3
Appendix B. Averaged cooperative region The averaged number of particles per cooperative region can be written as * !+ MI X 1 1 0 N lnh exp ac NI i N T ln exp ac g ni ac hMI ac ac i1
B:1
with the total number MI NIA NIB of cations per cooperative region I. The values ni represent the type of cation i i 1; . . . ; MI ). Especially, ni 1 if the cation i is of type A and ni ÿ1 if the cation is of type B. The second part of Eq. (B.1) can be expanded in terms of cumulants. Using the notation dNI g
MI X ni hMI ; i1
one obtains up to the second order of cumulant expansion 1 a ln h exp adNI i hdNI i h dNI 2 i ÿ hdNI i2 : B:2 a 2 The calculation of the averages can be realized in two steps. The ®rst step, denoted by h. . .in , is the calculation of the average over the composition considering a ®xed number of cations MI . The second step is the determination of the average over the number of cations per cooperative region. This average is de®ned by the symbol h. . .iM . Thus, one has the representation hAMI ; fni gi hhAMI ; fni gin iM : We obtain immediately hni in 2f H ÿ 1: The statistical independence of the cations inside the cooperative region leads further to ÿ 2 ÿ 2 ÿ hni nj in dij 1 ÿ dij 2f H ÿ 1 2f H ÿ 1 4dij f H 1 ÿ f H : The averaged number of cations MI can be written as
A B NI NI hMI i N cH with the concentration cH of the ®xed cations in the covalent network (note that each concentration is related to the total number of particles, e.g. cH is the number of cations per total number of atoms of the glass). N is the averaged number of particles in a cooperative region. The statistical independence of the particles leads to
cH Ud ÿ cH : MI2 ÿ hMI i2 N Ud
Thus one obtains * + * + * + MI MI MI X X X H ni hni in 2f ÿ 1 hMI i 2f H ÿ 1 N cH 2f H ÿ 1 i1
i1
M
i1
M
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and therefore
hdNI i N cH g 2f H ÿ 1 h :
B:3
The second term of Eq. (B.2) can be written as * + * + MI MI X X 2
ÿ 2 2 2 2 ni nj 2gh MI nj h2 MI2 ÿ N cH g 2f H ÿ 1h : h dNI i ÿ hdNI i g i;j1
i;j1
The consideration of * + * + MI MI D ÿ X X
ÿ E 2 ni nj ni nj n MI2 2f H ÿ 1 4MI f H 1 ÿ f H i;j1
i;j1
N c and
* MI
MI X
+ nj
ÿ 2 ÿ Ud ÿ cH N cH 2f H ÿ 1 4N cH f H 1 ÿ f H Ud
*
MI
MI X i
i;j1
M
M
+ h ni i n
MI2
ÿ
H
2f ÿ 1
M
Nc
H
M
ÿ
U d ÿ cH H 2f ÿ 1 Nc Ud H
leads to the relation D E 2 ÿ U d ÿ cH dNI 2 ÿ hdNI i2 N cH g 2f H ÿ 1 h 4g2 f H 1 ÿ f H : Ud
B:4
The substitution of Eqs. (B.2)±(B.4) into Eq. (B.1) yield 2 ÿ ac U d ÿ c H 0 H H H H 2 H H : N N T N c g 2f ÿ 1 h g 2f ÿ 1 h 4c g f 1 ÿ f 2 Ud Therefore, the averaged number of particles per cooperative region is given by N
n
1 ÿ cH g 2f H ÿ 1 h a2c
N 0 T Ud ÿcH Ud
2
g 2f H ÿ 1 h 4g2 f H 1 ÿ f H
o :
The ®nal result follows after an expansion in powers of g and h ÿ H ac H U d ÿ c H 2 ÿ N H 1 g 2f ÿ 1 h c c g 2f H ÿ 1 h 2ac g2 cH f H 1 ÿ f H 0 2 Ud N T 2 H2 H c g 2f ÿ1 h : Appendix C. Symbols Tg T1 c cc c A; B) cH cH c c A; B) cmob
glass transition temperature Vogel temperature total concentration cations in the glass concentration of component A and B, respectively cA cB c) total concentration of strongly trapped cations H H concentration of strongly trapped cations cH A cB c ) total concentration of weakly trapped cations (cmob c ÿ cH )
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f fH fmob Utot Ud Ec c A; B) EcH c A; B) DEc EcH ÿ Ec Dcmob c A; B) Dcimmob c A; B) Lc c A; B) s N r qc c A; B)
165
composition ratio, (f cA =cA cB composition strongly trapped cations composition ratio of relatively mobile cations total concentration of traps concentration of deep traps Ud P cH ) trapping energies of cations in weak traps trapping energies of strongly trapped cations energy dierence between strongly and weakly trapped cations diusion coecients of mobile fractions diusion coecients of relatively immobile fractions characteristic square length scale of diusion coecient characteristic time scale of diusion, average relaxation time average size (number of particles) of a cooperative region electrical conductivity electrical charge of the cations
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
G. Adam, J.H. Gibbs, J. Chem. Phys. 43 (1965) 139. D.E. Day, J. Non-Cryst. Solids 21 (1976) 343. M.D. Ingram, Phys. Chem. Glasses 28 (1987) 215. C.T. Moynihan, N.S. Saad, D.C. Tran, A.V. Lesikar, J. Am. Ceram. Soc. 63 (1980) 458. M.D. Ingram, Glastech. Ber. Glass Sci. Technol. 67 (1994) 151. J.E. Davidson, M.D. Ingram, A. Bunde, K. Funke, J. Non-Cryst. Solids 203 (1996) 246. A. Bunde, K. Funke, M.D. Ingram, Solid State Ionic 86±88 (1996) 1311. J.W. Haus, K.W. Kehr, Phys. Rep. 150 (1987) 263. J. Jackle, Rep. Prog. Phys. 49 (1986) 171. H. Vogel, Phys. Z. 22 (1921) 645; G.S. Fulcher, J. Amer. Chem. Soc. 8 (1925) 339; G. Tammann, G. Hesse, Z. Anorg. Allg. Chem. 156 (1926) 245. H. Baessler, Phys. Rev. Lett. 58 (1987) 767. M. Schulz, P. Reineker, Phys. Rev. B 48 (1993) 9369. R. B ohmer, K.L. Ngai, C.A. Angell, D.J. Plazek, J. Chem. Phys. 99 (1993) 4201. F.S. Howell, R.A. Bose, P.B. Macedo, C.T. Moynihan, J. Phys. Chem. 78 (1974) 639. G.N. Greaves, K.L. Ngai, Phys. Rev. B 52 (1995) 6358. C.C. Hunter, M.D. Ingram, Solid State Ionics 14 (1984) 31. W.C. Huang, H. Jain, J. Non-Cryst. Solids 188 (1995) 254. E.I. Kamitos, Y.D. Yiannopoulos, H. Jain, W.C. Huang, Phys. Rev. B 54 (1996) 9775. E.I. Kamitsos, A.P. Patsis, G.D. Chryssikos, J. Non-Cryst. Solids 152 (1993) 246. A.E. Owen, R.W. Douglas, J. Soc. Glass Technol. 43 (1959) 159. J.O. Isard, J. Non-Cryst. Solids 1 (1969) 235. H.M. Gar®nkel, C.B. King, J. Am. Ceram. Soc. 53 (1970) 686. G. Tomandl, H.A. Schaeer, in: G.H. Frischat (Ed.), Trans. Tech., Aedermannsdorf, 1977, p. 480. G.H. Frischat, H.E. Sch utz, Glastech. Ber. 49 (1976) 106. M. Tomozawa, J. Non-Cryst. Solids 152 (1993) 59. B. Gee, H. Janssen, H. Eckert, J. Non-Cryst. Solids 215 (1997) 41. E.W. Fischer, E. Donth, W. Steen, Phys. Rev. Lett. 68 (1992) 2344.
A cooperative model for the mixed mobile ion eect in covalent glasses Beatrix M. Schulz *, Manfred Dubiel, Michael Schulz Fachbereich Physik, Martin±Luther±Universit at Halle, 06099 Halle (Saale), Germany Received 2 October 1997; received in revised form 17 June 1998
Abstract A thermodynamical model of the long±standing problem of the mixed mobile ion eect (MMIE) or the mixed alkali eect, respectively, in glasses is presented. The basic idea of the model is an internal dynamic interaction between the cations and the dynamics of the cooperative regions in glasses. Two types of cation traps due to structural disorder in the glassy network are assumed, in which the deeper traps mainly determine the glass relaxation properties and the cations with a reduced binding energy are responsible for the movement of ions. The modi®cation of the usual diusion coecient of cations has been calculated under these considerations. That leads to a variation of dc-conductivity in dependence on the composition ratio, f , of the dierent cations introduced into a covalent network, i.e. the occurrence of the MMIE. The main result is a relation between the dc conductivity and temperature dependent relaxation time that is controlled by f . Furthermore, the conductivity properties of ion exchanged glasses can be predicted quantitatively if the exchange of ions take place either above or below the glass transition temperature. Ó 1998 Elsevier Science B.V. All rights reserved. PACS: 66.30.Dn; 72.80.Ng; 81.05 Kf
1. Introduction A large number of liquids have been observed to develop slower dynamics without the evolution of any observable long range correlated or ordered structure. This glass transition process shows a cooperativity [1] and can be described as a typical nonlocal process. On the other hand, the structure of glasses at the atomic level is relatively complex. It can be expected that both, local structure and nonlocal cooperative dynamics determine the ionic conduction in glasses and supercooled liquids. For example, the silicate glass system being the overwhelming application of glass products consists of randomly arranged SiO4 tetrahedrons. The network forms an in®nite large cluster of chemical bonds between these structural elements. Below the glass transition temperature, Tg , this random structure is almost ®xed, i.e. most of the chemical bonds are stable over the total observation time. The glass dynamics consist in the destruction of bonds, cooperative motion of the silicate network and the formation of new bonds.
*
Corresponding author. Tel.: +49-345 5527159; e-mail: [email protected].
0022-3093/98/$ ± see front matter Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 8 ) 0 0 7 6 1 - 3
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The introduction of alkali ions into a pure SiO2 glass leads to an increased electrical conductivity. Additionally, if the glass contains two types of cations simultaneously, e.g. Na and K , the conductivity has a non-linear dependence on the composition ratio of the cations [2,3]. This eect is denoted as mixed alkali eect, MAE, or more generally as mixed mobile ion eect, (MMIE). The decrease of conductivity is of the order about of 103 ±104 [4]. This decrease is in contradiction to the electrical conductivity expected from superposition of the single conductivities. The MMIE represents a property of ion-conducting disordered systems such as glasses. To explain this unusual eect in glasses a variety of models was developed. For a review see Day [2] or Ingram [3,5]. Till now, however, there exists no theoretical framework considering a nonlocal dynamical interaction between the alkali cations via cooperative rearrangements of the glass matrix, to our knowledge. Furthermore, the consideration of ion-exchanged glasses in the discussion of MMIE involves additional complications. Some measurements of the ionic conductivity or the diusion coecients in ion-exchanged silicate glasses indicate the presence of a reduced mobility of cations as measured for glasses produced by melting procedures, whereas other experiments do not show such eects (see for example Ref. [6]). Recently, the uni®ed site relaxation model (USRM) has been proposed by Bunde, Funke and Ingram [6,7]. The authors combined the relaxation of the Coulomb ®eld and the site relaxation during ion hopping processes to explain the frequency dependence of the ionic conductivity as well as eects such as the MMIE. According to this model the ions move to neighboring sites, if appropriate target sites exist. On the other hand, successful ion jumps require localized structural relaxations to match the environment for the speci®c cation. With respect to the ion-exchanged glasses, the authors predict the MMIE in a homogeneous mixture of dierent cations, whereas the eect disappears in a completely inhomogeneous mixture. In general, the cation mobility is reduced whenever two dierent cations are able to create their speci®c environment by local processes. A speci®c environment, the so-called site, is characterized by typical bond lengths and coordination numbers of surrounding oxygen atoms. These assumptions suggest a remarkable importance of glass dynamics below as well as above the glass transition temperature. The aim of the present work is to show a new phenomenological explanation of the mobility in mixed cation glasses by considering cooperative rearrangements of the introduced cations owing to their interactions with the covalent glassy network. The USRM is a typical local theory based on structural relaxations. In comparison to this model, our new theory is based on an eective dynamical interaction of cations situated in a volume of the order of magnitude of a cooperative region. This interaction will be calculated under the assumption of the existence of strongly and weakly trapped cations, i.e. the existence of two types of cation binding energies resulting from the disordered structure of glasses. The strongly trapped or relatively immobile cations should modify the relaxation process of the glass matrix, i.e. this type of cations has an aect on the mobility of all cations inside the glass matrix. This eect is described in terms of a conductivity relaxation time by which the change of the cation mobility and of the electrical conductivity can be explained. The results are discussed with respect to the aect of cation composition on the conductivity over a range cation concentrations in the temperature range below and above the glass transition temperature, Tg . Additionally, the introduction of cations into the glass network by an ion exchange will be considered. 2. Two-site model of cation bonds We restrict our investigations to glasses which form a network of covalent bonds. Typical materials are SiO2 , B2 O3 and GeO2 . Some polymers and polymer networks are also candidates. If such glasses are modi®ed by addition of salts or oxides, then the following scenario can be observed. Since all anions or oxygen atoms are tied to the network of covalent bonds, only the cations, i.e. alkali ions as Na or K ; remain as most mobile species. In this network the cations behave like ions in a potential landscape
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including traps of various depths. A large spectrum of local potential minima exist in the glass where cations can be trapped. We assume that there are only two types of potential minima: deep traps and shallow traps. Of course, this is a very simpli®ed assumption. In reality one has to consider a smooth distribution function with respect to the potential depths. The assumption of deep and shallow traps, however, is an approximation of the real distribution in disordered materials by a suciently simple discrete distribution. Because of the Coulombic attraction cation sites in the very neighborhood of negatively charged anions, e.g. of nonbridging oxygens, are thought to be deep traps in covalent networks. If this interaction is reduced, e.g. by neighboring cations or defects within the glass network, weaker traps are formed. For the following discussion we assume that strongly trapped cations determine the dynamics of the glass matrix. Obviously, a glass with two kinds of cation bonds has concentrations, cA and cB , of A and B components, respectively, or by the total concentration c cA cB and the composition ratio of cations f cA = cA cB . We introduce further quantities, such as the concentration of deep traps, Ud , the total concentration of strongly trapped cations, cH (with cH 6 Ud ), and the composition ratio, f H , of the strongly trapped cations. The concentration of deep traps depends on the structure of the glassy network which is mainly determined by the distribution of defects, e.g. of nonbridging oxygen atoms and other anions. Thus Ud implicitly becomes a function of the total cation concentration c and partially of the composition ratio f , i.e. Ud Ud c; f . In general, there is a monotonic relation between the total concentration of cations, c, and the concentration of the deep traps, Ud , i.e. the number of deep traps increases with increasing cation content. On the other hand, we assume that the dependence on the composition ratio, f , is weaker. Therefore, in the following discussion we will use the sucient assumption Ud Ud c, with the extension to the general case Ud Ud c; f being always possible. In general, the binding energy of a trapped cation depends also on the type of the cation, i.e. there are four energy levels: EA and EB ; the energies of the weakly trapped cations, and EAH and EBH , the energy levels of strongly trapped cations. Consequently, the concentration of the relatively immobile cations in deep traps is a function of the temperature, T , of the concentration of deep traps, Ud , and both cation contents, hence cH cH T ; Ud ; cA ; cB cH T ; Ud c; cf ; c 1 ÿ f or cH cH T ; c; f . The composition ratio f H of the cations in the deep traps is controlled by this parameters too, i.e. f H f H T ; c; f . An interesting case is given if the binding energies do not depend on the cation type EA EB and EAH EBH : Mostly, this speci®c borderline case is only an approximation for real glasses, but it will be very helpful for ®rst calculations. With that, we obtaine cH cH T ; Ud ; c cH T ; c and f H f . We will use this approximation in the further discussion (see below). Finally, we introduce the concentration of the weakly trapped cations, cmob c ÿ cH , and the corresponding composition ratio, fmob , with fmob f for the above mentioned speci®c borderline case. Considering the bonding of the two kinds of cations de®ned above, it is reasonable to identify the weakly trapped ions with the most mobile species because of the their smaller activation energies. On the other hand, the deep traps are assumed to be relatively immobile. 3. Calculation of relaxations time, diusion coecient and conductivity The fraction of cations occupying deep or shallow traps becomes a well de®ned quantity in the thermodynamical limit, i.e. thermodynamical ¯uctuations of cation concentrations are asymptotically irrelevant observables. Each cation occupies a trap only for a randomly distributed ®nite time and it moves to another trap during a shorter hopping time, i.e. the fraction of cations outside the traps may be neglected. The probability of a jump and, consequently the trapping time, is determined by the depth of the trap and the local mobility of the surrounding glass matrix. During a sucient time, however, each ion visits both kinds of traps. Thus, the average individual diusion coecients of a given kind of cation get equivalent
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values for sucient observation times, i.e. these coecients are weighted averages over the corresponding hopping rates for deep and weak traps [8]. On the other hand, conductivity is a collective property of all cations. In the thermodynamical equilibrium (principle of detailed balance) the number of cations escaping from deep traps is equivalent to the number of cations occupying such traps. Although the individual energetic situation of each cation is permanently changed, the collective state always remains conserved. Each cation leaving a fraction (i.e. the fraction of deeply trapped cations or weakly trapped cation, respectively) is replaced by another cation entering this fraction. Thus, the whole system can be described by well de®ned fractions of always strongly trapped cations and fractions of always weakly trapped cations, respectively. From a statistical point of view this consideration corresponds to the principle of exchangeable particles. Therefore the conductivity can be described by separate fractions of relatively mobile cations and of relatively immobile cations, respectively. Because of the energy dierence DEc EcH ÿ Ec between cations in deep and in shallow traps, whereas the index, c A; B, denotes the kind of cations, there is the relation Dc Dcmob Dimmob eDEc =kB T ;
1
between the diusion coecients of the mobile and immobile fractions of each type of cations. The conductivity of each component is given by the well known Nernst±Einstein relation composed of the contributions of the mobile and the immobile fractions q2 Dc q2c c c c mob c H ÿDEc =kB T Dmob cmob Dimmob cc cc cH ÿ 1 with c A; B 2 rc c e kB T kB T in which qc is the electrical charge of the cations. The Haven ratio is assumed to be unity to simplify the calculations. Usually, this ratio decreases with increasing concentration of cation from 1:0 to 0.2. The diusion coecients depend on the hopping ions and the mobility of the covalent network. Therefore, the diusion coecients, Dcmob , can be written as a product of two parts Dcmob Lc sÿ1 :
3
c
The ®rst factor, L , contains individual properties of the cation type c A, B, e.g. the mass of the cations, trapping energies, Ec , and possible interactions with the local environment of the cation. The second factor, sÿ1 , is mainly determined by relaxation of the glass matrix and depends on temperature and alkali content. This factor de®nes the true time scale, s, of the diusion. Therefore, s can be interpreted as the averaged conductivity relaxation time. As assumed above, the incorporation of relatively immobile cations into the glass matrix leads to a modi®cation of the covalent network. This modi®cation must be considered for calculating the averaged relaxation time, s. It can be expected that s depends on cH and f H . Obviously, a jump of a cation from one trap into another is hindered by the surrounding network acting like a cage. The averaged time in which an ion leaves its cage depends on the size of the corresponding cooperative region. In other words, a number, N , of atoms of the covalent network must move over a small distance, before the elementary hopping process of the captured cation can be realized. The modi®cation of the covalent network by the incorporation of the strongly trapped cations leads to a relation N N T ; cH ; f H . Following the arguments of Adam and Gibbs [1] (see also [9]) the conductivity relaxation time can be written as ÿ sc sc;1 exp ac N : 4 Here, a is an undetermined factor which is ®nally used as a ®t parameter. sc;1 is de®ned by a natural time scale which is given by sc;1 sc T ! 1. Using Eqs. (3) and (4) the conductivity of glass including cations is determined by ÿ 2 ÿ ÿ exp ÿ ac N H H ÿDEA =kB T 2 H H ÿDEB =kB T : 5 ÿ 1 qB LB c c 1 ÿ f e ÿ 1 r qA LA cf c f e sc;1 kB T
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Here LA and LB are temperature dependent quantities which consider in particular the trapping energies, EA and EB , respectively. For a further discussion of Eq. (5) we need an explicit relation N N T ; cH ; f H . The size of a cooperative region of a pure covalent network depends on temperature [1]. At temperatures suciently above the Vogel temperature, T1 [10], the size of cooperative regions can be described by [9] N 0 T
h0 : T ÿ T1
6
Obviously, this is an empirical representation which fails for T ! T1 . Near T1 other divergenceless approaches, e.g. ln s N 0 T T ÿ2 [11], should be used. Furthermore, it can be expected that close to and below T1 a hopping regime remains eective [12] which leads to an Arrhenius-like dependence of the relaxations time with a larger activation energy, e.g. ln s N 0 T T ÿ1 . A large part of ionically conducting glasses (based for example on SiO2 or GeO2 ) showing the MMIE belong to the strong glasses [13], i.e. the corresponding Vogel temperature is small or vanishes T1 ! 0). Therefore, the size of the cooperatively rearranging regions of strong glasses is approximately N 0 T ' T ÿ1 . Thus, Eq. (6) at temperatures suciently above T1 (fragile glasses) or over the whole temperature range (strong glasses, T1 ! 0) can be used as a reasonable approach. The detailed relation between N 0 and the temperature, however, is irrelevant for the following investigations. The modi®cation of the network by an incorporation of two kinds of cations (A,B) changes the size of the cooperative regions. We separate the volume of the glass into subvolumes, indicated by the index, I, of size, NI , of a cooperative region. Thus, one has the condition X NI Ntotal ; 7 I
where Ntotal is the number of all atoms of the glass. If the subvolume, I, contains NIA immobile cations of type A and NIB cations of type B, the size of the cooperative region is less changed as compared to a region without immobile cations. The consideration of only linear corrections leads to NI N 0 T g
MI X ÿ ÿ ni hMI N 0 T g NIA ÿ NIB h NIA NIB :
8
i1
From the present point of view g and h are empirical factors which should depend on the the underlying covalent network. In particular, the dierence, NIA ÿ NIB , denotes the eect of mass dierences and speci®c volume dierences on the mobility of the modi®ed glass. On the other hand, the cations inside a cooperative region can be assumed to be randomly distributed. The actual relaxation time of a cooperative region I is given by sc;I sc;1 exp ac NI : The time scale sc follows from the average over all possible distributions of A and B ÿ sc sc;1 exp ac N hsI i sc;1 h exp ac NI i:
9 10
Therefore, the averaged number of cations per cooperative region is given by N
1 ln h exp ac NI i: ac
11
This quantity can be obtained by a simple algebraic calculation, see Appendix B. We obtain ÿ H ac H U d ÿ c H 2 ÿ N H 1 g 2f ÿ 1 h c c g 2f H ÿ 1 h 2ac g2 cH f H 1 ÿ f H 2 Ud N 0 T 2 cH2 g 2f H ÿ1 h :
12
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The present relation allows the calculation of the relaxation time sc . For a better representation we determine the averaged number of particles of a cooperative region for only one component. The case of a pure component B f H 0) leads to ac cH Ud ÿ cH 2 2 H 0 H H2 13 N B c ; T N T 1 c h ÿ g h ÿ g c h ÿ g ; 2 Ud where the pure component A f H 1) corresponds to ac cH Ud ÿ cH 2 2 H 0 H H2 N A c ; T N T 1 c h g g h c g h : 2 Ud
14
Hence, the averaged number of ions per cooperative region for an available composition ratio becomes ÿ ÿ ac N N B cH ; T 1 ÿ f H N A cH ; T f H 2N 0 T ÿ 2 cH2 g2 f H 1 ÿ f H : 15 Ud Finally we obtain the averaged relaxation time s H H2 H ÿ H 1ÿf H s0 T lc f 1ÿf fH H sc sA c ; T sB c ; T sc;1
16
with the coecient l 2 ac =Ud ÿ 2g2 and the relaxation times of one component glasses sc cH ; T c A,B) given by ÿ ÿ sA cH ; T sc;1 exp ac N A and sB cH ; T sc;1 exp ac N B : 17 The third relaxation time, s0 T , of Eq. (16) corresponds to the extrapolated case of a covalent network with the same structure and distribution of deep and shallow traps as the underlying glass without any immobile cations. Note that this virtual relaxation time can be extrapolated only from experimental data, since an empty covalent network with the same structure as a network containing a ®nite fraction of cations is always unstable and experimentally unapproachable. s0 T , however, is never identical with the relaxation time of the pure covalent glass, e.g. amorphous SiO2 . It should be remarked also that the material dependent factor, ac , is a speci®c quantity for the investigated relaxation process. Another process in the same glass, e.g. the structural relaxation, has the same average size of cooperative regions, N , but a dierent factor, as . Therefore, dierent processes in the same glass show dierent relaxation times, e.g. structural relaxations s s exp a N , whereas the conductivity relaxation time is given by are described by s;1 s s sc sc;1 exp ac N . This dependence corresponds to the general assumption that the underlying dynamics of the glass are determined by the size of the cooperatively rearranging regions and is approximately universal whereas physical observables, e.g. electrical conductivity, dynamical structure factor, dielectric moduli, are coupled to this dynamic process by dierent strengths, e.g. as , ac . Usually, one obtains as > ac . Thus the decoupling index [14], Rs T ln ss =sc ln ss =sc as ÿ ac N , increases with decreasing temperatures. Rs T is small or nearly zero at high temperatures, i.e. structural and conductivity relaxation seem to be coupled at the same process. On the other hand, at low temperatures T 6 Tg ) the decoupling index becomes larger and the conductivity relaxation is apparently decoupled from the structural relaxation. It follows that a change of mechanical properties occurs close to or below Tg , whereas the conductivity shows only a smaller rate of decrease with decreasing temperature. Additionally, it should be remarked that sometimes both ac and as show a dependence on temperature. The substitution of Eq. (16) into Eq. (5) leads to the electrical conductivity ÿ ÿ 1 q2A LA cf cH f H eÿDEA =kB T ÿ 1 q2B LB c 1 ÿ f cH 1 ÿ f H eÿDEB =kB T ÿ 1 : 18 r fH 1ÿf H lcH2 f H 1ÿf H kB T s0 T =sc;1 sA cH ; T sB cH ; T
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155
Two additional remarks are necessary: At ®rst, the expansion to the ®rst order of cumulants leads to the mean ®eld approximation, hsI i sc;1 h exp ac NI i sc;1 exp ac hNI i. This approximation neglects all ¯uctuations of the composition ratio and the concentration. Thus, the non-linear exponent, lcH2 f H 1 ÿ f H , in Eq. (16) results from the deviation of the cation number from their average inside a cooperative region. Secondly, the result (16) can be obtained also by a phenomenological expansion of the averaged number of particles per cooperative region N N T ; cH ; f H in powers of the composition ratio up to the second order, i.e. N N 0 1 # 1 f H # 2 f H2 . The substitution of N in Eq. (4) by this power expansion yields an analogous dependence as Eq. (16), only the dependence on the concentration is unde®ned. 4. Discussion 4.1. Distribution of cation bonds The assumption of mobile and immobile cations, i.e. the existence of two dierent cation bonds, is similar to the model of ionic transport in silicate glasses by Greaves and Ngai [15]. These authors interpret the decrease of the macroscopic activation energy as well as binding energy with increasing alkali content by means of network hopping and intra-channel hopping of cations. For dilute concentrations the binding energy becomes equal to the Coulombic binding energy of an isolated alkali. The Coulombic interaction between cations and counter ions is to be interpreted in terms of deep traps following our ideas. Since the fraction of deeply trapped ions is determined by binding energies and corresponding Boltzmann factors, this fraction concerns the largest fraction of the cations. On the other hand, at higher alkali concentrations non-bridging oxygens channels are formed and the migration of cations is mainly controlled by the intrachannel hopping mechanism [15]. The channel sites can be considered as shallow traps. Hopping processes between these shallow traps, i.e. the motion along the channels, give the main contribution to the conductivity. It should be remarked, however, that the similarity of our model with the model of Greaves and Ngai is limited. The present model needs only the existence of deep and shallow traps, and speci®c structural properties, e.g. channels due to microsegregation of alkali, are not required. Therefore, our model can be generally applied to ionically conducting glasses, and should explain also the conductivity of borate or germanate glasses containing dierent cations. For these glasses a decreasing activation energy of the dc-ionic conductivity always was found with increasing fraction of alkali [16,17]. This dependence on the cation concentration supports the present assumptions, although its interpretation diers from that applied to silicates [18]. These authors calculate the activation energies for conductivity of Rb2 O±GeO2 glasses from data of far-infrared spectroscopy measurements as well as on the basis of an `unoccupied volume'. The long-range alkali motion is correlated to alkali ions occupying high-frequency IR active sites. An additional low frequency band, which was also assigned to vibrations of metal ions, gives experimental evidence for the presence of immobile cations as proposed by our thermodynamical model. Moreover, the presence of two IR bands which can be assigned to cation vibrations could be observed for many other glass systems as, e.g., silicates and borates (see Ref. [19]). However, no direct relationship of IR vibration frequencies with the binding energy of corresponding alkali ions is given. Especially, the sources of the low frequency vibrations are not completely known. Therefore, further investigations should try to identify dierent cations species which are responsible for the dynamical properties to examine this model. 4.2. Dependence of conductivity on cation concentration The qualitative dependence of conductivity on the concentration shall be discussed by assuming a covalent network with only one type of cation (e.g. A), i.e. we choose f f H 1. The conductivity then becomes
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ÿ q2A LA c ÿ cH cH eÿDEA =kB T : r kB T sA cH ; T
19
For the further discussion of Eq. (19) we need an approximate relation, cH cH c; T . This relation is usually obtained from experimental measurements. Here we will use an approximation leading to an almost qualitative result. A more detailed analysis requires better knowledge of cH , if possible, determined by direct experimental measurements. We assume that the covalent network is insensitive against changes of the cation content, i.e. the concentrations of deep and shallow traps do not depend on the cation concentration. Obviously, this is a very strong restriction, since it is well known that the number of traps increases continuously with increasing alkali content. Each addition of alkali cations implies simultaneously the incorporation of anions, e.g. nonbridging oxygens, into the covalent network. Thus, the physical properties of the network vary because of the changed cation concentration and it can be expected that the quantities describing the network, e.g. Ud and DEA , also change with alkali content. For a qualitative discussion, however, this assumption should be sucient. According to this consideration the concentration of deeply trapped cations is determined by U d ÿ c Ud ÿ c exp c Utot ÿ Ud ÿ c c
ÿ
DEA kB T
20
(see Appendix A), where Utot is the total concentration of traps in the modi®ed glass. From Eq. (20) a saturation at suciently high concentration c follows, i.e. limc!1 c c; T Ud . The opposite case c ! 0), however, can be described by the relation limc!0 c c; T c= 1 exp fÿDEA =kB T g, i.e. the main part of cations will be strongly trapped at suciently low concentrations. Eq. (19) is non-linear for smaller cation concentrations c>Ud ) r
q2A LA eÿDEA =kB T c ; ÿDE =k T A B kB T 1 e sA c; T
21
whereas at much larger concentrations a more linear dependence dominates: ÿ 1 q2A LA c ÿ Ud Ud eÿDEA =kB T r kB T sA Ud ; T
:
22
Note that this linear dependence is a result of the approximation applied. For real glasses we expect a non-linear dependence of the conductivity at smaller concentrations whereas the dependence at larger concentrations is less. Fig. 1 shows the electrical conductivity versus the concentration for the case of large energy dierences, DEA . This dependence is observed qualitatively for various materials [16]. In particular, the variation of the conductivity or the corresponding eective diusion coecient is a property of a glass network at smaller cation concentrations. SiO2 glasses, for example, show a nonlinear dependence on alkali content for suciently low (in the ppm range) concentrations of Na2 O [20,16], whereas at increased concentrations above (5 mol% Na2 O) an approximately linear behavior is observed. It should be remarked again, however, that the discussion of the concentration dependence is only of qualitative nature. As mentioned above, a thorough analysis is possible if the relation c c c; T is known in detail. Nevertheless, the qualitative behavior discussed above should be eective also for real glasses. Furthermore, the dependence on the composition ratio at a ®xed concentration discussed in the following is not in¯uenced by this restriction.
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Fig. 1. Qualitative behavior of the reduced conductivity rR r c=r c0 of a glass matrix modi®ed by one type of cations versus the reduced concentration x c=c0 c0 is the maximum allowed concentration of cations in the glass matrix). The reduced diusion coecient DR is de®ned by DR rR =x. The example is calculated by using the parameters aN0 T 80; a=c0 0:5, g hc0 ÿ0:5; c1 =c0 0:5 and exp ÿDE=kB T 0:005.
4.3. Mixed alkali glasses 4.3.1. Concentration of cations < Ud Here we will analyze the conductivity dependence on the composition ratio, f, for a low concentration of cations, i.e. the predominant part of cations is localized in deep traps. Using the above introduced speci®c borderline case EAH EBH EH and EA EB EH ÿ DE), the concentration of relatively immobile cations is given approximately by cH c= 1 exp fÿDE=kB T g and the composition ratio is de®ned by f H f . The exponent of s0 T =sc;1 (see Eq. (18)) can be neglected on condition that the cation content is suciently low, i.e. lc2 ! 0. We obtain 2e2 c LA f LB 1 ÿ f r : 23 kB T 1 eDE=kB T sA c; T f sB c; T 1ÿf
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Note that we have used the simpli®cation qA qB e. The conductivity shows an extremum at the composition ratio, fextr: : ÿ1 ÿ1 LA sB c; T ÿ ln 24 fextr: 1 ÿ LB sA c; T with the restriction that 0 6 fextr: 6 1. If this condition is not ful®lled, the conductivity increases or decreases monotonoicly with decreasing composition ratio over whole the range fextr: 2 0; 1. Fig. 2 shows the characteristic dependence on temperature and composition ratio for suciently small concentrations of cations. At suciently high temperatures a simple linear relation occurs. Only the low temperature regime (near the Vogel temperature, T1 < T < 1.2T1 ) shows a deviation from this dependence. It is evident that a MMIE for glasses with lower concentrations of cations cannot be observed. This result is in agreement with various experimental observations [21]. 4.3.2. Concentration of cations > Ud a. High temperature regime: Another case is given for high alkali content. It is well known [2,3] that the conductivity has a non-linear dependence on the composition ratio, f . Again we use the reasonable assumption EAH EBH EH and EA EB EH ÿ DE, which now corresponds to cH Ud and f H f . This simpli®cation leads to the conductivity ÿ e2 c Ud eÿDE=kB T ÿ 1 LA f LB 1 ÿ f : 25 r 2 f kB T sA Ud ; T sB Ud ; T 1ÿf s0 T =sc;1 lUd f 1ÿf
Fig. 2. Reduced electrical dc±conductivity r T =rB T versus the composition ratio f for a low cation concentration c. The used parameters are D0A =D0B 10, h 0 and h0 cH g=T1 0:1. The reduced temperature T 0 is de®ned by T 0 T =T1 . The dotted line corresponds to the case T 0 ! 1.
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Fig. 3. Reduced electrical dc-conductivity r T =rB T versus the composition ratio f for a high cation concentrations c. The used parameters are D0A =D0B 1, h 0 and h0 cH g=T1 0:1 and h0 a=c1 ÿ 2cH2 g2 =T1 8. The reduced temperature T 0 is given by T 0 T =T1 .
There is a non-linear dependence on f with a minimum, as shown in Fig. 3. As an example we consider the dependence on the composition ratio of the dc-conductivity of the glass system (K2 O)f (Na2 O) 1±f (SiO2 )3 obtained by usual melting procedures. The conductivity data measured by Moynihan [4] et al. ®tted by Eq. (25) are shown in Fig. 4. Within the accuracy of the measuring method there is a good agreement of our model with the experimental results. The conductivity shows the predicted minimum near the point of equal concentration of the two cations, sodium and potassium. This non-linearity with respect to the composition is a characteristic property of the mixed alkali eect. Finally, the MMIE becomes relevant if the quantity lcH2 cannot be neglected. The criterion for the observation of larger MMIE (or mixed alkali eect) can be approximated by using Eqs. (13)±(15). We obtain lcH g: As mentioned above, a detailed analysis of the dependence on the cation concentration should be discussed more carefully. In spite of the simplistic assumptions used the dependence on f for a given c, however, is approximately correct. It is an obvious result that the total conductivity is not a simple linear superposition of the single conductivity of cations of both types having the concentration cA fc and cB 1 ÿ f c, respectively. b. Low temperature regime T < Tg ). Eq. (25) is also valid after cooling below the glass transition temperature, i.e. the general dependence, with a extremum at f ' 1=2, can be observed below and above the glass transition temperature. Thus the two regimes are not strictly separated by Tg , since a continuous crossover occurs between both regimes. An interesting eect is to be expected, however, if a partial exchange of cations, e.g. by a low-temperature ion-exchange, is made below Tg . Such an exchange leads to a dierent change of the composition
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Fig. 4. Reduced electrical dc-conductivity of (K2 O)f (Na2 O)1ÿf (Si2 O)3 versus the composition ratio f (cation type A: K , cation type B: Na ). The experimental data are obtained from Ref. [14]. The dotted curves correspond to the theoretical prediction with D0A D0B , lg sA c; T =sB c; T 13:7, and lc21 lgs0 T =s1 0:8425. The last two ratios are ®tted at the temperature T 373 K. The second curve follows immediately using the reasonable Vogel temperature T1 122 K.
ratios for strongly trapped cations and weakly trapped cations, respectively. The change of f H and partially also of cH for strongly trapped cations is essentially smaller than the corresponding change for the relatively mobile cations. The system takes on a non-equilibrium state which approaches equilibrium only after a sucient relaxation time. It can be assumed that far below Tg and immediately after the exchange procedure both cH and f H have approximately the same magnitudes as before the exchange. Since the total concentration of cations is constant during the exchange procedure, the total composition ratio will be changed, i.e. f ! f 0 . Thus the conductivity after the cation exchange is given by ÿ ÿ e2 LA cf 0 Ud f H eÿDEA =kB T ÿ 1 LB c 1 ÿ f 0 Ud 1 ÿ f H eÿDEB =kB T ÿ 1 : 26 r fH 1ÿf H lU2 f H 1ÿf H kB T s0 T =sc;1 d sA Ud ; T sB Ud ; T Now we use again the assumption EAH EBH EH and EA EB EH ÿ DE. Therefore, the conductivity 0 . On the after the exchange of cations shows a linear dependence with respect to the composition ratio, fmob other hand, if the exchange is realized above Tg the system approaches equilibrium between strongly and weakly trapped cations in a shorter time, i.e. shortly after the exchange f H f 0 . Thus, the MMIE is to be observed as usual. We obtain the important result that with respect to the MMIE cation-exchanged glasses have a dependence that is determined by an exchange above or below Tg . An exchange suciently above Tg is equivalent to a preparation starting from the melt and usually leads to glasses with a larger MMIE. An exchange well below Tg prevents at least partially a thermodynamical equilibrium between the strongly and weakly trapped cations, i.e. no MMIE or almost no MMIE can be observed. Ion-exchanged alkali glasses show contrary dependence with respect to the deviation from linearity in dynamic and transport properties. On the one hand, the absence of the MMIE was proved for Na/Li exchanged aluminosilicate glasses [22] as well as for K/Na exchanged soda±lime glasses [23]. Therefore,
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according to our model, we assume that the ion exchange was carried out suciently below the glass transition temperature. Actually, the K/Na-exchange by Tomandl and Schaeer [23] was performed at utz [24] found for a Rb/Na substitution in binary 440°C, well below Tg . On the other hand, Frischat and Sch sodium silicates and for a Na/Rb substitution in binary rubidium silicates the respective conductivity dependence according to the MMIE. These Na2 Oá3SiO2 and Rb2 Oá3SiO2 glasses were cation-exchanged at 530°C, i.e. the exchange was at temperatures well above Tg of both glass systems. In addition, the samples were heat treated at 530°C after the cation exchange to achieve a homogeneous distribution of cations. After this procedure the complete non-linear conductivity dependence was observed. Note that to interpret these results, Tomandl and Schaeer assumed structural dierences for a cation exchange below the glass transformation temperature and the mixed-alkali structure as a result of melting processes. According to the thermodynamical model introduced here, deviations from the expected MMIE are only possible for an ion exchange below Tg since a total thermodynamical equilibration for the cation distribution shortly after the cation exchange is possible only for T > Tg . Note that the characteristic control parameter for a cation exchange well below Tg is the ratio n cH =c (with n Ud =c for c > Ud ) which determines mainly the behavior shortly after the exchange procedure. There exists a ®nite fraction of cations cmob c 1 ÿ n which can be exchanged without a signi®cant in¯uence on the dynamics of the covalent network. Thus, a remarkable deviation from the MMIE below Tg should be expected if n is suciently smaller than unity. 4.4. Excess enthalpy and MMIE It should be remarked that a connection between the MMIE in dc-conductivity and the enthalpy of mixing can be observed [25]. This connection leads to the suggestion that such an enthalpy shows a negative excess if the MMIE occurs. An excess enthalpy is correlated with the formation of a non-random distribution of the cations in equilibrium. In a number of papers [25,26] a considerable excess enthalphy is assumed resulting in a inhomogeneous distribution of cations. Such an inhomogeneous cation distribution may also explain the non-linear dependence of conductivity on the composition ratio of cations. Continuous changes of cation±cation correlations as a function of the Na/(Na+Li)-ratio [26] have been demonstrated recently by multi-nuclear single and double resonance magic-angle spinning nuclear magnetic resonance in Li2 O/Na2 O±SiO2 mixed cation glasses. It should be noted, however, that there is no direct experimental evidence of a preferred interaction between unlike cations, to our knowledge. The interaction between cations should be smaller than interaction with network ions because of their chemical equivalence assuming moderate dilution in the covalent network, i.e. the density ¯uctuations have a relatively short correlation length. Since in the modi®ed glass the average size of a cooperative region (1±10 nm) [27] is expected to be larger than this correlation length of the nonrandom cation distribution on the scale of the cooperative region an apparently random distribution of the cations is obtained. Thus, the existence of an excess enthalpy of mixing and a random distribution on the scale of the cooperative region do not contradict. 5. Conclusions The present model con®rms essential experimental results, concerning the mobility-fractions in ionconducting glasses. In contrast to other current theories our model considers the dynamic interaction between the cation and covalent network. This interaction is realized by the integration of a part of the cations into the covalent network of the underlying glass. From this point of view the main result of the present paper is deduced. The dc-conductivity is not a simple linear superposition of the conductivities of the single cations. The strong interaction of the cation with the dynamics of the covalent network leads to a typical many body system with internal restrictions. The MMIE with an extremum in the conductivity of
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mixed cation glasses is a result of a characteristic non-linear superposition, which was analyzed in the present paper. Future studies are required to con®rm of the present theory. The evidence of two types of cation binding energies or a corresponding asymmetric distribution of a simple binding energy represent one of the crucial question. Furthermore, for the ion exchange of cations above and below the glass transition temperature further experiments in proof of the characteristic non-linear dependence on the composition ratio of the conductivity above Tg as well as of a linear dependence of the conductivity on the composition ratio for a cation exchange below Tg are needed. Acknowledgements This work has been supported by the Deutsche Forschungsgemeinschaft, SFB 418 and schu 934/3-1. Appendix A. Concentration of immobile cations We suppose that the glass matrix contains Nd deep traps and Nw shallow traps. Furthermore, the number of cations should be de®ned by v. Thus, there exist Nw Nd k vÿk possibilities for a con®guration with k cations in deep traps in and v ÿ k weakly trapped cations. The vÿk with statistical weight of each con®guration is given by pk 1 ÿ p eDE=kB T : 1 eDE=kB T Therefore, the probability that k cations occupy deep traps is given by Nw vÿk ÿ1 Nd pk 1 ÿ p : A:1 P k N k vÿk P N realizes the normalization condition k P k 1. The thermodynamical limit Nd ; Nw ! 1) leads to a sharp maximum of P k at km . The maximum follows from p
oP k 0 ok
or
1 oP k o ln P k 0: P k ok ok
Using Stirling's formula, one obtains the approximation R exp f R ln R ÿ m ln m ÿ R ÿ m ln R ÿ mg m and therefore o ln P k DE 0: ln Nd ÿ k ÿ lnk ln v ÿ k ÿ ln Nw ÿ v k ok kB T Hence Nd ÿ k v ÿ k eÿDE=kB T : k Nw ÿ v k
A:2
The total number of particles in the modi®ed glass may be Np . Thus, the total concentration of the traps is given by Utot Nd Nw =Np , the concentration of the deep traps is de®ned by Ud Nd =Np . Furthermore,
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the concentrations of relatively immobile (or strong trapped) cations is cH k=Np and the total concentration of cations is c v=Np . Finally, the concentration of the relatively immobile cations is given by the equation U d ÿ cH c ÿ cH eÿDE=kB T : cH Utot ÿ Ud ÿ c cH
A:3
Appendix B. Averaged cooperative region The averaged number of particles per cooperative region can be written as * !+ MI X 1 1 0 N lnh exp ac NI i N T ln exp ac g ni ac hMI ac ac i1
B:1
with the total number MI NIA NIB of cations per cooperative region I. The values ni represent the type of cation i i 1; . . . ; MI ). Especially, ni 1 if the cation i is of type A and ni ÿ1 if the cation is of type B. The second part of Eq. (B.1) can be expanded in terms of cumulants. Using the notation dNI g
MI X ni hMI ; i1
one obtains up to the second order of cumulant expansion 1 a ln h exp adNI i hdNI i h dNI 2 i ÿ hdNI i2 : B:2 a 2 The calculation of the averages can be realized in two steps. The ®rst step, denoted by h. . .in , is the calculation of the average over the composition considering a ®xed number of cations MI . The second step is the determination of the average over the number of cations per cooperative region. This average is de®ned by the symbol h. . .iM . Thus, one has the representation hAMI ; fni gi hhAMI ; fni gin iM : We obtain immediately hni in 2f H ÿ 1: The statistical independence of the cations inside the cooperative region leads further to ÿ 2 ÿ 2 ÿ hni nj in dij 1 ÿ dij 2f H ÿ 1 2f H ÿ 1 4dij f H 1 ÿ f H : The averaged number of cations MI can be written as
A B NI NI hMI i N cH with the concentration cH of the ®xed cations in the covalent network (note that each concentration is related to the total number of particles, e.g. cH is the number of cations per total number of atoms of the glass). N is the averaged number of particles in a cooperative region. The statistical independence of the particles leads to
cH Ud ÿ cH : MI2 ÿ hMI i2 N Ud
Thus one obtains * + * + * + MI MI MI X X X H ni hni in 2f ÿ 1 hMI i 2f H ÿ 1 N cH 2f H ÿ 1 i1
i1
M
i1
M
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and therefore
hdNI i N cH g 2f H ÿ 1 h :
B:3
The second term of Eq. (B.2) can be written as * + * + MI MI X X 2
ÿ 2 2 2 2 ni nj 2gh MI nj h2 MI2 ÿ N cH g 2f H ÿ 1h : h dNI i ÿ hdNI i g i;j1
i;j1
The consideration of * + * + MI MI D ÿ X X
ÿ E 2 ni nj ni nj n MI2 2f H ÿ 1 4MI f H 1 ÿ f H i;j1
i;j1
N c and
* MI
MI X
+ nj
ÿ 2 ÿ Ud ÿ cH N cH 2f H ÿ 1 4N cH f H 1 ÿ f H Ud
*
MI
MI X i
i;j1
M
M
+ h ni i n
MI2
ÿ
H
2f ÿ 1
M
Nc
H
M
ÿ
U d ÿ cH H 2f ÿ 1 Nc Ud H
leads to the relation D E 2 ÿ U d ÿ cH dNI 2 ÿ hdNI i2 N cH g 2f H ÿ 1 h 4g2 f H 1 ÿ f H : Ud
B:4
The substitution of Eqs. (B.2)±(B.4) into Eq. (B.1) yield 2 ÿ ac U d ÿ c H 0 H H H H 2 H H : N N T N c g 2f ÿ 1 h g 2f ÿ 1 h 4c g f 1 ÿ f 2 Ud Therefore, the averaged number of particles per cooperative region is given by N
n
1 ÿ cH g 2f H ÿ 1 h a2c
N 0 T Ud ÿcH Ud
2
g 2f H ÿ 1 h 4g2 f H 1 ÿ f H
o :
The ®nal result follows after an expansion in powers of g and h ÿ H ac H U d ÿ c H 2 ÿ N H 1 g 2f ÿ 1 h c c g 2f H ÿ 1 h 2ac g2 cH f H 1 ÿ f H 0 2 Ud N T 2 H2 H c g 2f ÿ1 h : Appendix C. Symbols Tg T1 c cc c A; B) cH cH c c A; B) cmob
glass transition temperature Vogel temperature total concentration cations in the glass concentration of component A and B, respectively cA cB c) total concentration of strongly trapped cations H H concentration of strongly trapped cations cH A cB c ) total concentration of weakly trapped cations (cmob c ÿ cH )
B.M. Schulz et al. / Journal of Non-Crystalline Solids 241 (1998) 149±165
f fH fmob Utot Ud Ec c A; B) EcH c A; B) DEc EcH ÿ Ec Dcmob c A; B) Dcimmob c A; B) Lc c A; B) s N r qc c A; B)
165
composition ratio, (f cA =cA cB composition strongly trapped cations composition ratio of relatively mobile cations total concentration of traps concentration of deep traps Ud P cH ) trapping energies of cations in weak traps trapping energies of strongly trapped cations energy dierence between strongly and weakly trapped cations diusion coecients of mobile fractions diusion coecients of relatively immobile fractions characteristic square length scale of diusion coecient characteristic time scale of diusion, average relaxation time average size (number of particles) of a cooperative region electrical conductivity electrical charge of the cations
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
G. Adam, J.H. Gibbs, J. Chem. Phys. 43 (1965) 139. D.E. Day, J. Non-Cryst. Solids 21 (1976) 343. M.D. Ingram, Phys. Chem. Glasses 28 (1987) 215. C.T. Moynihan, N.S. Saad, D.C. Tran, A.V. Lesikar, J. Am. Ceram. Soc. 63 (1980) 458. M.D. Ingram, Glastech. Ber. Glass Sci. Technol. 67 (1994) 151. J.E. Davidson, M.D. Ingram, A. Bunde, K. Funke, J. Non-Cryst. Solids 203 (1996) 246. A. Bunde, K. Funke, M.D. Ingram, Solid State Ionic 86±88 (1996) 1311. J.W. Haus, K.W. Kehr, Phys. Rep. 150 (1987) 263. J. Jackle, Rep. Prog. Phys. 49 (1986) 171. H. Vogel, Phys. Z. 22 (1921) 645; G.S. Fulcher, J. Amer. Chem. Soc. 8 (1925) 339; G. Tammann, G. Hesse, Z. Anorg. Allg. Chem. 156 (1926) 245. H. Baessler, Phys. Rev. Lett. 58 (1987) 767. M. Schulz, P. Reineker, Phys. Rev. B 48 (1993) 9369. R. B ohmer, K.L. Ngai, C.A. Angell, D.J. Plazek, J. Chem. Phys. 99 (1993) 4201. F.S. Howell, R.A. Bose, P.B. Macedo, C.T. Moynihan, J. Phys. Chem. 78 (1974) 639. G.N. Greaves, K.L. Ngai, Phys. Rev. B 52 (1995) 6358. C.C. Hunter, M.D. Ingram, Solid State Ionics 14 (1984) 31. W.C. Huang, H. Jain, J. Non-Cryst. Solids 188 (1995) 254. E.I. Kamitos, Y.D. Yiannopoulos, H. Jain, W.C. Huang, Phys. Rev. B 54 (1996) 9775. E.I. Kamitsos, A.P. Patsis, G.D. Chryssikos, J. Non-Cryst. Solids 152 (1993) 246. A.E. Owen, R.W. Douglas, J. Soc. Glass Technol. 43 (1959) 159. J.O. Isard, J. Non-Cryst. Solids 1 (1969) 235. H.M. Gar®nkel, C.B. King, J. Am. Ceram. Soc. 53 (1970) 686. G. Tomandl, H.A. Schaeer, in: G.H. Frischat (Ed.), Trans. Tech., Aedermannsdorf, 1977, p. 480. G.H. Frischat, H.E. Sch utz, Glastech. Ber. 49 (1976) 106. M. Tomozawa, J. Non-Cryst. Solids 152 (1993) 59. B. Gee, H. Janssen, H. Eckert, J. Non-Cryst. Solids 215 (1997) 41. E.W. Fischer, E. Donth, W. Steen, Phys. Rev. Lett. 68 (1992) 2344.