Dielectric relaxation strength in ion conducting glasses caused by cluster polarization

Transcription

1 Chemical Physics xxx (2006) xxx xxx Dielectric relaxation strength in ion conducting glasses caused by cluster polarization Germà Garcia-Belmonte a, *, François Henn b, Juan Bisquert a a Departament de Ciències Experimentals, Universitat Jaume I, E Castelló, Spain b Laboratoire de Physicochimie de la Matière Condensée, UMR 5617 CNRS-Université Montpellier II, Place Eugène Bataillon-CC00, F Montpellier Cedex 5, France Received 27 February 2006; accepted 1 August 2006 Abstract A model for interpreting the dielectric relaxation strength De observed in ionic glasses is proposed. The model relates the dielectric strength to the polarization of nanometric structures formed by energetic clustering. The calculation of De relies exclusively upon equilibrium arguments, and its results are in agreement with previous derivations of the dc conductivity made from percolation approaches. The expression of De shows that its temperature and alkali content dependence is very sensitive to the underlying ion statistics so that no universal trend is expected. The model prediction is evaluated on the test case system of sodium conducting borate glass. The large variety in the dielectric strength dependences on modifier content is discussed in light of the inherent glassy structural complexity. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Ion conducting glasses; Dielectric strength; Cluster polarization; Percolation transport 1. Introduction * Corresponding author. Tel.: ; fax: address: garciag@uji.es (G. Garcia-Belmonte). Ion diffusion in amorphous materials is an issue of central interest because of its importance in technological uses. Defect diffusion in metals and alloys [1], ion conductivity in oxides [2,3], as well as insertion electrodes for batteries, electrochromism, fuel-cells, and oxygen membranes form a variety of technologies of potential application for glassy materials. In all these instances, ion transport mechanisms always lie at the heart of the system performance. Modeling the mechanisms accountable for the conductive and/ or dielectric properties of ionic conductors has thus been the subject of a broad literature among which many investigations have been devoted to the analysis and understanding of the complex conductivity ^rðxþ or permittivity ^eðxþ spectra. The intrinsic disordered environment of charge carriers in glasses is believed to give rise to distinct transport mechanisms such as the usually observed frequencydependent conductivity r 0 (x) [4 9]. In common to many other disordered materials, ac conductivity of oxide glasses is found to be frequency-dependent following an approximate power-law of the type r 0 (x) / x s (with 0.5 < s < 1). The onset of the ac contribution is marked by a characteristic frequency x c, which contains essential information about the ion conduction process. At low frequencies the conductivity exhibits the constant value r 0. For times t >1/x c, conduction appears to be homogeneous at all spatial scales as expected for ordinary (Fickean) diffusion, while for t <1/x c ions move following a subdiffusive regime which can be related to a polarization phenomenon. It is also experimentally observed that the transition frequency x c for the ac conductivity is located near the loss peak frequency of the complex permittivity spectra e 00 = [r 0 (x) r 0 ]/ie 0 x (e 0 represents the permittivity of the free space). Moreover, the dc conductivity and the dielectric constant increment De = e s e 1 (dielectric relaxation strength), which is often measured in ion conducting glasses, are correlated by means of the phenomenological, Barton Nakajama Namikawa (BNN) proportionality [10] /$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi: /j.chemphys

2 2 G. Garcia-Belmonte et al. / Chemical Physics xxx (2006) xxx xxx r 0 ¼ me 0 Dex c : ð1þ Here m is an experimental value of order 1. Eq. (1) would suggest that conduction and the excess of polarization share the same physical origin. Although ion conducting glasses have been deeply analyzed in order to explain the striking similarities among the conductivity spectra, much less effort has been paid to extract information from the variation of De with alkali concentration n and temperature T. A dielectric strength variation on composition different from the expected linear behavior De / n/t [11] has been reported to be De / n 1/3 / x 1/3 in case of sodium borates with compositions xna 2 O(1 x)b 2 O 3 [12] and other types of ionic glasses [13]. This scaling dependence is argued to lie behind nonuniversal features of the ac conductivity in several ion conducting glasses [14]. Likewise, the temperature dependence of De shows different behavior. While De increases with the inverse temperature for some germanates [15], it is often reported as almost temperature independent. It is therefore, important to recognize that experimental values of the dielectric strength might provide insight on the microscopic mechanisms of ion motion in glasses. Focusing on borate glasses there exist observations in the literature [10,12] that exhibit a variety of dependences of the type De / x a, the exponent a ranging from 1.5 to 1/3. Fig. 1 shows two limiting cases for borates glasses with sodium as alkali modifier. Also the case of a borosilicate is presented. Dielectric strength De has been associated [16] with the time at which the mean squared displacement hr 2 (1 )i of the mobile ions changes from sublinear hr 2 i/t b (b <1) to linear, ordinary diffusion hr 2 i/t. For a random walker in 3D system [5], De has been reported to be given by Fig. 1. Experimental values reported for the dielectric relaxation strength of borates glasses with composition xna 2 O(1 x)b 2 O 3. (m) extracted from Ref. [12] by assuming an average measuring temperature of 400 K, and (d) average value listed in Ref. [10] from previous work. (s) dielectric strength for a borosilicate glass xna 2 O(0.9 x)b 2 O 3 (0.1)SiO 2, also from Ref. [10]. Solid lines indicate dependences of the form De / x a. e 0 De ¼ e2 n 6k B T hr2 ð1þi: ð2þ From Eq. (2) and assuming that n / x, one can readily obtain that hr 2 (1)i /x a 1. The low value of the exponent (a = 1/3) reported in some cases has been regarded as indication of a correlation between the characteristic mean squared displacement and the average Na + Na + separation d. In this particular case hr 2 (1)i/x 2/3 / d 2 assuming that the mobile charges are uniformly distributed. It has been then suggested that the average distance to cover before entering the linear diffusion regime is proportional to the average mobile ion ion separation. Similar dependences have been discussed within the context of the MIGRATION concept [15,17]. The interpretation of Eq. (2) entails that hr 2 (1)i/x 0.5 in the case of exponent values as high as 1.5, which indicates a diffusing displacement increment with alkali modifier concentration. Other approaches interpret the dielectric strength in terms of purely dipolar polarizations [13]. The ion hop between neighboring sites is seen as a rotation of a permanent dipole. An expression analogous to Eq. (2) in which the characteristic distance corresponds to the elementary hopping distance was introduced as [13] e 0 De ¼ cn 3k B T ðedþ2 : ð3þ Here c stands for the fraction of cations which are mobile, and ed represents the effective dipole moment of the hopping ion. Deviations from the linear permittivity dependence with composition have been understood in terms of variations of the average elementary hopping distance. Similarly to that concluded from the approach of Eq. (2), the dipolar view predicts changes of the hopping distance in opposite ways depending on the experimental exponent value, 1/3 < a < 1.5. Although the previous models (Eqs. (2) and (3)) are based on different descriptions, implicitly both presuppose that the system can be heuristically divided, in time or space scale, into two subsystems: one giving rise to long range diffusion or dc conductivity, the other one being responsible for frequency dispersion and corresponding polarization. In the model presented in this work, based on fundamental findings for charge carrier dynamics in percolation systems, that splitting is explicitly taken into account. Our approach thus gives a meaningful interpretation to the dielectric strength from equilibrium arguments, without relying upon conductivity calculations via the Kramers Kronig transformation. The large diversity in the dielectric strength dependences on alkali content and on temperature is discussed in light of the structural complexity inherent to the considered sodium borate and borosilicate glasses. In the next sections of this paper we first describe briefly the basic features of percolation conductivity in a random potential model. We then calculate the consequences of such model for the dielectric polarization in quasi-equilibrium condition. A connection is established

3 between conduction and polarization, and the implications in relation with structural features and temperature and composition variations of borate glasses are discussed. G. Garcia-Belmonte et al. / Chemical Physics xxx (2006) xxx xxx 3 2. Percolation model for dielectric strength Percolation models of hopping conductivity have been used to interpret ion conduction in oxide glasses [7,9,18,19]. A key feature of hopping conductivity in strongly inhomogeneous media is that the microscopic energy relief forms large-scale geometric structures, or clusters, surrounded by energy barriers of large height n c (normalized to the thermal energy). Such barriers determine the activation energy of the dc conductivity DE a = k B Tn c. Long-range transport is constrained by the hopping at the bottlenecks between these clusters. Thus the effective length scale for transport resulting from the percolation theory is the correlation length of the infinite cluster L c and not the distance between adjacent sites l 0 (of the order of structural sizes 2 Å) [20]. L c may be understood as the typical (average) distance between barriers belonging to the percolation threshold n c [21]. Assuming the self-similar structure of the percolation cluster [22], it is derived that just a fraction of the inner sites will participate in dc conduction. Skal and Shoklovskii [23] proposed that within a box of size L c there is only one chain of bonds connecting high energy barriers n c. Effective transport occurs through the sublattice made up of threshold points rather than the microscopic lattice of structural spacing. This view implies that the dimensionality of the conducting path is approximately equal to one, although this has been recognized as a rough simplification [22]. In fact the dimensionality of the connected sites within the percolation cluster scales as L dc for L 6 L c (here L stands for the spatial coordinate), being d c 1.3 for three-dimensional systems because there actually exists a fraction of faster sites making up blobs (dense regions with more than one link between two points) [22,24]. A useful picture for the material electrical response as derived from the percolation approach could be as follows: material volume is split into conducting and polarizing regions (see Fig. 2) in such a way that both regions share the same size, i.e. the cluster length. Then the structural length responsible for both polarization and transport is determined by the underlying energetic clustering [21]. This is in fact a microscopic explanation of the BNN relation. The situation resembles that encountered in analyzing the conductivity of composites or mixed systems made up of insulating particles dispersed in an ionically conducting medium [25], but now the material is considered structurally homogeneous, the separation between polarizing zones and conducting paths being caused by percolation. The picture is also thoroughly consistent with the equivalent circuit representations [26], where the dielectric relaxation associated with the occurrence of De is placed in parallel with the dc conductivity. L c crossing point, ξ c percolation path polarizing region Fig. 2. Schematic representation of the effective percolation sublattice for ion conduction through crossing points belonging to the percolation threshold n c. Dimensionality of the conducting path is stated as one for the sake of simplicity. Polarizing regions develop within volumes limited by conducting paths. From our approach, it comes out that the dielectric strength De arises from the response to the electrical field of the polarizing zones, i.e. the clusters. Ions within polarizing regions in the cluster rearrange by effect of the applied electrical field. The occurrence of a perturbed concentration profile near-equilibrium (cluster polarization) is then considered as responsible for the observed De. A cluster is therefore seen as a nanoscopic dipole. We will next calculate a general expression for the cluster polarizability. From basic theory, it is stated that the electrochemical potential ~l before the electrical field application equals the chemical potential l, and it is related to the composition derivative of the free energy per site ~l ¼ l ¼ og ox : ð4þ Let us assume that ions remain basically in equilibrium after reaching the steady state under electrical field application. This entails that there is no change in the electrochemical potential but a change in the ion concentration profile, so as to achieve ~l 0 ¼ ~l. Now the electrochemical potential can be written as follows: ~l 0 ¼ l 0 efz ¼ og0 efz: ox ð5þ Here e represents the positive elementary charge, F stands for the electrical field which is assumed to be constant, and z is the spatial coordinate in the electrical field direction. Eq. (5) can be used to calculate the position-dependent composition function given that l 0 = l + efz. The composition becomes a function of the chemical potential by assuming the specific equilibrium statistics x = f(l) ofthe material. The elementary dipolar moment p associated with a cluster is determined by the ion rearrangement x 0 (z) with respect to the profile before electrical field application x,

4 4 G. Garcia-Belmonte et al. / Chemical Physics xxx (2006) xxx xxx which it is assumed to correspond to the molar alkali content. p ¼hx 0 ezi hxezi ¼ e Z Lc Z Lc x 0 zdz xzdz : ð6þ L c 0 0 By assuming that the electrical field introduces a small perturbation of the concentration profile one can restrict the calculation to the first-order expansion as x 0 ¼ f ðl 0 Þ¼fðlÞþ of of F : ð7þ The partial derivative is rewritten in useful terms as of/ of =(of/ol)ez. The elementary dipolar moment then results p ¼ e2 FL 2 c of : ð8þ 3 ol The material polarizability P is defined as the product of the elementary dipolar moment p and the dipole density N. By considering that all the volume contributes to the polarization the cluster density is given by N ¼ L 3 c, and therefore the polarization becomes P ¼ Np ¼ e2 F of : ð9þ 3L c ol Finally, we obtain an expression for the dielectric strength because in the linear regime it is known that P = e 0 DeF. Therefore, the dielectric strength takes the general form e 0 De ¼ e2 3L c of ol : ð10þ Note that the term of/ol relates to the particularities of the ion thermodynamics. For ions obeying simple Fermi Dirac statistics, 1 x ¼ ; ð11þ 1 þ exp E 0 l k B T as in the case of random-barrier models for which E 0 stands for the bottom of the potential wells, Eq. (10) can be easily evaluated and simply results, e 0 De ¼ e2 xð1 xþ : ð12þ 3k B TL c Note here that Eq. (12) is in good agreement with expressions derived from percolation theory for dc conductivity that also assumed such ion statistics [7], r 0 ¼ e2 xð1 xþ x c : ð13þ k B TL c By comparing Eqs. (12) and (13) one obtains a value of m = 3 for the BNN relation in Eq. (1). It is important to stress that the dielectric strength derivation presented here is based in purely equilibrium arguments whereas previous percolative calculations relied on the application of the Kramers Kronig relationship to the frequency-dependent conductivity [19]. 3. Discussion Assuming a random barrier model, i.e. same bottom energy level for all available sites for hopping, we can give a value corresponding to the percolation length variation with alkali content (Fig. 3). It should be stressed that the use of Eq. (12) must be taken with caution. The actual free-energy landscape that moving ions experience is unknown. Hence the proper calculation by means of Eq. (10) is simply impossible. This entails that the conclusions we are able to give are rather speculative. In any case, the percolation approach predicts the existence of cluster structures of nanometric size governing both the long-range transport and the excess of polarization. The size of such structures can vary with the amount of modifier ions in different, and even opposite, fashions depending on the samples. Various behaviors can also be found when investigating the temperature dependence of De. In most cases the dielectric strength is considered as almost temperature independent whereas it has also been reported for germanates [15] to be proportional to 1/T. Such behavior can be ascribed to different forms of the term of/ol in Eq. (10). For instance, simply by assuming a temperature-dependent concentration of mobile ions the term x(1 x)/t in Eq. (12) can change the derivative ode/ot, which would result positive, negative or roughly constant depending on the considered temperature domain. It must be noticed here that the temperature dependence of De will always appear negligible in comparison to the temperature activation of r 0. This leads to the conclusion that r 0 and x c have roughly the same activation energy, in accordance with the BNN relation. Concerning glass structuring, it is well known that borate glasses develop super-structures of intermediate range order (IRO) like boroxol ring, pentaborate, tetraborate, diborate groups, etc. The concentration of these Fig. 3. Average percolation length calculated by assuming ions obey Fermi Dirac statistics within polarizing cluster regions (Eq. (11)). Same data as Fig. 1. Solid lines indicate dependences of the form L c / blnx.

5 G. Garcia-Belmonte et al. / Chemical Physics xxx (2006) xxx xxx 5 IRO structures depends heavily on the alkali modifier content and temperature. At temperatures below T g they can be considered as stable. It has been reported that smaller structures like boroxol rings are progressively substituted by larger ones as the Na content increases from 10 mol% up to 40 mol% [27]. The other structural feature of importance for borate glasses is their tendency toward immiscibility [28]. For sodium content below 26 mol% Na 2 O, it is known that glasses develop small sodium-rich immiscibility regions in a B 2 O 3 -rich matrix phase. The size of these regions changes with the preparation method, particularly with the cooling rate after melting. As recently demonstrated [29], the Na 2 O concentration determines the formation of IRO in the melt. Therefore small fluctuations in the sodium concentration can lead to develop complex spatial distributions of IRO, owing to the tendency toward immiscibility [29]. Accordingly, developed IRO structures can strongly influence alkali content dependence of L c (and De) by modifying in a rather complex fashion the energy barrier landscape. The previous comments lead us to consider that only by combining information extracted from electrical as well as structural analyses (NMR and Raman spectroscopy) a deeper knowledge about the underlying phenomenon responsible for the dielectric relaxation strength will be accessible. An alternative way to check the polarization model presented would be the analysis of computer simulations in which both the energy landscape and the ion statistics are regarded as inputs of the problem. For instance, Porto and co-workers [11] studied the conductivity spectra resulting from a Gaussian site-energy disorder model, assuming Fermi Dirac statistics for mobile ions. Using a percolation approach these authors concluded that conductivity data fairly collapse into a single curve for typical concentration values and sufficiently low temperature. Unfortunately, such analysis did not explore the variation of the permittivity on concentration and temperature at low enough frequencies. The simulation approach could be highly informative when ion statistics departs from the Fermi Dirac function. When Coulomb interactions are introduced in the formulation of the system energy [30], the term of/ol in Eq. (10) may vary abruptly and exert a great influence on the variation of De. It should be noted that the term of/ol can be found from a series of grand canonical Monte Carlo simulations using a fluctuation approach [31], and De relates to the low-frequency limit of the imaginary part of the ac complex conductivity which is derived from the time-dependent diffusion coefficient [11]. 4. Conclusion We have analyzed the dielectric relaxation strength of several ionic conductive glasses. In particular, the alkali content and temperature dependence of De has been interpreted using a percolation approach. The model assumes that De arises from the rearrangement of mobile ions into polarizing regions or clusters. It has been emphasized that the variation of De results from the structural complexity, i.e. inhomogeneity in the cation distribution throughout the glass. Taking such structural complexity into account, we are inclined to relate the variety in the cluster size variation with composition to specific structural features of each particular glass. 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