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Diffusion in Concentrated Jahn–Teller Systems
Diffusion in Concentrated Jahn– Teller Systems A. Ya. Fishman,1 M. A. Ivanov,2 T. E. Kurennykh,3 A. S. Lahtin,1 L. L. Surat4 and V. B. Vykhodets3 1
Institute of Metallurgy, Ural Branch of Russian Academy of Sciences, Amundsen Str. 101, Ekaterinburg 620016, Russia 2 Institute of Metal Physics, National Academy of Sciences of Ukraine, Vernadskogo Str. 36, Kiev-142 03680, Ukraine 3 Institute of Metal Physics, Ural Branch of Russian Academy of Sciences, S. Kovalevskaya Str. 18, Ekaterinburg 620219, Russia 4 Institute of Solid State Chemistry, Ural Branch of Russian Academy of Sciences, Pervomaiskaya Str. 91, Ekaterinburg 620218, Russia
Abstract The effect of strontium on the diffusion coefficients D of oxygen tracers in the La12y Sry MnO32d , temperature dependence of the oxygen tracer diffusion coefficient in La12y Sry MnO32d ð y ¼ 0:025Þ were determined. The D values were measured using 18O tracers and the nuclear microanalysis method. Analysis of the concentration dependences of the diffusion coefficients D was made using the approach, which takes into account variation of the number of orbitally degenerate (JT) states of 3d-ions during non-isovalent alloying. Contents 1. Introduction 2. Experimental: samples and methods 3. Experimental results 4. Theory 5. Conclusions Acknowledgements References
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1. INTRODUCTION The study of oxygen diffusion in oxides with mixed-valence 3d-ions presents great interest both in theoretical and practical terms. Such systems with Jahn –Teller (JT) 3d-cations are suitable model objects for analysis of the diffusion process in degenerate or pseudo-degenerate condensed systems. The mechanism of multi-well potential formation has been explored well for JT ions [1,2] and it is possible to give a simple microscopic description of the inter-center interactions and different properties of these systems. The practical interest paid to diffusion properties of the ADVANCES IN QUANTUM CHEMISTRY, VOLUME 44 ISSN: 0065-3276 DOI 10.1016/S0065-3276(03)44032-X
q 2003 Elsevier Inc. All rights reserved
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said oxides (including the La12ySryMnO32d compound with a perovskite structure) is determined by the use of mixed-conductivity oxides in separators to convert available hydrocarbons into more desirable products, in electrochemical reactors, for the separation of oxygen from mixed gases, in chemical sources of current, to utilize materials for many catalysis applications, etc. It was found that the oxygen self-diffusion coefficients can be changed by several orders of magnitude by alloying of the cation sublattice of perovskites (see, for example, Ref. [3]). Unfortunately, theoretical concepts, which would allow the prediction of variation regularities of the oxygen diffusion coefficients, have not been formulated so far. This situation is explained to a large extent by the lack of experimental data, which can be related reliably to the effect of JT interactions on the diffusion coefficients. Let us clarify this statement in terms of a universally acknowledged model when the diffusion activation energy Q is viewed as the sum of the energy of vacancy formation in the oxygen subsystem Qf and the migration energy Qm. It is known that alloying of the metal sublattice with lower-valence cations may lead to formation or an increase in the concentration of structural vacancies and, correspondingly, an increase in the oxygen diffusion coefficients. This process is observed, for example, in Sm12ySryCoO32d, Sm12yCdyCoO32d [4], La12ySryCoO32d [5] and La12ySryFeO32d [6]. When Sr2þ or Cd2þ ions are added instead of Sm3þ, Fe3þ or La3þ, some oxygen sites (proportional to d ¼ y=2) become vacant and the contribution of Qf to Q decreases or reduces to zero. Such substitution should have a strong effect on the diffusion coefficients, because the activation energy Q ø Qm usually is , 1 eV for oxygen self-diffusion in oxide structures with anion structural vacancies and Q . 2 eV for diffusion with thermal vacancies [7]. Other weaker mechanisms, through which substitutional atoms (including JT ones) influence the activation energy and the oxygen diffusion coefficients, can hardly be distinguished against this background. Remarkably, the above examples refer to systems, in which only one mechanism of charge compensation, that is, formation of structural vacancies, can be realized during alloying. It is known that an alternative mechanism of charge compensation, which is related to appearance of different valence states of 3dcations, operates in some other systems upon alloying. One of such valence states of the cation in an octahedral configuration is, as a rule, orbitally degenerate. The number of these orbitally degenerate (JT) states of 3d-ions is changed during nonisovalent alloying. This is accompanied by a rearrangement of the crystal structure and variation of magnetic conduction, and other properties. One of the systems, in which all these effects are pronounced most, is La12ySryMnO32d (see, for example, Refs. [8 –10]). Therefore, it may be expected that variation regularities of the diffusion coefficients in strontium-alloyed La12ySryMnO32d will differ from those observed in other perovskite systems. Unfortunately, experimental evidence obtained for the La12ySryMnO32d system proved to be contradictory. Firstly, the oxygen self-diffusion coefficients D, which were determined by the secondary ion mass spectroscopy (SIMS method), differed by several orders of magnitude from those evaluated from the chemical diffusion coefficients [5] (the latter were
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measured using a potentiostatic step technique). Secondly, strontium had an absolutely different effect on the diffusion coefficients: D increased with growing strontium concentration in SIMS experiments and decreased in potentiostatic experiments. Considering what has been said above, the main goals of this study were as follows: obtain reliable data concerning the effect of strontium on the diffusion coefficients D of oxygen tracers in the La12ySryMnO32d system; analyze concentration dependences of the diffusion coefficients D using the approach, which takes into account variation of the number of orbitally degenerate (JT) states of 3d-ions during non-isovalent alloying. In view of the theoretical nature of this study (analysis of the effect of JT displacements on the oxygen diffusion coefficients), we thought it was reasonable to take strontium-diluted solutions. The D values were measured using 18O tracers and the nuclear microanalysis method, which provides, similarly to SIMS, a direct determination of concentration profiles of diffusing atoms.
2. EXPERIMENTAL: SAMPLES AND METHODS Powders of La12ySryMnO32d oxides (y ¼ 0; 0:025; and 0.075) were prepared by annealing of a mixture of pure initial reagents La2O3, Mn2O3 and SrCO3 at 900 8C for 20 h, at 1000 8C for 50 h and at 1300 8C for 15 h, one after the other. The powders were mixed thoroughly before and after the annealings. X-ray diffraction indicated that the samples had perovskite structure. The concentrations of metal components in the samples were not controlled. They were calculated by weighing the initial powders. The powder was pressed in cylinders 10 mm in diameter and 3 mm high. The durability of the samples was provided by their annealing at 1300 8C for 6 h. The samples were cooled in air to 650 8C, were homogenized additionally at this temperature for 6 h, and were cooled slowly to room temperature. Special tests for porosity and microcracks were not performed, but relevant qualitative information was obtained from the analysis of concentration profiles c(x) of 18O tracers after diffusion annealing. The samples contained some quantity of pores or microcracks. This circumstance in combination with some other factors narrowed considerably the temperature interval for measurement of the volume diffusion coefficients in the present study. At temperatures below 500 8C, the profiles c(x) generally were not connected with the volume diffusion, but were due to diffusion of oxygen atoms along pores and microcracks or grainboundary diffusion. Therefore, the study was performed only in the temperature interval from 850 to 950 8C. The volume diffusion zone was identified reliably for this interval. Diffusion annealings were performed in a quartz tube under an oxygen atmosphere enriched to 70% with the 18O isotope at a pressure of 0:21 £ 105 Pa: The isotope composition of the gaseous phase was maintained constant throughout
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the annealing time. Immediately before the diffusion annealing, the samples were annealed in the same quartz tube in air. The latter annealing process was longer than the diffusion annealing, while temperatures of the two annealing processes were equal. When changing to the diffusion annealing, the air atmosphere in the quartz tube was replaced quickly by the oxygen atmosphere with the 18O isotope. This procedure eliminated an oxygen concentration gradient (16O þ 18O) in the diffusion zone throughout the annealing time. The annealed samples were cooled for about 1 min. A chromel –alumel thermocouple measured the temperature to within ^ 1 8C. The quartz tube was placed in a bulky metal cylinder during annealing. As a result, the zone with the samples had no temperature gradients within the measurement accuracy. The concentration of oxygen isotopes in the samples was measured using the nuclear microanalysis technique [11,12]. A 2-MV Van der Graaf accelerator was used. The sample was fitted in an evacuated chamber and was exposed to a beam of accelerated deuterons or protons having an energy of 900 and 762 keV, respectively. The flat surface of the samples was perpendicular to the axis of the primary beam. The recording angle of nuclear reaction products was 1608. Energy spectra of reaction products were measured using a surface-barrier silicon detector about 10 mm in diameter having an energy resolution of 20 keV. The detector was covered with an absorber (a mylar film) of backscattered deuterons and protons 16 and 10 mm thick, respectively. The diameter of the incident beam was 1– 2 mm, i.e., the measured concentrations were averaged over a large number of crystallites. The number of the incident beam ions was determined using a secondary monitor [13] to within 0.5%. The oxygen concentration was measured to a depth of , 1.5 mm without destruction of the samples. The 16O (d,p1) 17O and 18 O (p,a) 15N reactions were used. Profiles of the 16O isotope were measured only to check the total (16O þ 18O) oxygen concentration in the diffusion zone. The corresponding data are not reported below. The concentration profile was calculated by comparing the spectra of the test sample and a standard sample containing a constant oxygen concentration (the Fe3O4 oxide was used). The concentrations were determined using data on the hydrogen and helium stopping powers in elements. The stopping powers for the test oxides were calculated using tabulated data for pure components and Bragg’s additive rule [14]. The meanroot-square measurement error of the oxygen concentrations did not exceed a few percent. The method for calculation of the concentration profiles was described in more detail elsewhere [15 – 17]. A typical concentration profile of 18O oxygen is shown in Fig. 1. It is characterized by some specific features as compared to a simple case of the diffusion equation for a constant source. Firstly, it is seen that the concentration cs on the surface of the sample is lower than the equilibrium concentration in the test samples. Therefore, a solution of the diffusion equation depending on the boundary conditions must be used [18]. Secondly, the concentration c1 of the 18O isotope was almost constant at a large depth x and was much higher than the initial concentration of the 18O isotope in samples. It is not improbable that this
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Fig. 1. Penetration plot obtained after 1.23 h anneal at 900 8C for La12ySryMnO32d ðy ¼ 0:025Þ: c1 is the concentration of 18 O at a large depth; cs is the surface concentration of 18 O.
feature of the profile c(x) was determined not only and not so much by the grainboundary diffusion as by the presence of pores and cracks in the samples. Correctly substantiated methods for processing of experimental data are lacking in this case. Therefore, in what follows, the c1 values were subtracted from the experimental profiles c(x) and the data CðxÞ ; cðxÞ 2 c1 were fit to theoretical solutions of the diffusion equations in order to calculate the diffusion parameters. Such solutions were for a constant source: x C ¼ C0 erfc pffiffiffiffi ð1Þ 2 Dt and a solution of the diffusion equation depending on boundary conditions: pffiffiffiffi x x C ¼ C0 erfc pffiffiffiffi 2 expðHx þ H 2 DtÞerfc pffiffiffiffi þ H Dt ð2Þ 2 Dt 2 Dt where C0 is an equilibrium 18O concentration in the samples if an oxygen enriched to 70% with the 18O isotope to use; H is a fitting parameter, accounting the influence of surface potential barriers on diffusion. From general considerations, the use of equation (2) seems to be more reasonable. However, we used both approximations (1) and (2) to understand how the choice of boundary conditions affected the D value. The data in Fig. 2 (these data were sufficiently typical) and in Table 1 show that boundary conditions had a weak effect on the diffusion coefficients. Therefore, results obtained with the approximation (2) will be discussed below. It may be noted, however, that the approximation
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Fig. 2. (a) Penetration plot (top) obtained after 1.23 h anneal at 900 8C for system La12ySryMnO32d ðy ¼ 0:025Þ: Solid line represents a fit to solution of diffusion equation, accounting for volume (equation (1)). (b). Penetration plot (top) obtained after 1.23 h anneal at 900 8C for system La12ySryMnO32d ðy ¼ 0:025Þ: Solid line represents a fit to solution of diffusion equation, accounting for volume and surface exchange limited (equation (2)).
Table 1. Oxygen diffusion experimental results for La12ySryMnO32d Composition La12ySryMnO32d
Temperature (8C)
Time (h)
Da (cm2/s)
Db (cm2/s)
y¼0
900
2
2.21 £ 10213
3.22 £ 10213
y ¼ 0:025
850 850 870 900 900 950
2 3.72 1.82 1.23 3.23 0.48
7.87 £ 10214 6.58 £ 10214 1.68 £ 10213 1.96 £ 10213 1.24 £ 10213 5.47 £ 10213
1.28 £ 10213 9.94 £ 10214 2.24 £ 10213 2.87 £ 10213 1.97 £ 10213 8.27 £ 10213
y ¼ 0:075
900
2
9.64 £ 0214
1.49 £ 10213
a
Calculated from equation (1). Calculated from equation (2).
b
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(2) described the experimental results more adequately. This fact showed up as a lower value of the normalized average deviation from the analytical curve in the case of equation (2).
3. EXPERIMENTAL RESULTS As was already noted in the foregoing discussion, diffusion coefficients of oxygen atoms in perovskites, which were measured in various studies, differ by several orders of magnitude and, therefore, cannot be systematized. Let us consider this problem more comprehensively in connection with the obtained results. From Fig. 3 and Table 1 it is seen that a discrepancy between results of two independent measurements on samples having the same compositions at 1173 and 1123 K is systematic: a longer annealing time t corresponds to a lower D value. Moreover, in both cases the D1t1/D2t2 ratios are very close (indices 1 and 2 correspond to two independent measurements at one and the same annealing temperature). On one hand, the observed differences (1.46 times at 1173 K and 1.3 times at 1123 K) may be viewed as standard for ceramic samples [5]. On the other hand, they
Fig. 3. Temperature dependence of the oxygen tracer diffusion coefficient in the crystals La12ySryMnO32d ðy ¼ 0:025 – circleÞ: Also shown are this work data for La12ySryMnO32d ðy ¼ 0:075 – squareÞ and SIMS data for La0.65Sr0.35MnO32d (diamond) [21] and extrapolated Arrhenius diffusivity plot for La0.9Sr0.1MnO32d (lower line) [22].
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are much larger than the expected scatter of experimental D values if one takes into account a high accuracy of the concentration profiles, which were measured using the nuclear microanalysis technique. We believe that the systematical scatter of the diffusion coefficients is due to an improper fulfillment of boundary conditions of the diffusion equations. In turn, an improper fulfillment is connected with a poor quality of ceramic samples. Considering these factors, we thought it was possible to use different criteria for comparison of diffusion measurement results. If the diffusion coefficients are measured at largely different annealing times and, more so, in samples of different quality, a 2-fold difference of the measured values is quite regular and reasonable for ceramic samples. Otherwise, in this study it was reasonable to head for the measurement accuracy of the diffusion coefficients equal to a few percent. In terms of these criteria, the D values determined in this study agree well with all literature data known to us, which were obtained for ceramic samples with a similar composition using the SIMS method (Fig. 3). Therefore, this group of data on the volume diffusion coefficients of oxygen tracers in La12ySryMnO32d may be assumed to be reliable. The main result of the experimental part of the study, which is shown in Fig. 4, confirms the supposition, which was made in the Introduction above, that one cannot expect formation of structural vacancies in the oxygen sublattice of strontiumalloyed La12ySryMnO32d perovskites and, correspondingly, a considerable increase
Fig. 4. Variation of the oxygen diffusion coefficient D with y in La12ySryMnO32d.
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in the diffusion coefficients of oxygen tracers. Moreover, it may be viewed as a reliably established fact that D values decrease with growing concentration of strontium. Let us note two additional factors pointing to reliability of this result. The samples with y ¼ 0 and y ¼ 0:075 were annealed simultaneously ðt ¼ 2 hÞ; i.e., these data are free of systematic errors caused by different temperatures and annealing times. The scatter of the D values for the compound with y ¼ 0:025 will be much smaller if we take into account a different duration of the diffusion anneal of these samples in accordance with the condition D1 t1 =D2 t2 ø const: Thus, the result obtained for D(y) is due only to a decrease in the concentration of JT ions in the strontium-alloyed samples (some Mn3þ ions change their valence to Mn4þ). It will be used below when we discuss the effect of JT deformations on the oxygen diffusion coefficients.
4. THEORY Considering the experimental findings presented above, we shall assume that when Sr2þ is substituted for La3þ, the mechanism of charge compensation is connected with a change in the valence of Mn3þ ions to Mn4þ, rather than formation of structural vacancies. In this case, the effect of alloying on the diffusion coefficients and the diffusion activation energy is not related to the vacancy formation energy and is determined largely by the variation of the concentration ratio of Mn3þ and Mn4þ ions. In high-temperature crystalline La12ySryMnO32d the mean (quantum-statistical) value of JT deformations of the nearest oxygen neighbors of Mn3þ ions is zero in the absence of random crystalline fields. In this case, a multi-well character of the potential at a JT ion may be reflected in the diffusion process if the oxygen neighbors of JT ions – a multi-level system – do not manage to relax (average themselves thermodynamically) during the time of a diffusion jump. This ratio between the relaxation time of a degenerate center and the diffusion jump time is possible only if splitting of degenerate levels is relatively small. If, on the contrary, splitting energies are large and, correspondingly, the ratio between the said times is reverse, the JT characteristics are lost to a large extent. A relevant theoretical analysis of the diffusion coefficients of anions in crystals was performed on the assumption that an equilibrium occupancy of degenerate levels is not established during a diffusion jump in the JT subsystem [18]. In this case, the symmetry of the anion environment of the diffusing atom at the saddle point is determined by a random configuration of vibronic states of the nearest JT centers. Displacements of anions from symmetric sites, which correspond to these states, cause changes in the form of the diffusion potential barrier and, hence, the activation energy of jump frequencies. Consequently, configurations appear, leading to both a decrease and an increase in the activation energy as compared to the case
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when JT displacements of anions in the crystal lattice are absent: X Wi exp{ 2 DEi =kB T} ø D0 exp{ðD=2kB TÞ2 } D ¼ D0
ð3Þ
i
where Wi is the probability of an ith configuration of vibronic states of JT centers nearest to the saddle point; DEi is a change of the potential barrier height in the ith configuration caused by displacements of anions; D is the dispersion of the distribution DEi, D 0 is the diffusion coefficient in the absence of JT interactions.For simplicity, we shall restrict ourselves to analysis of changes in the activation energy DEi in terms of the elastic model of diffusion potential barriers [19,20]: ð4Þ DEi ¼ Keff veff ðRipol 2 Ranion Þ2 2 ðR0pol 2 Ranion Þ2 =R2anion where R ipol is the passage radius in an ith configuration; R 0pol is the same radius in a configuration not distorted by JT displacements; Ranion is the radius of a diffusing anion; Keff and veff denote effective values of the elastic modulus and the volume describing the deformation work during the diffusion migration. If the difference between dimensions of the anion and the passage at the saddle point, Ranion 2 R0pol is comparable with local JT displacements uJT , ð1021 – 1022 ÞRanion ; the D value should increase considerably as compared to D 0 at temperatures kB T # DðD $ 1021 eVÞ: Let us turn to the possibility of observing this effect. Most frequently, the diffusion of anions may be observed in high-symmetry phases of cooperative JT systems having a relatively high temperature of the JT transition TD , ð102 – 103 ÞK: The cooperative JT effect ensures diffusion in a crystal through fast channels of the anion migration, while a relatively high temperature of the structural JT transition provides local JT displacements, which are large enough. One may expect that non-isovalent substitutions in these systems should cause a considerable decrease in the diffusion coefficients. This is due to a drop in the total number of degenerate centers and a simultaneous decrease in the concentration of degenerate centers having a relatively long relaxation time. Then for dilute systems with y ,, 1 the following value of ratio between the diffusion coefficients D(y)/D(0) takes place: DðyÞ=Dð0Þ ø 1 2 yz; where z is some coordination number. As is seen from Fig. 4, for y ¼ 0:075 that value agrees fairly well with an experimental result Dðy ¼ 0:075Þ=Dðy ¼ 0Þ ¼ 0:46 at a reasonable values of z (for example, Dð0:075Þ=Dð0Þ ¼ 0:4 at z ¼ 8). When some critical concentration of substitutional centers is reached, the percolation through centers (saddle points) with a relatively small potential barrier should stop. This change of the diffusion coefficient with the concentration should correlate with the inhibition of the structural phase transformation during alloying. It is not inconceivable that similar situation took place at temperatures higher than the temperature of the structural phase transformation in a pseudo-JT perovskite-like La22ySryCuO4 system [3], where in the region of critical concentration for structural phase transition the oxygen diffusion coefficient was changing anomalously.
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5. CONCLUSIONS 1. The obtained diffusion data confirmed the existing ideas about mechanisms of charge compensation in oxides with 3d-ions. When the cation sublattice is alloyed with lower-valence ions, formation of anion structural vacancies and a large increase in the diffusion coefficients of oxygen atoms should be observed only if the cation sublattice is free of ions whose valence may increase upon alloying. Otherwise, the diffusion coefficients should not increase considerably upon alloying in the region of dilute solutions. 2. It was shown that the presence of degenerate or pseudo-degenerate states in the systems may have a considerable effect on the oxygen diffusion coefficients in oxides. In the high-temperature (high-symmetry) crystalline phase this effect is due to the influence of random JT deformations on parameters of diffusion potential barriers. In terms of the elastic model of formation of potential barriers, the effect is determined by a change (caused by JT deformations) of the size of the cavity, through which a diffusing atom passes at the saddle point. If dimensions of the cavity and the diffusing atom differ little, even small deformations may considerably influence the diffusion coefficients. 3. A correlation was predicted in terms of this model between the behavior of the concentration dependence of the temperature of the JT (or pseudo-JT) structural phase transition and the concentration dependence of the oxygen diffusion coefficient.
ACKNOWLEDGEMENTS The work is performed at support of Russian Foundation for Basic Research, grant 00-03-32362.
REFERENCES [1] F. S. Ham, Jahn – Teller effects in electron paramagnetic resonance spectra, in Electron Paramagnetic Resonance (ed. S. Geshwind), Plenum Press, New York, 1972. [2] M. D. Sturge, Solid State Phys., 1967, 20, 91. [3] S. J. Rothman, J. L. Routbort, J. L. Novicki, K. C. Goretta, L. J. Thompson, J. N. Mundy and J. E. Baker, Def. Diff. Forum, 1989, 66–69, 1081. [4] R. Doshi, J. L. Routbort and C. B. Alcock, Def. Diff. Forum, 1995, 127 and 128, 39. [5] J. L. Routbort, R. Doshi and M. Krumpelt, Solid State Ionics, 1996, 90, 21. [6] T. Ishigaki, S. Yamauchi, K. Kishio, J. Mizusaki and K. Fueki, J. Solid State Chem., 1988, 73, 179. [7] M. Runde, J. L. Routbort, J. N. Mundy, S. J. Rothman, C. L. Wiley and X. Xu, Phys. Rev., 1992, B46, 3142. [8] (a) V. E. Naish, Fiz. Met. Metalloved., 2001, 92 (5), 5; (b) V. E. Naish, Fiz. Met. Metalloved., 1998, 85 (6), 5. [9] E. L. Nagaev, Uspehi Fizicheskih Nauk, 1996, 166 (8), 833. [10] V. M. Loktev and Yu. G. Pogorelov, Fiz. Nizkih Temp., 2000, 26, 231.
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[11] G. Amsel and D. Samuel, Anal. Chem., 1967, 39, 1689. [12] J. M. Mayer and E. Remini, Ion Beam Handbook for Materials Analysis, Academic Press, New York, 1977. [13] V. N. Volkov, V. B. Vykhodets, I. K. Golubkov, S. M. Klotsman, P. V. Lerkh and V. A. Pavlov, Nucl. Instr. Methods, 1983, 205, 73. [14] V. B. Vykhodets, S. M. Klotsman and A. D. Levin, Fiz. Met. Metalloved., 1987, 64, 920. [15] J. F. Ziegler (ed.), The Stopping and Ranges of Ions in Matter, Pergamon Press, New York, 1977. [16] V. B. Vykhodets, S. M. Klotsman, T. E. Kurennykh, L. D. Kurmaeva, A. D. Levin, V. A. Pavlov, M. A. Plekhanov and L. V. Smirnov, Fiz. Met. Metalloved., 1987, 63, 974. [17] V. B. Vykhodets, I. K. Golubkov, S. M. Klotsman, T. E. Kurennykh, G. N. Tatarinowa and A. N. Timofeer, Fiz. Met. Metalloved., 1988, 66, 303. [18] M. A. Ivanov, E. A. Pastukhov and A. Ya. Fishman, Doklady Academii Nauk, 2002, 387, 1. [19] Y. A. Bertin, J. Parisot and J. L. Gacougnolle, J. Less-Common Met., 1980, 69, 121. [20] J. D. Eshelby, Solid State Phys., 1956, 3, 79. [21] S. Carter, A. Selcuk, R. J. Chater, J. Kajda, J. A. Kilner and C. H. Steel, Solid State Ionics, 1992, 53–56, 597. [22] J. L. Routbort, J. Wolfenstine, K. C. Goretta, R. E. Cook, T. R. Armstrong, C. Clauss and A. Dominguez-Rodniguez, Diff. Dif. Forum, 1997, 143–147, 1201.
Institute of Metallurgy, Ural Branch of Russian Academy of Sciences, Amundsen Str. 101, Ekaterinburg 620016, Russia 2 Institute of Metal Physics, National Academy of Sciences of Ukraine, Vernadskogo Str. 36, Kiev-142 03680, Ukraine 3 Institute of Metal Physics, Ural Branch of Russian Academy of Sciences, S. Kovalevskaya Str. 18, Ekaterinburg 620219, Russia 4 Institute of Solid State Chemistry, Ural Branch of Russian Academy of Sciences, Pervomaiskaya Str. 91, Ekaterinburg 620218, Russia
Abstract The effect of strontium on the diffusion coefficients D of oxygen tracers in the La12y Sry MnO32d , temperature dependence of the oxygen tracer diffusion coefficient in La12y Sry MnO32d ð y ¼ 0:025Þ were determined. The D values were measured using 18O tracers and the nuclear microanalysis method. Analysis of the concentration dependences of the diffusion coefficients D was made using the approach, which takes into account variation of the number of orbitally degenerate (JT) states of 3d-ions during non-isovalent alloying. Contents 1. Introduction 2. Experimental: samples and methods 3. Experimental results 4. Theory 5. Conclusions Acknowledgements References
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1. INTRODUCTION The study of oxygen diffusion in oxides with mixed-valence 3d-ions presents great interest both in theoretical and practical terms. Such systems with Jahn –Teller (JT) 3d-cations are suitable model objects for analysis of the diffusion process in degenerate or pseudo-degenerate condensed systems. The mechanism of multi-well potential formation has been explored well for JT ions [1,2] and it is possible to give a simple microscopic description of the inter-center interactions and different properties of these systems. The practical interest paid to diffusion properties of the ADVANCES IN QUANTUM CHEMISTRY, VOLUME 44 ISSN: 0065-3276 DOI 10.1016/S0065-3276(03)44032-X
q 2003 Elsevier Inc. All rights reserved
498
A. Ya. Fishman et al.
said oxides (including the La12ySryMnO32d compound with a perovskite structure) is determined by the use of mixed-conductivity oxides in separators to convert available hydrocarbons into more desirable products, in electrochemical reactors, for the separation of oxygen from mixed gases, in chemical sources of current, to utilize materials for many catalysis applications, etc. It was found that the oxygen self-diffusion coefficients can be changed by several orders of magnitude by alloying of the cation sublattice of perovskites (see, for example, Ref. [3]). Unfortunately, theoretical concepts, which would allow the prediction of variation regularities of the oxygen diffusion coefficients, have not been formulated so far. This situation is explained to a large extent by the lack of experimental data, which can be related reliably to the effect of JT interactions on the diffusion coefficients. Let us clarify this statement in terms of a universally acknowledged model when the diffusion activation energy Q is viewed as the sum of the energy of vacancy formation in the oxygen subsystem Qf and the migration energy Qm. It is known that alloying of the metal sublattice with lower-valence cations may lead to formation or an increase in the concentration of structural vacancies and, correspondingly, an increase in the oxygen diffusion coefficients. This process is observed, for example, in Sm12ySryCoO32d, Sm12yCdyCoO32d [4], La12ySryCoO32d [5] and La12ySryFeO32d [6]. When Sr2þ or Cd2þ ions are added instead of Sm3þ, Fe3þ or La3þ, some oxygen sites (proportional to d ¼ y=2) become vacant and the contribution of Qf to Q decreases or reduces to zero. Such substitution should have a strong effect on the diffusion coefficients, because the activation energy Q ø Qm usually is , 1 eV for oxygen self-diffusion in oxide structures with anion structural vacancies and Q . 2 eV for diffusion with thermal vacancies [7]. Other weaker mechanisms, through which substitutional atoms (including JT ones) influence the activation energy and the oxygen diffusion coefficients, can hardly be distinguished against this background. Remarkably, the above examples refer to systems, in which only one mechanism of charge compensation, that is, formation of structural vacancies, can be realized during alloying. It is known that an alternative mechanism of charge compensation, which is related to appearance of different valence states of 3dcations, operates in some other systems upon alloying. One of such valence states of the cation in an octahedral configuration is, as a rule, orbitally degenerate. The number of these orbitally degenerate (JT) states of 3d-ions is changed during nonisovalent alloying. This is accompanied by a rearrangement of the crystal structure and variation of magnetic conduction, and other properties. One of the systems, in which all these effects are pronounced most, is La12ySryMnO32d (see, for example, Refs. [8 –10]). Therefore, it may be expected that variation regularities of the diffusion coefficients in strontium-alloyed La12ySryMnO32d will differ from those observed in other perovskite systems. Unfortunately, experimental evidence obtained for the La12ySryMnO32d system proved to be contradictory. Firstly, the oxygen self-diffusion coefficients D, which were determined by the secondary ion mass spectroscopy (SIMS method), differed by several orders of magnitude from those evaluated from the chemical diffusion coefficients [5] (the latter were
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measured using a potentiostatic step technique). Secondly, strontium had an absolutely different effect on the diffusion coefficients: D increased with growing strontium concentration in SIMS experiments and decreased in potentiostatic experiments. Considering what has been said above, the main goals of this study were as follows: obtain reliable data concerning the effect of strontium on the diffusion coefficients D of oxygen tracers in the La12ySryMnO32d system; analyze concentration dependences of the diffusion coefficients D using the approach, which takes into account variation of the number of orbitally degenerate (JT) states of 3d-ions during non-isovalent alloying. In view of the theoretical nature of this study (analysis of the effect of JT displacements on the oxygen diffusion coefficients), we thought it was reasonable to take strontium-diluted solutions. The D values were measured using 18O tracers and the nuclear microanalysis method, which provides, similarly to SIMS, a direct determination of concentration profiles of diffusing atoms.
2. EXPERIMENTAL: SAMPLES AND METHODS Powders of La12ySryMnO32d oxides (y ¼ 0; 0:025; and 0.075) were prepared by annealing of a mixture of pure initial reagents La2O3, Mn2O3 and SrCO3 at 900 8C for 20 h, at 1000 8C for 50 h and at 1300 8C for 15 h, one after the other. The powders were mixed thoroughly before and after the annealings. X-ray diffraction indicated that the samples had perovskite structure. The concentrations of metal components in the samples were not controlled. They were calculated by weighing the initial powders. The powder was pressed in cylinders 10 mm in diameter and 3 mm high. The durability of the samples was provided by their annealing at 1300 8C for 6 h. The samples were cooled in air to 650 8C, were homogenized additionally at this temperature for 6 h, and were cooled slowly to room temperature. Special tests for porosity and microcracks were not performed, but relevant qualitative information was obtained from the analysis of concentration profiles c(x) of 18O tracers after diffusion annealing. The samples contained some quantity of pores or microcracks. This circumstance in combination with some other factors narrowed considerably the temperature interval for measurement of the volume diffusion coefficients in the present study. At temperatures below 500 8C, the profiles c(x) generally were not connected with the volume diffusion, but were due to diffusion of oxygen atoms along pores and microcracks or grainboundary diffusion. Therefore, the study was performed only in the temperature interval from 850 to 950 8C. The volume diffusion zone was identified reliably for this interval. Diffusion annealings were performed in a quartz tube under an oxygen atmosphere enriched to 70% with the 18O isotope at a pressure of 0:21 £ 105 Pa: The isotope composition of the gaseous phase was maintained constant throughout
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the annealing time. Immediately before the diffusion annealing, the samples were annealed in the same quartz tube in air. The latter annealing process was longer than the diffusion annealing, while temperatures of the two annealing processes were equal. When changing to the diffusion annealing, the air atmosphere in the quartz tube was replaced quickly by the oxygen atmosphere with the 18O isotope. This procedure eliminated an oxygen concentration gradient (16O þ 18O) in the diffusion zone throughout the annealing time. The annealed samples were cooled for about 1 min. A chromel –alumel thermocouple measured the temperature to within ^ 1 8C. The quartz tube was placed in a bulky metal cylinder during annealing. As a result, the zone with the samples had no temperature gradients within the measurement accuracy. The concentration of oxygen isotopes in the samples was measured using the nuclear microanalysis technique [11,12]. A 2-MV Van der Graaf accelerator was used. The sample was fitted in an evacuated chamber and was exposed to a beam of accelerated deuterons or protons having an energy of 900 and 762 keV, respectively. The flat surface of the samples was perpendicular to the axis of the primary beam. The recording angle of nuclear reaction products was 1608. Energy spectra of reaction products were measured using a surface-barrier silicon detector about 10 mm in diameter having an energy resolution of 20 keV. The detector was covered with an absorber (a mylar film) of backscattered deuterons and protons 16 and 10 mm thick, respectively. The diameter of the incident beam was 1– 2 mm, i.e., the measured concentrations were averaged over a large number of crystallites. The number of the incident beam ions was determined using a secondary monitor [13] to within 0.5%. The oxygen concentration was measured to a depth of , 1.5 mm without destruction of the samples. The 16O (d,p1) 17O and 18 O (p,a) 15N reactions were used. Profiles of the 16O isotope were measured only to check the total (16O þ 18O) oxygen concentration in the diffusion zone. The corresponding data are not reported below. The concentration profile was calculated by comparing the spectra of the test sample and a standard sample containing a constant oxygen concentration (the Fe3O4 oxide was used). The concentrations were determined using data on the hydrogen and helium stopping powers in elements. The stopping powers for the test oxides were calculated using tabulated data for pure components and Bragg’s additive rule [14]. The meanroot-square measurement error of the oxygen concentrations did not exceed a few percent. The method for calculation of the concentration profiles was described in more detail elsewhere [15 – 17]. A typical concentration profile of 18O oxygen is shown in Fig. 1. It is characterized by some specific features as compared to a simple case of the diffusion equation for a constant source. Firstly, it is seen that the concentration cs on the surface of the sample is lower than the equilibrium concentration in the test samples. Therefore, a solution of the diffusion equation depending on the boundary conditions must be used [18]. Secondly, the concentration c1 of the 18O isotope was almost constant at a large depth x and was much higher than the initial concentration of the 18O isotope in samples. It is not improbable that this
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Fig. 1. Penetration plot obtained after 1.23 h anneal at 900 8C for La12ySryMnO32d ðy ¼ 0:025Þ: c1 is the concentration of 18 O at a large depth; cs is the surface concentration of 18 O.
feature of the profile c(x) was determined not only and not so much by the grainboundary diffusion as by the presence of pores and cracks in the samples. Correctly substantiated methods for processing of experimental data are lacking in this case. Therefore, in what follows, the c1 values were subtracted from the experimental profiles c(x) and the data CðxÞ ; cðxÞ 2 c1 were fit to theoretical solutions of the diffusion equations in order to calculate the diffusion parameters. Such solutions were for a constant source: x C ¼ C0 erfc pffiffiffiffi ð1Þ 2 Dt and a solution of the diffusion equation depending on boundary conditions: pffiffiffiffi x x C ¼ C0 erfc pffiffiffiffi 2 expðHx þ H 2 DtÞerfc pffiffiffiffi þ H Dt ð2Þ 2 Dt 2 Dt where C0 is an equilibrium 18O concentration in the samples if an oxygen enriched to 70% with the 18O isotope to use; H is a fitting parameter, accounting the influence of surface potential barriers on diffusion. From general considerations, the use of equation (2) seems to be more reasonable. However, we used both approximations (1) and (2) to understand how the choice of boundary conditions affected the D value. The data in Fig. 2 (these data were sufficiently typical) and in Table 1 show that boundary conditions had a weak effect on the diffusion coefficients. Therefore, results obtained with the approximation (2) will be discussed below. It may be noted, however, that the approximation
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Fig. 2. (a) Penetration plot (top) obtained after 1.23 h anneal at 900 8C for system La12ySryMnO32d ðy ¼ 0:025Þ: Solid line represents a fit to solution of diffusion equation, accounting for volume (equation (1)). (b). Penetration plot (top) obtained after 1.23 h anneal at 900 8C for system La12ySryMnO32d ðy ¼ 0:025Þ: Solid line represents a fit to solution of diffusion equation, accounting for volume and surface exchange limited (equation (2)).
Table 1. Oxygen diffusion experimental results for La12ySryMnO32d Composition La12ySryMnO32d
Temperature (8C)
Time (h)
Da (cm2/s)
Db (cm2/s)
y¼0
900
2
2.21 £ 10213
3.22 £ 10213
y ¼ 0:025
850 850 870 900 900 950
2 3.72 1.82 1.23 3.23 0.48
7.87 £ 10214 6.58 £ 10214 1.68 £ 10213 1.96 £ 10213 1.24 £ 10213 5.47 £ 10213
1.28 £ 10213 9.94 £ 10214 2.24 £ 10213 2.87 £ 10213 1.97 £ 10213 8.27 £ 10213
y ¼ 0:075
900
2
9.64 £ 0214
1.49 £ 10213
a
Calculated from equation (1). Calculated from equation (2).
b
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(2) described the experimental results more adequately. This fact showed up as a lower value of the normalized average deviation from the analytical curve in the case of equation (2).
3. EXPERIMENTAL RESULTS As was already noted in the foregoing discussion, diffusion coefficients of oxygen atoms in perovskites, which were measured in various studies, differ by several orders of magnitude and, therefore, cannot be systematized. Let us consider this problem more comprehensively in connection with the obtained results. From Fig. 3 and Table 1 it is seen that a discrepancy between results of two independent measurements on samples having the same compositions at 1173 and 1123 K is systematic: a longer annealing time t corresponds to a lower D value. Moreover, in both cases the D1t1/D2t2 ratios are very close (indices 1 and 2 correspond to two independent measurements at one and the same annealing temperature). On one hand, the observed differences (1.46 times at 1173 K and 1.3 times at 1123 K) may be viewed as standard for ceramic samples [5]. On the other hand, they
Fig. 3. Temperature dependence of the oxygen tracer diffusion coefficient in the crystals La12ySryMnO32d ðy ¼ 0:025 – circleÞ: Also shown are this work data for La12ySryMnO32d ðy ¼ 0:075 – squareÞ and SIMS data for La0.65Sr0.35MnO32d (diamond) [21] and extrapolated Arrhenius diffusivity plot for La0.9Sr0.1MnO32d (lower line) [22].
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are much larger than the expected scatter of experimental D values if one takes into account a high accuracy of the concentration profiles, which were measured using the nuclear microanalysis technique. We believe that the systematical scatter of the diffusion coefficients is due to an improper fulfillment of boundary conditions of the diffusion equations. In turn, an improper fulfillment is connected with a poor quality of ceramic samples. Considering these factors, we thought it was possible to use different criteria for comparison of diffusion measurement results. If the diffusion coefficients are measured at largely different annealing times and, more so, in samples of different quality, a 2-fold difference of the measured values is quite regular and reasonable for ceramic samples. Otherwise, in this study it was reasonable to head for the measurement accuracy of the diffusion coefficients equal to a few percent. In terms of these criteria, the D values determined in this study agree well with all literature data known to us, which were obtained for ceramic samples with a similar composition using the SIMS method (Fig. 3). Therefore, this group of data on the volume diffusion coefficients of oxygen tracers in La12ySryMnO32d may be assumed to be reliable. The main result of the experimental part of the study, which is shown in Fig. 4, confirms the supposition, which was made in the Introduction above, that one cannot expect formation of structural vacancies in the oxygen sublattice of strontiumalloyed La12ySryMnO32d perovskites and, correspondingly, a considerable increase
Fig. 4. Variation of the oxygen diffusion coefficient D with y in La12ySryMnO32d.
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in the diffusion coefficients of oxygen tracers. Moreover, it may be viewed as a reliably established fact that D values decrease with growing concentration of strontium. Let us note two additional factors pointing to reliability of this result. The samples with y ¼ 0 and y ¼ 0:075 were annealed simultaneously ðt ¼ 2 hÞ; i.e., these data are free of systematic errors caused by different temperatures and annealing times. The scatter of the D values for the compound with y ¼ 0:025 will be much smaller if we take into account a different duration of the diffusion anneal of these samples in accordance with the condition D1 t1 =D2 t2 ø const: Thus, the result obtained for D(y) is due only to a decrease in the concentration of JT ions in the strontium-alloyed samples (some Mn3þ ions change their valence to Mn4þ). It will be used below when we discuss the effect of JT deformations on the oxygen diffusion coefficients.
4. THEORY Considering the experimental findings presented above, we shall assume that when Sr2þ is substituted for La3þ, the mechanism of charge compensation is connected with a change in the valence of Mn3þ ions to Mn4þ, rather than formation of structural vacancies. In this case, the effect of alloying on the diffusion coefficients and the diffusion activation energy is not related to the vacancy formation energy and is determined largely by the variation of the concentration ratio of Mn3þ and Mn4þ ions. In high-temperature crystalline La12ySryMnO32d the mean (quantum-statistical) value of JT deformations of the nearest oxygen neighbors of Mn3þ ions is zero in the absence of random crystalline fields. In this case, a multi-well character of the potential at a JT ion may be reflected in the diffusion process if the oxygen neighbors of JT ions – a multi-level system – do not manage to relax (average themselves thermodynamically) during the time of a diffusion jump. This ratio between the relaxation time of a degenerate center and the diffusion jump time is possible only if splitting of degenerate levels is relatively small. If, on the contrary, splitting energies are large and, correspondingly, the ratio between the said times is reverse, the JT characteristics are lost to a large extent. A relevant theoretical analysis of the diffusion coefficients of anions in crystals was performed on the assumption that an equilibrium occupancy of degenerate levels is not established during a diffusion jump in the JT subsystem [18]. In this case, the symmetry of the anion environment of the diffusing atom at the saddle point is determined by a random configuration of vibronic states of the nearest JT centers. Displacements of anions from symmetric sites, which correspond to these states, cause changes in the form of the diffusion potential barrier and, hence, the activation energy of jump frequencies. Consequently, configurations appear, leading to both a decrease and an increase in the activation energy as compared to the case
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when JT displacements of anions in the crystal lattice are absent: X Wi exp{ 2 DEi =kB T} ø D0 exp{ðD=2kB TÞ2 } D ¼ D0
ð3Þ
i
where Wi is the probability of an ith configuration of vibronic states of JT centers nearest to the saddle point; DEi is a change of the potential barrier height in the ith configuration caused by displacements of anions; D is the dispersion of the distribution DEi, D 0 is the diffusion coefficient in the absence of JT interactions.For simplicity, we shall restrict ourselves to analysis of changes in the activation energy DEi in terms of the elastic model of diffusion potential barriers [19,20]: ð4Þ DEi ¼ Keff veff ðRipol 2 Ranion Þ2 2 ðR0pol 2 Ranion Þ2 =R2anion where R ipol is the passage radius in an ith configuration; R 0pol is the same radius in a configuration not distorted by JT displacements; Ranion is the radius of a diffusing anion; Keff and veff denote effective values of the elastic modulus and the volume describing the deformation work during the diffusion migration. If the difference between dimensions of the anion and the passage at the saddle point, Ranion 2 R0pol is comparable with local JT displacements uJT , ð1021 – 1022 ÞRanion ; the D value should increase considerably as compared to D 0 at temperatures kB T # DðD $ 1021 eVÞ: Let us turn to the possibility of observing this effect. Most frequently, the diffusion of anions may be observed in high-symmetry phases of cooperative JT systems having a relatively high temperature of the JT transition TD , ð102 – 103 ÞK: The cooperative JT effect ensures diffusion in a crystal through fast channels of the anion migration, while a relatively high temperature of the structural JT transition provides local JT displacements, which are large enough. One may expect that non-isovalent substitutions in these systems should cause a considerable decrease in the diffusion coefficients. This is due to a drop in the total number of degenerate centers and a simultaneous decrease in the concentration of degenerate centers having a relatively long relaxation time. Then for dilute systems with y ,, 1 the following value of ratio between the diffusion coefficients D(y)/D(0) takes place: DðyÞ=Dð0Þ ø 1 2 yz; where z is some coordination number. As is seen from Fig. 4, for y ¼ 0:075 that value agrees fairly well with an experimental result Dðy ¼ 0:075Þ=Dðy ¼ 0Þ ¼ 0:46 at a reasonable values of z (for example, Dð0:075Þ=Dð0Þ ¼ 0:4 at z ¼ 8). When some critical concentration of substitutional centers is reached, the percolation through centers (saddle points) with a relatively small potential barrier should stop. This change of the diffusion coefficient with the concentration should correlate with the inhibition of the structural phase transformation during alloying. It is not inconceivable that similar situation took place at temperatures higher than the temperature of the structural phase transformation in a pseudo-JT perovskite-like La22ySryCuO4 system [3], where in the region of critical concentration for structural phase transition the oxygen diffusion coefficient was changing anomalously.
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5. CONCLUSIONS 1. The obtained diffusion data confirmed the existing ideas about mechanisms of charge compensation in oxides with 3d-ions. When the cation sublattice is alloyed with lower-valence ions, formation of anion structural vacancies and a large increase in the diffusion coefficients of oxygen atoms should be observed only if the cation sublattice is free of ions whose valence may increase upon alloying. Otherwise, the diffusion coefficients should not increase considerably upon alloying in the region of dilute solutions. 2. It was shown that the presence of degenerate or pseudo-degenerate states in the systems may have a considerable effect on the oxygen diffusion coefficients in oxides. In the high-temperature (high-symmetry) crystalline phase this effect is due to the influence of random JT deformations on parameters of diffusion potential barriers. In terms of the elastic model of formation of potential barriers, the effect is determined by a change (caused by JT deformations) of the size of the cavity, through which a diffusing atom passes at the saddle point. If dimensions of the cavity and the diffusing atom differ little, even small deformations may considerably influence the diffusion coefficients. 3. A correlation was predicted in terms of this model between the behavior of the concentration dependence of the temperature of the JT (or pseudo-JT) structural phase transition and the concentration dependence of the oxygen diffusion coefficient.
ACKNOWLEDGEMENTS The work is performed at support of Russian Foundation for Basic Research, grant 00-03-32362.
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[11] G. Amsel and D. Samuel, Anal. Chem., 1967, 39, 1689. [12] J. M. Mayer and E. Remini, Ion Beam Handbook for Materials Analysis, Academic Press, New York, 1977. [13] V. N. Volkov, V. B. Vykhodets, I. K. Golubkov, S. M. Klotsman, P. V. Lerkh and V. A. Pavlov, Nucl. Instr. Methods, 1983, 205, 73. [14] V. B. Vykhodets, S. M. Klotsman and A. D. Levin, Fiz. Met. Metalloved., 1987, 64, 920. [15] J. F. Ziegler (ed.), The Stopping and Ranges of Ions in Matter, Pergamon Press, New York, 1977. [16] V. B. Vykhodets, S. M. Klotsman, T. E. Kurennykh, L. D. Kurmaeva, A. D. Levin, V. A. Pavlov, M. A. Plekhanov and L. V. Smirnov, Fiz. Met. Metalloved., 1987, 63, 974. [17] V. B. Vykhodets, I. K. Golubkov, S. M. Klotsman, T. E. Kurennykh, G. N. Tatarinowa and A. N. Timofeer, Fiz. Met. Metalloved., 1988, 66, 303. [18] M. A. Ivanov, E. A. Pastukhov and A. Ya. Fishman, Doklady Academii Nauk, 2002, 387, 1. [19] Y. A. Bertin, J. Parisot and J. L. Gacougnolle, J. Less-Common Met., 1980, 69, 121. [20] J. D. Eshelby, Solid State Phys., 1956, 3, 79. [21] S. Carter, A. Selcuk, R. J. Chater, J. Kajda, J. A. Kilner and C. H. Steel, Solid State Ionics, 1992, 53–56, 597. [22] J. L. Routbort, J. Wolfenstine, K. C. Goretta, R. E. Cook, T. R. Armstrong, C. Clauss and A. Dominguez-Rodniguez, Diff. Dif. Forum, 1997, 143–147, 1201.