CONDENSED MATTER

STATISTICS OF SIZE DISTRIBUTION OF RARE EARTHAGGLOMERATIONS IN CERAMIC MATRIX

A. CALBOREANUa, E. DIMITRIUb, R. RAMERc

a National Institute of Physics and Nuclear Engineering, Bucharest, Romaniab REGO COM SRL, Bucharest, Romania

c School of Electrical Engineering and Telecommunication, University of South Wales,Sydney 2052, Australia

Received April 12, 2005

An image analysis of the scanning electron micrographs was performed inorder to study the concentration distribution of doped rare earth ions in dopedPT-type electroceramics. This distribution was further compared with predictions ofvarious statistical approaches. A good agreement with experimental data is obtainedfrom an application of Tsallis non-extensive statistics. This is in agreement withmany other clustering phenomena in nature.

INTRODUCTION

A piezoelectric material is composed from grains and pores with differentshapes and sizes. Each grain contains a number of domains with the netpolarization zero or non-zero in both un-poled or poled ceramics, respectively.[1] In a poled material the grains have at least one-grain boundary adjacent to agrain with different orientation. [2] Most of the research papers are focused onthe size of the grains and pores, on the various phases formed in the process ofthe crystallization to perovskite phase, on the domains orientation, because allthese influence the dielectric and piezoelectric properties of the materials. [3]Besides these influences, the homogeneity of the composition is of greatimportance not only on the electromechanical proprieties but even in thepreparation of a piezoelectric material. Even if the material is homogeneous atthe macroscopic scale, the microscopic, elemental distribution of a PZT- orPT-sample is not homogeneous [4, 5].

We present in this paper the variation of the elemental concentration in aEu-doped PT-type ceramics. A statistical analysis of experimental data of theconcentration distribution of Eu ions in the ceramic matrix was performed.

EXPERIMENTAL

A PT-type material with the composition Pb0.88Eu0.08Ti0.98Mn0.02O3 ++ 0.01% Bi2O3 was prepared from pure oxides by ceramic technique. The

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546 A. Calboreanu, E. Dimitriu, R. Ramer 2

powder was pressed in pellets of 10 mm diameter and sintered in the temperaturerange 1050°C–1200°C. The microstructural examination was performed onsamples sintered at 1050°C, using a Cameca SX50 microprobe equipped withfour wavelength dispersive spectrometers and one energy dispersive X-rayanalyzer. Elemental area maps have been obtained on sample areas of4.5 × 4.5 mm2 for the poled sample. On this area map path lines were randomlyselected and, along it, the elemental concentration has been recorded.

RESULTS AND DISCUSSIONS

The elemental concentration map and the elemental distribution on arandom line drawn on the concentration map for Eu, Pb and Ti are presented inthe Figs. 1, 3 and 4. A high non-uniformity in the concentration is observed for all

Fig. 1. – The distribution of Eu concentration(at %) on a scanned line.

elements. The stoichiometric concentration of Eu of our piezoelectric element is4.96 at %. The experimental Eu concentration in the ceramic matrix (see Fig. 1)is between 3 and 23 at %. This result shows that there are areas in the ceramic oflow concentration of Eu, such as the calculated compound cannot be obtained. Inother places greater amounts of Eu are observed. A part of the Eu (4.96%) isconsumed in order to obtain the compound desired, but the other part of it isagglomerated in clusters. In Fig. 2, one can see the map of the whole surfaceinvestigated, with frequent peaks of europium concentration, plotted in at %.Great fluctuation of the lead and the titanium concentrations are observed inFigs. 3 and 4. In the as prepared compound, acceptable concentrations are 17.74at % (62.864 wt%) for Pb, and 19.758 at % (16.016 wt%) for Ti. Theexperimental results show Pb concentrations between 50 and 80 wt%, and Ticoncentration between 11.5 and 19.7 wt%. It is obvious that in the as preparedmaterial coexist the phases with the desired composition together with other

3 Rare earth agglomerations in ceramic matrix 547

Fig. 2. – 3D distribution of Eu-concentration (at %).

Fig. 3. – The distribution of lead concentration (wt %). The global Pb concentration is 62.86 wt%.

compounds and non reacted raw materials. Such an inhomogeneous materialdoes not present piezoelectric properties, after poling, suggesting that the sinteringtemperature is too low. For this material, sintered at 1050°C, a new series ofinvestigations have been carried out. The Eu concentration was measured inabout 300 points, each one situated at a distance of 20 µm from another. Theconcentrations were selected in groups, and a statistical analysis of experimentaldata of the size distribution of Eu ions in the ceramic matrix was performed.

This statistics is important for both practical purposes and for answeringfundamental questions related to the statistics of some physical phenomena. It isclear from a simple examination of the data that larger sizes are more frequentlyrealized than one expects from the classical Boltzman distribution. This fact

548 A. Calboreanu, E. Dimitriu, R. Ramer 4

Fig. 4. – The distribution of titanium concentration (wt %). The global Ti concentration is 16.016%.

concords with many other physical phenomena such as the cluster formation. Infact it is a rather common feature not only in physics; it appears that wheneveragglomerations, diffusion, dissipation or fluctuations appear, they present peculiarfeatures not encompassed by the traditional statistics [6, 7]. The failure ofBoltzman-Gibbs statistics in description of a host of physical and non-physicalphenomena conducted to questioning the very basis of classical thermodynamics,namely the extensivity [8]. An attractive, simple approach was achieved byTsallis [9], who formulated a versatile relation capable to cover a large body ofphenomena depending on a parameter q that at the limit q = 1 renders theBoltzman statistics. If pi is the weight of a certain event, Tsallis generates theentropy of an ensemble Sq as

1 ( )1

qi

q

pS k

q

−=

−∑ ( )1 ,ip =∑ When 1q → qS S→ (Boltzman)

Obviously, the entropies of two subsystems of the ensemble are not additive, if qdiffers from the value 1:

S(A + B) = S(A) + S(B) + (1 – q)S(A)S(B)/k

If q >1, the tail of the distribution is favored.A first application of the Tsallis statistics to analyze the size distribution of

the clusters was made by Calboreanu [7, 10].Now, based on Tsallis surmise, we generate a probability of events of the

form:1/( 1)

0( ) [1 ( 1) ( )]qw n q n n − −= + − β −

where n is the size parameter. The non-extensivity parameter q as well as β andn0 are determined from a fit of the experimental data. The physical significanceof these numbers will be discussed elsewhere.

5 Rare earth agglomerations in ceramic matrix 549

The result of the fit, are shown in Fig. 5 along with the probability for eachsize. The numerical values of the parameters are:

q = 1.97β = 9.96n0 = 0.97

Fortunately, the size of agglomerations is smaller than the resolution of themicroscope, while the average distance between grains allows a clearidentification of grains. Only the statistical errors are indicated in the graph.

Fig. 5. – Eu grain size distributionon a randomly selected line path.

CONCLUSION

A statistical analysis of experimental data of the Eu distribution in theceramic matrix is performed, using the Tsallis formula. The Eu distributiondisplays a specific feature suggesting a non extensive behaviour of clusterformation. The favorisation of the larger size (longer tail), is common to most ofthe phenomena of this type (clustersization, fragmentation). Whether this isreally a new statistics, or somehow is contained in traditional statistics is still amatter of research and debate.

Our data can shed some light upon fundamental questions regarding thearrangements of particles in a given space, or matrix. An extension of such

550 A. Calboreanu, E. Dimitriu, R. Ramer 6

analysis to a variety of concentrations and type of impurity and material base ofceramics would certainly be of great importance for both theory and applications.

REFERENCES

1. Nava Setter, “ABC of Piezoelectricity and Piezoelectric Materials”, Piezoelectric Materialsfor the End User”, Conference Notes, Interlaken Switzerland, 24th–27th February, 2002.

2. R. D. Roseman, “High Temperature Poling Effects on Conductive Barium TitanateCeramics”, Ferroelectrics, 215, 31–45 (1998).

3. J. Ricote, C. Alemany, L. Pardo and C. E. Millar, “Microstructure – Properties Relationshipsin Samarium Modified Lead Titanate Piezoceramics – II. Dielectric, Piezoelectric andMechanical Properties”, Acta mater, 44, 1167–1179 (1996).

4. R. Ramer, E. Dimitriu, L. Boerasu, A. Iuga, “Microanalysis of Pb(Zr, Ti, W, Li)O3Piezoceramic”, Key Engineering Materials, 264–268, 1173–1176 (2004).

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2000.8. P. Levy, Theorie de l’Addition des Variables Aleatoires, GV- Paris, 1937.9. C. Tsallis, J. Stat. Phys, 52, 479 (1988).

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