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August 11, 2009

Barium titanate stannate functionally graded materials


Functionally graded materials (FGMs) are established as an attractive class of materials in which it is possible to create a gradient of properties that cannot be attained in any spatially-homogeneous materials. FGMs have properties which vary as a function of position; continuous changes in the characteristics such as: chemical composition, grain size, porosity, etc., result in the gradient of electrical and/or magnetic features, and also, in enhanced structural performances, such as mechanical and thermal expansion. FGMs have been used for the fabrication of various technological components, such as electrical devices (piezoelectric ceramics, thermoelectric semiconductors, etc.), electrochemical ones (for example solid oxide fuel cells - SOFC, or high-efficiency hybrid direct energy conversion systems - HYDECS), as well as biomaterials.

An important processing goal for FGMs is to obtain high-quality microstructure with desired grain size and density. The fabrication of FGMs by powder technology is accompanied by significant problems of component shape distortion. During the thermal treatment, graded layers in FGM show different shrinkage rates and extents of shrinkage during sintering, as well as different final density. This phenomenon can lead to excessive shape distortion, warping, delamination, development of cracks and micro-structural damage in the sintered FGMs. Therefore, it is desirable to predict the sintering process for every graded layer in FGM and design sintering strategies to achieve high-quality FGM free from any form of deformation. 

Quantitative predictions of the sintering process of most systems of industrial importance and, furthermore, designing strategies to control sintering in order to prepare high-quality products, can be achieved through the concept of the master sintering curve (MSC).

The MSC is defined as the relationship between experimental density (ρ) and Θ:



For a constant heating rate, the Θ is:


where: t is instantaneous time, which is a function of temperature, c is the heating rate, Ea is the activation energy, R is the gas constant and T is the absolute temperature, and To is the temperature below which no sintering takes place.

Thus, for the construction of MSC the integral of Eq. (2) and the experimental density should be known; a series of runs at different temperatures or constant heating rate over a range of heating rates are needed.

We used the concept of MSC to estimate the activation energy for the sintering for barium titanate stannate (BaTi1-xSnxO3, BTS [1]) graded layers in BaTi0.975Sn0.025O3/BaTi0.85Sn0.15O3 (noted as BTS2.5/BTS15) functionally graded material. Generally, because of relatively high dielectric permittivity in a wide temperature range and lead-free relaxor behavior, BTS FGMs have practical application as electroceramics in electronic industry (i.e. ceramic capacitors, bending actuators, microwave phase shifters, sensors, etc.). Moreover, the electrical characteristics of BTS FGMs can be tailored by modifying tin/titanium concentration gradient [2,3].

Table 1 The main characteristics of used BTS powders.




Nominal Sn content (mol%)
Crystal structure
Theoretical density (g/cm3)
Crystallite size (nm)
V (A3)
Average particle size (nm)
Average green density (%)



The BTS2.5/BTS15 FGMs were fabricated by the powder-stacking method and uniaxially-pressing process, followed by sintering in air up to 1420°C [4]. The FGMs were sintered in a heating microscope in order to determine the sintering shrinkage. The shrinkage data obtained at four constant heating rates, 2, 5, 10 and 20°/min was used for the construction of the MSCs. The diameters of the investigated samples were photographed during the sintering process at defined time intervals, and measured on appropriate images - Fig. 1 (a). Fig.1 (b) shows schematically the layered structure of the samples.


Fig. 1 (a) Photograph of sintered cylindrical sample as observed in heating microscope and (b) scheme of uni-axially pressed layered sample (after sintering denoted as BTS FGM), with marked diameters measured during sintering (Image: J. Eur. Ceram. Soc. 29 (2009) 2309-2316).


From experimental data for dbottom (BTS2.5), and dtop (BTS15) recorded during the sintering and using Eq. (3), the percentage of shrinkage was calculated for the diameter of graded BTS layers in FGMs:


where Δd denotes the difference between the initial value of diameter do (at time to) and the values of di (at time ti).

The calculated values of shrinkage were used to determine the sintering behavior of FGMs graded layers. In Fig. 2, the sintering behavior of FGMs was represented by the shrinkage curves of the diameter of each graded layer, BTS2.5 (dbottom), and BTS15 (dtop) versus temperature during sintering.

Fig. 2 Shrinkage of BTS graded layers in FGMs during different heating rates of sintering: (a) 2; (b) 5; (c) 10 and (d) 20°/min (Image: J. Eur. Ceram. Soc. 29 (2009) 2309-2316).


According to Fig. 2, it can be noticed that the kinetics of densification of the graded layers within the FGMs depend on the chemical composition and the heating rate. However, regardless the heating rate, there is always a difference in shrinkage between graded layers BTS2.5 and BTS15. Graded layer BTS2.5 shows a higher value of shrinkage than graded layer BTS15 in the FGMs. Moreover, the shrinkage of graded layer BTS2.5 occurs at the lower temperature interval. The different sintering shrinkage of the graded layers BTS2.5 and BTS15 is attributed to the difference in stoichiometry, and can be explained by different sintering activation energy. Thus, it is necessary to estimate the sintering activation energies for the two graded layers, and to find whether they are suitable for the preparation of deformation free FGMs.

The values of experimental relative density that are necessary for the construction of MSCs, were calculated from the shrinkage values using the Eq. (4):


where r and ro are the densities of the sintered and green layer, respectively.

Fig. 3 shows relative density vs. temperature for (a) BTS2.5, and (b) BTS15 graded layers in FGMs sintered up to 1420°C at the heating rates of 2, 5, 10 and 20°C/min.


Fig. 3 Relative density (% TD) versus temperature for: (a) graded layer BTS2.5 and (b) graded layer BTS15, in BTS2.5/BTS15 FGM (Image: J. Eur. Ceram. Soc. 29 (2009) 2309-2316).

Once the experimental density values are determined, for the further construction of MSC, we have to calculate the values of Θ. One of the essential data for the calculation of Θ is activation energy, Ea. If the activation energy is unknown, it can be estimated with good precision from Θversus ρ data. For this purpose, a particular value of activation energy should be chosen and ρ-Θ curves should be constructed for each heating rate. If the curves fail to converge, a new value of activation energy should be chosen and the calculations repeated. This procedure should be continued until all the curves converge, showing which activation energy is acceptable for sintering. A curve can be then fitted through all the data points, and then the convergence of the data to the fitted line can be quantified through the sum of residual squares of the points with respect to the fitted line. The best estimate of Ea will be the value of the minimum in the plot of the activation energy versus mean residual squares.

Therefore, the experimental density data for graded layers and values calculated from Eq. (2) are used for the estimation of Ea. Firstly, the ρ-Θ curves were constructed for BTS2.5 graded layer, for the four heating profiles, for a chosen value of activation energy (200 kJ/mol) as shown in Fig. 4 (a). It can be seen that the curves for different heating rates are not converging. So, a new value of Ea was chosen and the calculation was repeated. The curves for 300, 400 and 500 kJ/mol are shown in Fig. 4 (b)-(d), respectively.

Fig. 4 , for different Ea, for graded layer BTS2.5 (Image: J. Eur. Ceram. Soc. 29 (2009) 2309-2316).

The same procedure was repeated for graded layer BTS15. The ρ-Θ curves were constructed for a chosen value of activation energy (200, 300, 400 and 500 kJ/mol) as shown in Fig. 5 (a)-(d).

Fig. 5 , for different Ea, for graded layer BTS15 (Image: J. Eur. Ceram. Soc. 29 (2009) 2309-2316).

Fig. 6 presents the mean residual squares for the activation energies, used for the construction of plots , for graded layers BTS2.5 and BTS15. After fitting, the value of 359.5 kJ/mol was obtained as the minimum for BTS2.5 graded layer, while the value of 340.5 kJ/mol was obtained as the minimum for BTS15 graded layer.

From the knowledge of the sintering activation energies, MSCs for graded layers BTS2.5 and BTS15 were constructed and are shown in Fig. 7.

Fig. 6 Mean residual square versus Ea for graded layers
BTS2.5 and BTS15 in BTS2.5/BTS15 FGM (Image: J. Eur. Ceram. Soc. 29 (2009) 2309-2316).
Fig. 7 MSCs () of BTS2.5
and BTS15 graded layers in BTS2.5/BTS15 FGM (Image: J. Eur. Ceram. Soc. 29 (2009) 2309-2316).
According to these two very close and similar MSCs for graded layers BTS2.5 and BTS15 in BTS2.5/BTS15 FGM, and due to the estimated results for sintering activation energies (359.5 and 340.5 kJ/mol, respectively), delamination or distortion of FGMs are not expected during sintering up to 1420°C. These assumptions were confirmed by a SEM analysis of BTS2.5/BTS15 FGMs in cross-section view, as shown in Fig. 8 (white dashed line marks the boundary region of the FGM). There is no distinct boundary between layers in BTS2.5/BTS15 FGM (Fig. 8a). Also, there is no significant change of densification behavior nor microstructure in the boundary region (Fig. 8b). Clearly, delamination or distortion of FGMs does not exist. The absence of delamination and/or distortion of BTS2.5/BTS15 FGMs are provided by the use of similar effective activation energies of BTS2.5 and BTS15.
Fig. 8 Scanning electron micrographs showing: (a) macroscopic shape of the FGM and (b) magnifier BTS2.5/BTS15 boundary region. SEMs were done on polished cross-section of BTS2.5/BTS15 FGM sintered at 1420°C, heated by rate of 10°/min (Image: J. Eur. Ceram. Soc. 29 (2009) 2309-2316).

Electrical properties of BTS FGMs

The electrical characteristics of several different BTS FGMs, as well as microstructural and/or macrostructural defects were determined by an impedance spectroscopy (IS) analysis, using an Gamry EIS300 Impedance Analyzer at frequencies of 1 Hz-100 kHz. Measurements were done in cooling from 320 to 25°C [5].


Fig. 9 IS spectra of BTS FGMs fired at 1420°C, measured in air at room temperature [5].


IS spectra of BTS FGMs measured at room temperature are shown in Fig. 9. No point dissipation is observed, indicating high-quality ceramics. This is especially important for FGMs, meaning no insulator interfaces (cracks and/or delamination) between graded layers were formed during the processing and high-temperature sintering of the powders. High specific resistivity, ρdc, of the BTS FGMs at room temperature (Table 2) indicates low leakage currents; making these materials suitable for practical application as capacitors.

Table 2 Electrical and microstructural characteristics of BTS FGMs [5].









In order to separate grain and grain boundary contribution of the ceramics' specific resistivity, the IS data collected above 210°C were analyzed. Fig. 10 shows the impedance spectra of: (a) 2.5/15 FGM and (b) its ingredients (BTS2.5 and BTS15) at temperatures between 320 and 210°C. The impedance spectra of all the investigated FGMs are very similar, exhibiting two arcs -- the high-frequency arc is ascribed to grain-interior resistance, ρgi, while the low-frequency arc is ascribed to grain boundary resistance, ρgb. In addition, the resistivity of grain-interior and grain boundary increase with decreasing temperature.

Fig. 10 IS spectra and microstructure of (a), (d) BTS2.5; (b), (e) 2.5/15 FGM; and (c), (f) BTS15, sintered at 1420°C [5].
The recorded IS spectra were fitted in Z-View2 computer program, by an equivalent circuit consisting of two parallel RCPE elements connected in series (Fig. 11); as output values of ρgi and ρgb were obtained.
Fig. 11 Model of equivalent circuit [5].

The values of specific resistivities were converted to specific conductivities; separately for grain-interior (σgi = 1/ρgi) and grain boundary (σgb = 1/ρgb), and used for the calculation of activation energies. Apropos, ionic conductivity in BTS occurs due to the migration of oxygen ions through oxygen vacancies, which are produced due to slight loss of oxygen during sintering at 1420°C.

The data for ionic conductivity were fitted as a function of temperature T following the Arrhenius law:

σ= (σo/T) exp(-Ea/kBT) (5)

where Ea is the activation energy for ionic migration, kB is the Boltzmann constant, and σo is a constant related to the density of charge carriers.

Arrhenius plots of grain-interior and grain boundary conductivity, for BTS FGMs are shown in Fig. 12.

Fig. 12 Arrhenius plots of ln(σT) versus 1000/T of different FGMs. Contributions of grain interior and grain boundary are separated [5].

All plots obey the Arrhenius law and therefore activation energies (Egi and Egb) can be estimated from the slopes of these diagrams. The values of the activation energy determined by the least square fitting are listed in Table 2. The activation energy for grain boundary ionic conductivity is typically higher than that for the grain-interior transport, for BTS FGMs.

The activation energies estimated from grain-interior conductivity are very similar, ranging from 0.74 to 0.78 eV. Since Egi is determined by chemical composition, it can be concluded that the concentration gradient in FGMs does not influence the intrinsic conductivity.

Quite the contrary, the activation energy for grain boundary conductivity is different for different FGMs; Egb vary in the range of 1.03 to 1.29 eV, influenced by the varying concentration gradient. The highest Egb of 1.29 eV, exhibits FGM 2.5/15, material that possesses the highest concentration and microstructural/porosity gradient. Air gaps on the contact between grains of different stoichiometry were produced due to different theoretical density of the used powder, so high porosity gradient is produced in applied sintering conditions. These air gaps act as potential barriers for the transposrt of oxygen ions. Accordingly, FGM 15/5/7 with the smallest concentration, microstructural and porosity gradient, has the smallest Egb (1.03 eV). Therefore, Egb is influenced by microstructural/porosity gradient, which are direct consequences of concentration gradient. Precisely, it is deduced that an increase of concentration gradient promotes the increase of microstructural and porosity gradients, as well as the increase of Egb.



[1] S. Marković, M. Mitrić, N. Cvjetićanin and D. Uskoković, "Structural and dielectric properties of BaTi1-xSnxO3 ceramics", Mater. Sci. Forum  518 (2006) 241-246.

[2] S. Marković, M. Mitrić, Č. Jovalekićand M. Miljković, "Dielectric and ferroelectric properties of BaTi1-xSnxO3 multilayered ceramics", Mater. Sci. Forum  555 (2007) 249-254.

[3] S. Marković, M. Mitrić, N. Cvjetićanin and D. Uskoković, "Preparation and properties of BaTi1-xSnxO3 multilayered ceramics", J. Eur. Ceram. Soc. 27 (2007) 505-509, DOI:10.1016/j.jeurceramsoc.2006.04.066.

[4] S. Markovićand D. Uskoković, "The master sintering curves for BaTi0.975Sn0.025O3/BaTi0.85Sn0.15O3 functionally graded materials", J. Eur. Ceram. Soc. 29 (2009) 2309-2316, doi:10.1016/j.jeurceramsoc.2009.01.027.

[5] S. Marković, Č. Jovalekić, Lj. Veselinović, S. Mentus and D. Uskoković, "Electrical properties of barium titanate stannate functionally graded materials", to be published.

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Dr. Smilja Marković Institute of Technical Sciences of SASA, Belgrade
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Milica Ševkušić (On-line editor)
Institute of Technical Sciences of SASA, Belgrade
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