Theoretical Investigation of the Electrocaloric Properties of Lead-free Ferroelectric Ceramic

BaTi 0.91 Sn 0.09 O 3 (BST) sample was prepared by the solid-state reaction method. The structural, morphological and electrocaloric properties were studied. The sample crystallized in the tetragonal structure with P4mm space group. Based on mapping image, the sensitivity and spatial resolution of the different elements in our sample were improved. According to the variation of polarization as function of temperature, our sample had a paraelectric-ferroelectric phase transition, around room temperature. The electrocaloric properties of our sample were investigated using theoretical approaches. The important parameters such as maximum entropy change, relative cooling power and full width at half maximum were explained qualitatively. These results make our sample promising candidate for refrigeration domain.


Introduction
The refrigeration market has grown considerably because of the constant expansion of the industry, rising living standards, and climate change [1]. This has resulted in a lack of control over consumer energy expenditure. It should be noted that the extensive use of refrigeration is a major factor of excessive energy consumption resulting in the depletion of non-renewable energy resources, which exacerbates the effect of global warming. Nowadays, the open debate is mainly focused on the energy transition towards a green development focused on protecting the environment, preserving human health and reducing global warming. The necessity to improve energy performance has become a major concern for industrial and scientific communities. This fragment of innovation has, therefore, become under great pressure to produce more sustainable technological solutions predicated on promising cooling In this work, we present a detailed study of the structure, morphology and ECE for the BaTi0.91Sn0.09O3 sample. Here, ECE was simulated based on a phenomenological model. This simulation allowed us to predict EC properties under different applied electric fields. The theoretical model predicted the experimental results in order to minimize the cost of experimental study. Here, different EC parameters were explained such as; example entropy (−∆S E ), relative cooling power (RCP), heat capacity (∆C P,E ) and temperature changes (∆T).

Experimental Details
BaTi0.91Sn0.09O3 polycrystalline sample was prepared by the solid-state reaction method. They used high purity precursors with high purity were BaCO3, TiO2, SnO2. The initial powder was prepared by grounding starting materials in ethanol with an agate mortar for 2 h. The resulting powder was calcined in two stages: at 900°C for 24 h and then at 1200°C for 12 h. Between the two stages the powder was well ground and for the second stage the powder was calcined in pellet form of 12 mm diameter. The obtained powder was again ground for 2 h and was pressed under a pressure of 5 tons per cm 2 into pellets with a diameter of 8 mm. Subsequently, these pellets were sintered at 1400°C for 2h to get dense ceramic. The pellet sample was immersed in a thermostatic oil bath. X-ray diffraction (XRD) pattern of the sintered BST ceramic was recorded on a Philips diffractometer using CuKα radiation (λ = 1.54056 Å) in the angle range 20° ≤ 2θ ≤ 90°. The microstructure was determined, at room temperature, by a scanning electron microscope (SEM) using a TS QUATA 250. The compositions of our sample was obtained by a semi-quantitative analysis performed at 15 kV accelerating voltage using energy dispersive X-ray analysis (EDX). Field emission scanning electron microscopy (FE-SEM) (JSM-7500, Japan) was used to investigate the surface morphologies and element mapping of the studied sample. The hysteresis measurements were determined for different electric fields and different temperatures by current amplifier Keithley 428 and high voltage amplifier TREK Model 20/20C at a switching frequency of 1 Hz. The polarization (P) versus temperature (T) data for different fields was determined from hysteresis loops.

Results and discussion
To describe the structural properties of our sample, we carried out XRD analysis, at room temperature. Fig.1 shows the dependence of XRD patterns of BST ceramic. It crystallized in the tetragonal structure with P4mm space group with cell parameters: a = b = 6.854 (2) Å and Regarding the SEM micrograph, the inset (a') of Fig. 2. (a), shows that our sample featured a relatively dense microstructure. The average particle size of the compounds was estimated using Image J software. Then, we can adjusted the data obtained with the log-normal function [22]: where 0 and σ are the median diameter obtained from the SEM and data dispersions, respectively. The inset (a'') of Fig. 2. (a) shows the dispersion histogram. The mean diameter ) and stand deviation =< > [ 2 − 1] 1 2 were determined using fit results. The average particle size is summarized in Table II. In addition, Fig.3. (b) shows the composition dependence of element mapping of BST ceramic, in order to clearly display the distribution of elements. One can see that all elements are homogenously distributed into the ceramics matrix.

Theoretical considerations
The present work was undertaken to measure ECE, near room temperature, in BST ceramic. A phenomenological model was applied to predict significant EC properties of our sample. where ρ is the mass density of the sample.

The variation of P vs.T and TC is presented by
At T = TC, -∆S E becomes maximum so that Eq. 3 may be written as follows: Another very important parameter for magnetic refrigeration is RCP, which present the product of ∆S Max E and δT FWHM . It is defined as: Here, δTFWHM is an expression of the extent of a function, given by the difference between the two extreme values of the independent variable at which the dependent variable is equal to half of its maximum value. It can be defined as: A polarization-related change of heat capacity is given by: According to this phenomenological model, a change of heat capacity is given by:

Simulation
In order to apply the phenomenological model, the numerical calculations were performed with parameters displayed in Table. 1. Fig. 3 shows P vs. T for our studied sample under different electric field values, ranging from 0 to 30 kV.cm -1 . The symbols signify the experimental data and the red lines indicate the modeled data based on the model given by Eq. 2. We note that the modeled data are in good agreement with the experimental ones. We can point out that P presents a continuous change, for different electric fields applied, around TC and that TC increases considerably with the increase of the electric field. One of the main parameters of the ECE is which has been was plotted for different applied electric fields based on Eq. 3. Fig. 4 shows that −∆S E is positive for all the applied electric fields and throughout the temperature range, which confirms the FE character [26].−∆S max is an important parameter that is spotted around TC. These values of for different applied fields are summarized in Table. 2. In the framework of electrocaloric refrigeration, it is essential to take into account two other parameters, having the same importance of −∆S max E , namely RCP and δTFWHM, which are defined in Eqs. 5 and 6, respectively. All EC parameters are recorded in Table. 2 and are compared to those found in other works [27][28][29][30].The TC values for each applied electric field are close to the ambient temperature, which allows us to suggest that our sample can be considered as a promoter candidate for the electrocaloric refrigeration.  Table. 2.
To verify the nature of the magnetic phase transition, it is necessary that the plots ΔSM(T), measured indifferent µ0H, should collapse on a single curve with a second-order phase transition, which suggested by Franco et al. [32,33]. So by analogy with the MCE, the universal phenomenological ∆S ′ curve can be determined by the normalization of −∆S E [34,35]: Therefore, to construct the universal curve, it is important to resize the temperature axis, below and above TC, by a new parameter on two clearly separated reference temperatures, represented by the following equation: whereθ, Tr1 and Tr2 are, respectively, the rescaled temperature and the temperatures lower and higher than TC of each curve which should satisfy the relation ∆S E (T r 1,2 ) = −∆S max The curves of ΔS ' (θ) for different applied electric fields are shown in Fig. 6. It is worth noting that all the data are dispersed on a single universal curve, which indicates that the nature of the transition in our sample is second-order [36].

Conclusion
To sum up, BaTi0.91Sn0.09O3 sample was prepared by solid-state method. Based on mapping image, the sensitivity and spatial resolution of the different elements in our sample were improved. According to polarization vs. temperature analysis, our sample admits a FE-PE phase transition around room temperature. ECE properties of our studied sample were investigated.
The different EC parameters were comparable to those obtained in the literature. This allows our sample to present as a potential candidate in the refrigeration domain.     Absolute values of electrocaloric entropy changes vs. temperature, obtained by Eq. (4) at different electric eld.

Supplementary Files
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