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Stochastic optimization of ferroelectric ceramics for piezoelectric applications

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Abstract

An optimization model and numerical framework is developed to identify the optimal microstructure of ferroelectric (FE) materials. Piezoelectricity in polycrystalline ceramic FEs differs significantly from that of single crystals because of the presence of crystallites (grains) possessing crystallographic axes aligned imperfectly. The polarization and for that matter the piezoelectric properties of FEs are inextricably related to the orientation of individual crystallographic grains that constitute the polycrystal. The orientation of the grains fall in a Gaussian distribution after the poling process. The orientation distribution parameters, (viz., the standard deviation σ and mean μ) hence, would play the crucial role in the effective piezoelectric properties of FE polycrystals. We have applied a stochastic optimization combined with homogenization to determine the optimal distribution parameters which dictates the orientation distribution (texture) of an ideal ferroelectric polycrystal. This procedure would be used to find out the optimum piezoelectric properties characterised by the piezoelectric coefficients \(e_{\it ijk}\). The method is applied with an objective to maximize the piezoelectricity of ferroelectric BaTiO3 which in turn maximizes the electromechanical coupling. Convergence studies are made in order to arrive at a meaningful representative volume element for the computation. Focussing on the polar component of piezoelectric coefficient, e 33, the optimization course is perfected. Apparent enhancement of piezoelectric coefficient \(e_{\it ijk}\) is observed in an optimally oriented BaTiO3 single crystal. In polycrystalline ceramics, optimal grain configurations that shows better piezoelectric performance than polar ferroelectrics are derived. Our optimization model provides designs for materials with better piezoelectric performance, which would stimulate further studies involving materials possessing higher spontaneous polarization.

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Notes

  1. Throughout this text, we have used Einstein summation convention that repeated indices are implicitly summed over.

  2. Nonpolar in the sense that a direction other than the spontaneous polarization direction.

  3. μ θ qualifies as a design variable owing to the freedom of global rotation (e.g., Jayachandran et al. 2008) of the polycrystal which would in turn changes the average of Euler angles.

  4. Since the strains \(\varepsilon_{\it jk}\) are symmetric, \(e_{\it ijk}=e_{\it ikj}\) which implies that \(e_{\it ijk}\equiv e_{i\nu}\). Here we use the matrix (or compressed) notation where the Latin indices run from 1 to 3 and Greek indices run from 1 to 6 (Nye 1985).

  5. Individual perturbations.

  6. Nonetheless, the percentage deviations from the mean values of all the e iν s are similar.

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Acknowledgement

KPJ acknowledges the award of Ciência2007 by FCT, Portugal.

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  1. IDMEC, Instituto Seperior Técnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    K. P. Jayachandran, Jose M. Guedes & Helder C. Rodrigues

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  1. K. P. Jayachandran
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  2. Jose M. Guedes
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  3. Helder C. Rodrigues
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Correspondence to K. P. Jayachandran.

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Jayachandran, K.P., Guedes, J.M. & Rodrigues, H.C. Stochastic optimization of ferroelectric ceramics for piezoelectric applications. Struct Multidisc Optim 44, 199–212 (2011). https://doi.org/10.1007/s00158-011-0626-y

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