An optimization-based computational model is proposed to study domain evolution in polycrystalline ferroelectrics composed of numerous grains, each of which consists of multiple domains. Domain switching is realized by an optimization process to minimize the free energy of each grain. Similar to phase field modeling, no priori domain-switching criterion is imposed in the proposed model. Moreover, by focusing on the volume fractions of domains only, the computational complexity of this model becomes much smaller and the domain textures evolution can be captured. Simulation results on both tetragonal and rhombohedral lead titanate zirconate ceramics illustrate the efficiency of this model. © 2010 American Institute of Physics.published_or_final_versio

Li, FX

Soh, AK

Zhou, XL

Applied Physics Letters

Crossref

An optimization-based computational model is proposed to study domain evolution in polycrystalline ferroelectrics composed of numerous grains, each of which consists of multiple domains. Domain switching is realized by an optimization process to minimize the free energy of each grain. Similar to phase field modeling, no priori domain-switching criterion is imposed in the proposed model. Moreover, by focusing on the volume fractions of domains only, the computational complexity of this model becomes much smaller and the domain textures evolution can be captured. Simulation results on both tetragonal and rhombohedral lead titanate zirconate ceramics illustrate the efficiency of this model.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000276794100051&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=8e1609b174ce4e31116a60747a720701Physics, AppliedSCI(E)EI9ARTICLE15null9

Li, F. X.

Zhou, X. L.

Soh, A. K.

Institutional Repository of Peking University

An optimization-based &quot;phase field&quot; model for polycrystalline   ferroelectrics

HKU Scholars Hub

Title An optimization-based "phase field" model for polycrystallineferroelectricsAuthor(s) Li, FX; Zhou, XL; Soh, AKCitation Applied Physics Letters, 2010, v. 96 n. 15Issued Date 2010URL http://hdl.handle.net/10722/65474Rights Applied Physics Letters. Copyright © American Institute ofPhysics.An optimization-based “phase field” model for polycrystalline ferroelectricsF. X. Li,1,2,a X. L. Zhou,1 and A. K. Soh3,b1State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University,Beijing 100871, China2Center for Applied Physics and Technology, Peking University, Beijing 100871, China3Department of Mechanical Engineering, The University of Hong Kong, Hong Kong Special AdministrativeRegion, ChinaReceived 10 January 2010; accepted 11 March 2010; published online 13 April 2010An optimization-based computational model is proposed to study domain evolution inpolycrystalline ferroelectrics composed of numerous grains, each of which consists of multipledomains. Domain switching is realized by an optimization process to minimize the free energy ofeach grain. Similar to phase field modeling, no priori domain-switching criterion is imposed in theproposed model. Moreover, by focusing on the volume fractions of domains only, the computationalcomplexity of this model becomes much smaller and the domain textures evolution can be captured.Simulation results on both tetragonal and rhombohedral lead titanate zirconate ceramics illustratethe efficiency of this model. © 2010 American Institute of Physics. doi:10.1063/1.3377899Ferroelectric materials have been widely used in modernindustries as sensors, actuators, transducers, etc. This type ofmaterials shows excellent linear response at low electric/stress fields but exhibits significant nonlinearities under alarge electric field or high stress due to domain switching.1Modeling of domain switching in ferroelectric materials hasreceived intensive attentions in recent years,2–11 amongwhich the phase field model PFM seems morepromising.6–11 No imposition of any priori domain-switchingcriterion is required in PFM, and domain switching is a natu-ral process during minimization of the whole material sys-tem’s total free energy. Although PFM has achieved greatsuccess in modeling domain evolutions in ferroelectric singlecrystals, it has hardly been used for modeling polycrystallineferroelectrics.9–11 In our recent works,12 domain switchingproblems in polycrystalline ferroelectrics have been ad-dressed by an analytical constrained domain-switchingmodel, in which a priori domain-switching criterion must beprescribed and only uniaxial loading is allowed to determinethe analytical solution. Thus, it is not suitable for a generalcomputational study of domain evolution in polycrystallineferroelectrics under arbitrary loading. In this letter, we pro-pose an optimization-based computational model to studydomain switching in polycrystalline ferroelectrics. Similar toPFM, no priori domain-switching criterion is imposed in theproposed model, whose computational complexity is, how-ever, much smaller than that of PFM and, therefore, it is ableto handle three-dimensional 3D cases where numerousgrains are modeled. Moreover, the domain texture evolutionprocess under applied loading can be illustrated.In the present model, a polycrystalline ferroelectric ismade up of numerous randomly oriented grains each ofwhich contains N types of domains where N=6 for the te-tragonal case and N=8 for the rhombohedral case, etc., thevolume fraction of each type of domains is denoted by f ii=1,2 . . .N and i=1N fi=1. Both the electric and elastic inter-actions between grains are considered in a self-consistentinclusion manner.13 In other words, each grain is regarded asan inclusion surrounded by an infinite large matrix with thematerials properties same as the whole material, as shown inFig. 1, which is two-dimensional. As spherical shape is thesimplest one and no other shapes are more appropriate, suchshape is employed for modeling of inclusions. Moreover, byadopting the concept of Eshelby’s inclusion,13 both the elec-tric and stress fields are assumed to be uniform inside eachgrain inclusion, i.e., the interactions between domains inthe same grain are neglected. Furthermore, in order to ac-count for the major characteristics of domain switching andto make computation manageable, a polycrystalline ferro-electric is assumed to exhibit dielectric and elastic isotropyand shows linear dielectric and elastic behavior unless do-main switching occurs.In the conventional PFM, the free energy of the wholematerial system subjected to external loading is minimizedusing the time-dependent Ginzburg–Landau equation to de-termine the final stabilized domain structures. As the compu-tational complexity of this minimization process increasesdrastically with increasing number of computing grids,6 PFMis usually used to simulate two-dimensional 2D single crys-tals. In the present model, to make the computational com-plexity manageable in 3D case, the focus is not placed on theenergy minimization process but only on the final stabilizedstate. Besides, as the interactions between grains have beenconsidered in an Eshelby inclusion manner in this model, thefree energy of each grain is used as the optimization objec-tive and the optimization process is carried out for all grainsto determine the domain structures and properties of thewhole material system. The free energy of a specific graincan be expressed asaElectronic mail: lifaxin@pku.edu.cn. Tel.: 86 10 62757454. FAX: 86 1062751812.bElectronic mail: aksoh@hkucc.hku.hk. Tel.: 852 28598061. FIG. 1. 2D Illustration of material model for polycrystalline ferroelectrics.APPLIED PHYSICS LETTERS 96, 152905 20100003-6951/2010/9615/152905/3/$30.00 © 2010 American Institute of Physics96, 152905-1Downloaded 03 Aug 2010 to 147.8.21.43. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jspUE, = − E · Pr − :r +12kE · E +1212:−12v21 + vtr2 +1227 − 5v151 − v2r− :r −  +12 L13kPr − Pr · Pr − Pr+ f180W180 + fnon-180Wnon-180, 1where E,  are the applied electric field tensor, and appliedstress tensor, respectively; k is the isotropic dielectric con-stant;  ,v denote the isotropic shear modulus and Possion’sratio, respectively; , Pr are the average strain and polariza-tion of the whole material system or the matrix, respectively;L 0L1 is the electric depolarization factor, which de-pends on the extent of charge screening;14 Pr and r are theremnant polarization and remnant strain of a grain, whichcan be expressed as linear functions of the volume fractionsof domains as follows:Pr = i=1NfiPi, r = i=1Nfii, 2where Pi and i are the spontaneous polarization and spon-taneous strain tensor of the ith domain.W180 and Wnon-180 are the energy barrier per unit vol-ume for 180° and non-180° domain switching, respectively;and f180 and fnon-180 are the corresponding volume fractions,which can be calculated as follows:fnon-180 =12i=1N/2f2i−1 + f2i − f2i−10 + f2i−10  , 3af180 =12i=1Nf i − f i0 − fnon-180, 3bwhere f i0 is the volume fraction of the ith domain calculatedat last loading step.Referring to the expression of the free energy given inEq. 1, the first two items on the right hand side is thepotential energy;1 the third item is the linear dielectric en-ergy, the fourth and fifth items present the linear strain en-ergy; the sixth and seventh items are the elastic and electricinclusion energies, respectively; and the last two items arethe dissipation energies during domain switching, which aresimilar to the domain wall energy in the conventional PFM.6The volume fractions of domains, i.e., the optimizationvariables, at a specific loading are obtained by minimizingthe total free energy of each of the grains in a ferroelectricpolycrystalline. The mathematical optimization problem forthis model can be summarized by the following:Problem IMin Uf1, f2, . . . , fN ,s . t . 0 f i 1 i = 1,2, . . . ,N ,i=1Nfi = 1.As the free energy is a nonlinear function of the volumefractions of domains, Problem I becomes a constrained non-linear optimization problem. To solve this type of optimiza-tion problems, the sequential quadratic programmingmethod15 is a very effective algorithm with high accuracyand quick convergence. As the expression of the free energyvaries from grain to grain and depends on the applied load-ing, at each loading step, the optimization process involvesall grains to determine the domain structures, polarizations,and strains.The behavior of both tetragonal and rhombohdral PZTferroelectric ceramics under cyclic electric loading are stud-ied using the proposed optimization model consisting of10 000 grains, in which the charge screening that exists inreal ceramics12,16 is taken into account. Thus, the depolariza-tion field induced by polarization switching vanishes, i.e.,L=0 in Eq. 1. The material constants used are listed inTable I, which are taken from Hoffmann et al.17 but withTABLE I. Material constants of tetragonal and rhombohedral PZT used inthe model.Material constants Rhombohedral TetragonalShear modulus  GPa 30 30Poisson’s ratio v 0.3 0.3Dielectric constant k F/m 1.010−8 1.010−8Spontaneous polarization P0 C /m2 0.52 0.52Lattice deformation Slattice 0.65% 2.77%Coercive field EC kV/mm 1.0 1.0FIG. 2. Color online Simulated a polarization and b strain curves oftetragonal and rhombohedral PZT ceramics under applied electric loading.152905-2 Li, Zhou, and Soh Appl. Phys. Lett. 96, 152905 2010Downloaded 03 Aug 2010 to 147.8.21.43. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jspslight modifications because of the isotropic dielectric andelastic assumptions. For easy comparison, the elastic and di-electric properties, spontaneous polarization, and the nomi-nal coercive field of these two types of ceramics are taken tobe the same as they do not differ much in reality. Note that inTable I, the lattice deformation Slattice=c /a−1 and Slattice=d111 /d111¯−1 for the tetragonal and rhombohedral types,respectively.12 In addition, the energy barriers for both 180°and non-180° switching are taken to be 2P0EC.1Figure 2 shows the plots of calculated polarization andlongitudinal strain under cyclic electric loading up to 5EC forboth tetragonal and rhombohderal PZT ceramics. The curvesoverlap themselves very well during cyclic loading whichconfirms good convergence of this model. The computationtime required on a Pentium 3.0 GHz PC for 61 loading stepsis about 7 h for the tetragonal PZT and about 12 h for therhombohedral PZT, and it increases only linearly with in-creasing number of grains. In Fig. 2, both the remnant polar-ization and strain of the rhombohedral PZT ceramics are ob-viously larger than those of the tetragonal PZT ceramics,which are consistent with previous experiments17 andpredictions.12 The rather flat strain curve of the tetragonalceramics indicates that little 90° domain switching has oc-curred, even though the 180° switching is almost complete,which can be estimated from the polarization curve in Fig.2a. The large remnant strain of rhombohedral PZT ceram-ics shows that considerable amount of non-180° switchinghas occurred during electric poling.The evolution process of the domain textures repre-sented by the pole figures of polar axes in this paper canalso be captured by this optimization model. Under electricloading or uni-axial compression tension, the pole figure ofpolar axis is axisymmetric around the field direction and,thus, can be described by an angular distribution functiong 0 /2.18 Note that g1 for a random uni-form distribution and the normalization condition requires0/2gsin d=1. Obviously, g completely depends onnon-180° domain switching. In this paper, g is calculatedby dividing the interval of 0, 90° into eighteen subintervalseach of which is 5°. The accumulated volume fractions ofdomains at all these subintervals give rise to g at discretevalues of .Figure 3 shows the domain textures in both tetragonaland rhombohedral PZT ceramics corresponding to the elec-tric loading points indicated in Fig. 2a. It can be seen fromFig. 3a that during electric loading, the domain textures oftetragonal PZT ceramics change little from the initial un-poled state. This indicates that little 90° domain switchingoccurs which is consistent with the small remnant strainshown in Fig. 2b. Whereas, the domain textures of rhom-bohedral PZT ceramics change a lot from the unpoled stateduring electric poling R1→R2, as shown in Fig. 3b,which shows that upon removing the electric field, domainswitching reverses causing domain texture variations R2→R3. Nevertheless, in rhombohedral PZT ceramics, the do-main textures cannot return to the initial unpoled state insubsequent electric loading after poling.F.L. gratefully thanks Professor X. Guo Dalian Univer-sity of Technology for his helpful discussions. This work issupported by the Natural Science Foundation Grant No.10872002, the 985 Project Foundation of Peking University,and the Research Grants Council of the Hong Kong SpecialAdministrative Region, China Project No. HKU 716007E.1S. C. Hwang, C. S. Lynch, and R. M. McMeeking, Acta Metall. Mater. 43,2073 1995.2J. E. Huber, N. A. Fleck, C. M. Landis, and R. M. McMeeking, J. Mech.Phys. Solids 47, 1663 1999.3C. M. Landis, J. Mech. Phys. Solids 50, 127 2002.4F. X. Li and D. N. Fang, Mech. Mater. 36, 959 2004.5W. Tang, D. N. Fang, and J. Y. Li, J. Mech. Phys. Solids 57, 1683 2009.6L. Q. Chen, Annu. Rev. Mater. Res. 32, 113 2002.7A. K. Soh, Y. C. Song, and Y. Ni, J. Am. Ceram. Soc. 89, 652 2006.8J. Wang and M. Kamlah, Appl. Phys. Lett. 93, 042906 2008.9Y. U. Wang, Y. M. Jin, and A. G. Khachaturyan, J. Appl. Phys. 92, 13512002.10S. Choudhury, Y. L. Li, C. E. Krill III, and L. Q. Chen, Acta Mater. 53,5313 2005.11D. Schrade, R. Mueller, B. Xu, and D. Gross, Comput. Methods Appl.Mech. Eng. 196, 4365 2007.12F. X. Li and R. K. N. D. Rajapakse, Acta Mater. 55, 6472 2007; 55,6481 2007.13J. D. Eshelby, Proc. R. Soc. London, Ser. A 241, 376 1957.14M. E. Lines and A. M. Glass, Principles and Applications of Ferroelec-trics and Related Materials Clarendon, Oxford, 1977, p. 87.15S. P. Han, J. Optim. Theory Appl. 22, 297 1977.16D. A. Hall, A. Steuwer, B. Cherdhirunkorn, P. J. Withers, and T. Mori, J.Mech. Phys. Solids 53, 249 2005.17M. J. Hoffmann, M. Hamme, A. Endriss, and D. C. Lupascu, Acta Mater.49, 1301 2001.18F. X. Li and R. K. N. D. Rajapakse, J. Appl. Phys. 101, 054110 2007.FIG. 3. Color online Calculated a 001 pole figures in tetragonal PZTceramics and b 111 pole figures in rhombohedral PZT ceramics corre-sponding to different electric loading points in Fig. 2a.152905-3 Li, Zhou, and Soh Appl. Phys. Lett. 96, 152905 2010Downloaded 03 Aug 2010 to 147.8.21.43. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp