Effective magnetoelectric effect in multicoated circular fibrous multiferroic composites
Hsin-Yi Kuo and Ernian Pan
Citation: Journal of Applied Physics 109, 104901 (2011); doi: 10.1063/1.3583580 View online: http://dx.doi.org/10.1063/1.3583580
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/109/10?ver=pdfcov Published by the AIP Publishing
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Effective magnetoelectric effect in multicoated circular fibrous multiferroic
composites
Hsin-Yi Kuo1,a)and Ernian Pan2
1
Department of Civil Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, Republic of China
2
School of Mechanical Engineering, Zhengzhou University, Henan, People’s Republic of China and Department of Civil Engineering, University of Akron, Ohio 44325, USA
(Received 25 January 2011; accepted 26 March 2011; published online 16 May 2011)
Rayleigh’s formalism is generalized for the evaluation of the effective material properties in multicoated circular fibrous multiferroic composites. The derived solution is applied to the special three-phase composite in which coated fibers are embedded in a matrix. For composites made of piezoelectric (BaTiO3) and piezomagnetic (CoFe2O4 or Terfenol-D) phases, we find
that the magnetoelectric effect in the composite made of CoFe2O4 coated BaTiO3 in matrix
Terfenol-D is five times larger than that in the composite made of BaTiO3coated Terfenol-D in
matrix CoFe2O4. Furthermore, in each case, with appropriate coating to the circular fiber, the
magnetoelectric effect in the coated composites can be enhanced by more than one order of magnitude as compared to the corresponding noncoating composite.VC 2011 American Institute of
Physics. [doi:10.1063/1.3583580]
I. INTRODUCTION
This work is concerned with the magnetoelectric (ME) effect of a periodic array of multicoated circular fibrous pie-zoelectric-piezomagnetic composites. ME materials, which simultaneously show two or more types of ferroelectric, magnetic, or elastic orderings, have been the focus of research due to the varieties of their microstructural phenom-ena and macroscopic properties. These make them promising for a wide range of applications, such as magnetic field sensors, electrically controlled microwave phase shifters, four-state memories, etc.1,2However, this coupling is weak in single-phase materials, and is often observed only at very low temperatures. For instance, Cr2O3 has a ME voltage
coefficient of 0.02 V/cmOe below the antiferromagetic Ne´el temperature of 307 K.3This is insufficient for practical appli-cations and thus has motivated the study of composites of piezoelectric and piezomagnetic media, as reviewed recently by Spaldin and Fiebig,4 Eerensteinet al.,1Nan et al.,2 and Srinivasan.5The “product property” causes the ME effect in multiferroic composites: an applied electric field creates a strain in the piezoelectric material, which in turn creates a strain in the piezomagnetic material, resulting in a magnetic polarization.
A number of micromechanical models for two phase composites were proposed to predict the effective moduli of multiferroic composites. Among them, Nan6and Huang and Kuo7used the Green’s function approach to study a fibrous composite consisting of BaTiO3and CoFe2O4. Benveniste
8 derived an exact expression for the effective magnetoelec-troelastic moduli for transversely isotropic fibrous compo-sites based on a formalism of Milgrom and Shtrikman.9 Harsheet al.10used a cubic model, and Aboudi11employed
a homogenization micromechanical method to investigate the particulate composites. The classical Eshelby’s equiva-lent inclusion approach and the Mori-Tanaka mean-field model have been generalized to multiferroic composites by Li and Dunn,12,13Huang,14Li,15Wu and Huang,16and Srini-vas et al.17 The frequency-dependence of the ME coeffi-cients of multiferroic laminates was studied by Bichurin et al.18–20A complete review of all of this literature can be found in Nanet al.2and Bichurinet al.19
Recently, some multi-phase piezoelectric and piezomag-netic composites were made experimentally to enhance the ME coupling. Nanet al.21,22prepared three-phase ME partic-ulate composites with Terfenol-D particles in a lead zircon-ate titanzircon-ate (PZT)-polyvinylidene fluoride (PVDF) mixture, and the measured ME coefficient was enhanced to 45 mV/ cm. Gupta and Chatterjee23 made a three-phase BaTiO3
-CoFe2O4-PVDF particulate composite and showed a
maxi-mum ME voltage coefficient around 26 mV/cmOe. Chau et al.24investigated the ME behavior of composites consist-ing of Terfenol-D and PZT with the polymer PMMA, poly-merized ethylene oxide (PEO), or Liþ-PEO and demonstrated that higher matrix conductivity could enhance the ME signals. Jadhavet al.25prepared a three-phase com-bination of Ni0.5Cu0.2Zn0.3Fe2O4/BaTiO3/PZT and measured
a maximum ME coefficient of 975 lV/cmOe.
The inhomogeneity-coating-matrix composite is a special and interesting three-phase heterogeneous material. The coat-ing, a thin layer of the third phase intervening between an inclusion (or inhomogeneity) and the matrix, creates some im-portant applications. For instance, to reduce the heat or stress concentration along the interface, interphase layers are often introduced to act as thermal barriers. To enhance the electric conductivity of the electric composites, coated fibers are designed to serve as reinforcements. Such coatings may have constant or spatially varying properties.26 Research into graded multiferroics has primarily been confined to the study
a)Author to whom correspondence should be addressed. Electronic mail:
of bilayer and multilayer structures. Among them, piezoelec-tric or piezomagnetic coefficients are assumed to have linear variation in the direction of the thickness,27–29 although an exponentially graded assumption was also adopted recently.30,31Apart from these laminate structures, Wang and Pan32investigated how the imperfect interface affects the ME effect in a multiferroic fibrous composite. Panet al.33showed that the nonclassical interface condition exerts a significant influence on the local and overall ME responses, especially when the fibers are at the nanoscale. Wanget al.34enhanced the ME effect via the curvature of a heterogeneous cylinder. Thostensonet al.35experimentally coated carbon fibers with carbon nanotubes (CNTs), which were then embedded in a polymer matrix. They showed that CNT-coated carbon fiber composites could improve the local interfacial load transfer, and therefore were likely to reinforce the local strength along the interface of the carbon fiber and the polymer matrix. Good effective thermal conductivity for coated fiber filler compo-sites was also analytically predicted by Hatta and Taya.36 They theoretically showed that the use of highly conductive or resistive coating, even if its thickness is small, is quite effi-cient in enhancing the overall thermal conductivity or resistiv-ity. Nicrovici et al.37 studied the equivalence between a coated three-phase composite and the corresponding two-phase composite on the dielectric constants or the transport coefficients.
In this paper, by generalizing Rayleigh’s classic approach,38 we investigate the ME properties of circular fi-brous composites under the generalized anti-plane shear de-formation. The solution can be for any multicoated circular fibers in a matrix. As a numerical example, we apply our so-lution to the coated fiber in a matrix made of BaTiO3,
CoFe2O4, and Terfenol-D. This article is organized as
fol-lows: We consider, in Sec.II, a composite made of piezo-electric and piezomagnetic phases arranged on a microstructure consisting of parallel cylinders in a matrix. The phases are transversely isotropic and under anti-plane shear with in-plane electromagnetic fields. In this situation, the fields are decoupled in the interior of every phase, and the coupling between the fields occurs only through the inter-face conditions. We exploit this in detail in Sec.IIIin order to obtain a representation of the solution for the multicoated circular cylinder. We obtain the effective properties in Sec.
IV, and show that the macroscopic (or overall effective) properties depend only on a single expansion coefficient. This methodology is illustrated in Sec.V using composites made of BaTiO3, CoFe2O4, and Terfenol-D.
II. FORMULATION
Let us consider a composite consisting of a periodic rec-tangular array of parallel and separated circular cylinders. The domain of the cylinder is denoted byV. We assume that the cylinders and the matrix are made of distinct phases. Fur-ther, we assume that each phase is either piezoelectric or pie-zomagnetic with transversely isotropic symmetry (i.e., has 6mm symmetry) about the fiber axis. We introduce a Carte-sian coordinate system with thex- and y-axes in the plane of the cross-section and thez-axis along the axes of the
cylin-ders. The origin of the coordinate is positioned at the center O of one of the cylinders (Fig.1). The sides of the unit cell X parallel to the x- and y-axes are, respectively, denoted by a and b, and the cylinders are of the same size.
Let the composite be subjected to anti-plane shear strains ezx;ezy, in-plane electric fields Ex; Ey, and magnetic
fields Hx; Hy at infinity. Thus the heterogeneous material is
in a state of anti-plane shear deformation8 and can be described by
ux¼ uy¼ 0; uz¼ w x; yð Þ;
u¼ u x; yð Þ; w ¼ w x; yð Þ; (1) whereux,uy, anduzare the elastic displacements along thex-,
y-, and z-axes, and u and w are, respectively, the electric and magnetic potentials.
The general constitutive laws for the nonvanishing field quantities can be written in a compact form as
Rj¼ LZj; j¼ x; y; (2) where Rj¼ rzj Dj Bj 0 @ 1 A; L¼ Ce1544 je1511 kq1511 q15 k11 l11 0 @ 1 A; Zj¼ ezj Ej Hj 0 @ 1 A: (3) In Eq.(3), rzj,Dj,Bj, ezj,Ej, andHjare the stress, electric
dis-placement, magnetic flux, strain, electric field, and magnetic field, respectively. C44, j11, l11, and k11 are the elastic
FIG. 1. A schematic representation of the square-arrays composite in which the unit cell is made of a coated cylindrical fiber within the matrix. The vol-ume fraction of the inclusionf, is defined as the ratio of the volume of the fiber plus the coating layer over the total volume of the unit cell (fiber plus coating layer plus matrix). The radius ratio of the fiber over the coating shell is defined as c.
modulus, dielectric permittivity, magnetic permeability, and ME coefficient, whilee15 andq15 are the piezoelectric and
piezomagnetic coefficients. The shear strains ezx and ezy,
in-plane electric fieldsExandEy, and in-plane magnetic fields
HxandHyare related to the gradient of the elastic
displace-ment, electric potential, and magnetic potential.
Making use of the equilibrium equations (in the absence of body force, electric charge density, and electric current density), the elastic displacementw and the electric and mag-netic potentials u and w are found to satisfy
C44r2wþ e15r2uþ q15r2w¼ 0; e15r2w j11r2u k11r2w¼ 0; q15r2w k11r2u l11r 2 w¼ 0; (4)
where r2 ¼ @2=@x2þ @2=@y2 represents the
two-dimen-sional Laplace operator for the variablesx and y. Because L is, in general, a nonsingular matrix, we can decouple Eq.(4)
into three independent Laplace equations, r2
w¼ 0; r2u¼ 0; r2w¼ 0; (5)
which should be satisfied in the interior of each phase. In other words, the three fields—elastic displacement, electro-static potential, and magnetoelectro-static potential—are completely decoupled in the interior of each phase.
These differential equations can be solved, subject to suitable interface and boundary conditions. We assume that the interfaces are perfectly bonded, and therefore the field quantities satisfy Rjnj ¼ LZj nj ¼ 0; Zjtj ¼ 0; (6)
where ½ denotes the jump in the associated quantities½ across the interface, n is the unit outward normal to the inter-face, t is the unit tangent to the interinter-face, and the repeated indexj denotes the summation over the components x and y. Because L is different in each phase, the fields w, u, and w are generally coupled by the interface equations.
III. MULTICOATED CIRCULAR CYLINDERS
We consider the case in which the fibers are multicoated circular cylinders with an outer radiusa1. We denote the
ma-trix as phase 0, with material parametersC44(0),e15(0),q15(0),
j11(0), l11(0), and k11(0). The multicoated cylinder consists of
a core with radius r¼ aM, surrounded by (M 1) coating
layers. The jth layer of the coatings occupies the annulus Vj: ajþ15r5aj; j¼ 1; 2; :::; M, in which V¼ V1[ V2[
:::[ VM. Because the innermost core is solid, we have
aMþ1¼ 0. We assume that the material properties of the jth
layer of the multicoated cylinder are C44(j), e15(j), q15(j),
j11(j), l11(j), and k11(j).
Furthermore, without a loss of generality, we consider the situation in which the composite is subjected to a macro-scopically uniaxial loading
wext¼ ezxx; uext¼ Exx; wext¼ Hxx (7)
for constants ezx; Ex, andHx. We may write this in short as
Uext¼ Z U
xx; (8)
where U represents the appropriate field: the anti-plane de-formation w, the electric potential u, or the magnetic poten-tial w.
The potential field (the elastic deformation w, electric potential u, or magnetic potential w) for each layer of the multicoated circular cylinder and its surrounding matrix can be expanded with respect to its centerO as39
UðjÞðr; hÞ ¼ AUðjÞ0 þX 1 n¼1 AUðjÞn rnþ BUðjÞ n r n cosnh (9)
for thejth layer, and Uð0Þðr; hÞ ¼ AUð0Þ0 þ X1 n¼1 AUð0Þn r n þ BUð0Þn r n cosnh (10)
for the matrix. Here (r,h) is the polar coordinate centered on the origin of the cylinder. The coefficientsAUðjÞn andBUðjÞn are
unknowns, to be determined from the interface and boundary conditions. Note that thesine terms that would be present in a general expansion are missing because we impose a uniax-ial loading along thex-direction only. Further, Uðr; hÞ has to be antisymmetric with respect to the y-axis, and thus only terms with an odd number are included. In addition, because the potential atr! 0 should be finite, we can set BUðMÞ
n ¼ 0.
Using the orthogonality properties of trigonometric functions, the interface conditions in Eq.(6)provide
aðj1Þn bðj1Þn ¼ kðjÞn aðjÞn bðjÞn ; j¼ 1; 2; :::; M; (11) where aðjÞn ¼ AwðjÞn AuðjÞn AwðjÞn 0 B @ 1 C A; bðjÞn ¼ BwðjÞn BuðjÞn BwðjÞn 0 B @ 1 C A; kðjÞn I a 2n j I Lðj1Þ a2n j L ðj1Þ !1 I a2nj I LðjÞ a2n j L ðjÞ ! ; (12)
and I is the 3 3 identity matrix. Now repeated use of Eq.
(11)gives að0Þn bð0Þn ¼ KðjÞn aðjÞn bðjÞn ; j¼ 1; 2; :::; M; (13) with KðjÞn kð1Þ n k ð2Þ n k ðjÞ n : (14) Forj¼ M, we have að0Þn bð0Þn ¼ KðMÞn aðMÞn bðMÞn : (15)
Furthermore, according to Eq.(15), and keeping in mind that BUðMÞn ¼ 0, we have að0Þn ¼ KðMÞn h i 11 K ðMÞ n h i1 21b ð0Þ n ; (16)
where [Kn(M)]11 and [Kn(M)]21 are, respectively, the
upper-left and lower-upper-left (3 3) submatrices of Kn(M).
Finally, imposing the periodicity conditions yields a generalized Rayleigh’s identity,39
AUð0Þn þ X1 m¼1 SmþnBUð0Þm ¼ Z U xdn;1; (17) with Sm¼ X l6¼o ReðXlþ iYlÞm (18)
being the lattice sums characterizing the geometry of the per-iodic structure, and (Xlþ iYl) the center of the lth cylinder
when measured at the central pointO. The index l runs over all the cylinders underlying the periodic array except for the central one. A list of nonzero normalized lattices for square arrays can be found in Berman and Greengard.40
Equations (16)and(17)constitute an infinite set of lin-ear algebraic equations. Upon appropriate truncation of the expansion terms, we can determine the expansion coeffi-cientsAUðjÞn and BUðjÞn . Once these coefficients are obtained,
we have the solutions for the elastic deformationw, electric potential u, or magnetic potential w. By taking the deriva-tives, we can finally obtain the field solutions in each phase of the composite.
IV. EFFECTIVE MODULI
Our solutions above are now applied to derive the effec-tive properties. Here we concentrate on a square array, i.e., a¼ b. Although in the case of elasticity a square arrange-ment of circular cylinders results, in general, in a square symmetry,41 it turns out that in the case of conduction, square symmetry and transverse isotropy become identical.42 This statement is also correct for magnetoelectricity under the generalized anti-plane shear deformation, which is the case in our study. Therefore, there is no distinction between the effective properties of thex- and y-axes.
We first recall the basic definition of the effective mag-netoelectroelastic parameter L*, given by
Rj
¼ L Zj
; (19)
where the angular brackets denote the area averages over the unit cell X, i.e.,
Rj ¼1 X ð X Rjdv; Zj ¼ 1 X ð X Zjdv: (20)
For the given far-field in Eq.(7), we can compute the aver-age Zx by noting that each component is a gradient and
applying the divergence theorem. We obtain
ZxU
¼ ZUx: (21)
Next, in order to find RU x
, we again use the divergence the-orem, equilibrium condition, and interface conditions to arrive at RUx ¼1 X ð X RUxdv¼ 1 X ð X r xR Udv ¼1 X ð @X x R Um nds; (22) where Rw¼ ezx;ezy ;Ru¼ Dx; Dy ;Rw¼ Bx; By : (23) We then use field expansions (9) and (10) to obtain
1 X ð @X x Z Um nds ¼ ZUx 2pBU 01ð Þ ab (24) Here Zw¼ ezx;ezy ; Zu¼ Ex; Ey ; Zw¼ Hx; Hy : (25) Putting Eqs.(22)and(24)together, and recalling the consti-tutive relation (2) for the matrix, we obtain
rzx h i Dx h i Bx h i 0 @ 1 A ¼ Ce1544 ej1511 qk1511 q15 k11 l11 0 @ 1 A ð0Þ ezx 2pBwð0Þ1 ab Ex 2pBuð0Þ1 ab Hx 2pBwð0Þ1 ab 0 B B B @ 1 C C C A: (26) Putting together Eqs.(19)and(26), and noting that the coef-ficientBUð0Þ1 depends linearly on the applied field, we obtain
the equations for the effective property L*.
V. RESULTS AND DISCUSSION
As a numerical example, we apply our solution to a sin-gle coated fiber, i.e.,M¼ 2, for which the radii of the (fiber) core and coating shell are, respectively, a2and a1. For the
piezoelectric material, we consider the widely used BaTiO3
(BTO). For the piezomagnetic material we consider CoFe2O4
(CFO) as well as the Terfenol-D alloy (TD). All of these are transversely isotropic. The material properties are listed in Table I in Voigt notation, where thexoy plane is isotropic TABLE I. Material parameters of BaTiO3, CoFe2O4(Ref.12), and
Terfe-nol-D (Refs.43and44).
Property BaTiO3 CoFe2O4 Terfenol-D
C44(N/m2) 43 109 45.3 109 13.6 109 e15(C/m 2 ) 11.6 0 0 q15(N/Am) 0 550 108.3 j11(C 2 /Nm2) 11.2 109 0.08 109 0.05 109 m11(Ns2/C2) 5 106 590 106 5.4 106 k11(Ns/VC) 0 0 0
and the unique axis is along thez-direction. Note that in all materials the ME coefficients are zero, i.e., k11¼ 0.
The ratio of the radius between the circular fiber and the coating shell is defined as c¼ a2/a1, and the coated fibers are
embedded in the matrix in a square array pattern. It is obvious that if c¼ 0, then a2¼ 0. In other words, there is no
fiber phase. On the other hand, if c¼ 1, it means that there is no coating shell. In our study, we are particularly interested in the ME voltage coefficient, which is the important figure of merit for magnetic field sensors. It relates the overall elec-tric field that is generated in the composite when it is sub-jected to a magnetic field. It combines the coupling and dielectric coefficients, and is defined by
a11 ¼ k11=j
11: (27)
Figure 2 shows the dependence of the ME coefficient in coated fibrous composites on both the volume fractionf and the ratio of the radii of the fiber and the shell c¼ a2/a1. The
volume fraction is defined as the volume of the fiber and the coated shell over the total volume (i.e., the fiber, plus the coated shell and the matrix). Figure2(a)is for (fiber/coating/
matrix)¼ (BTO/TD/CFO). It is observed that, for a fixed volume fraction, the ME effect increases when the radii’s ra-tio c increases from 0 to 0.70; then it decreases with increas-ing c. Furthermore, for a fixed c, the ME effect increases with increasing volume fraction and, in most cases, reaches its maximum where the volume fraction f is around 0.74. Figure 2(b) shows the corresponding results when the fiber and coating shell in Fig.2(a)are switched. Compared to Fig.
2(a), it is obvious that although the magnitude of the ME is about the same, the direction or the sign has been changed. Furthermore, the maximum magnitudes are all reached aroundf¼ 0.30, and the magnitude increases with increasing c from 0 to 0.94 (the maximum ME effect is about 0.740 V/cmOe, slightly larger than in the first case).
Figure2(c)shows the ME effect for the composite made of (fiber/coating/matrix)¼ (BTO/CFO/TD). The ME effect is positive, and for fixed f, it increases with increasing c (from 0 to 0.84). It reaches its maximum value 3.304 V/cm Oe at c¼ 0.84, and then it decreases. Furthermore, for any fixed c, the ME effect reaches its maximum whenf is around 0.70. Figure 2(d) shows the ME effect in the composite when the coating shell and fiber in Fig. 2(c) are switched.
FIG. 2. (Color online) The effective ME voltage coefficient a11 *
vs the volume fraction of the inclusionf and the radius ratio c. The composite is in a square array where coated cylindrical fibers are embedded in the matrix. (a) BTO fiber coated by TD, with both in a CFO matrix. (b) TD fiber coated by BTO, with both in a CFO matrix. (c) BTO fiber coated by CFO, with both in a TD matrix. (d) CFO fiber coated by BTO, with both in a TD matrix.
Similarly, the ME effect increases from c¼ 0 to 0.96, and then decreases. Also, in Figs.2(c) and 2(d), the maximum ME value is about 3.5 V/cmOe, and in Figs.2(a)and2(b)it is only about 0.7 V/cmOe. Furthermore, as compared to the uncoated case, where the ME value is either zero or very small, the ME effect in the coated fibrous composites can be enhanced by 10 times. Similar trends are also observed if we replace BTO with PZT-5A.
VI. CONCLUSIONS
We have extended Rayleigh’s formalism on periodic conductive composites to a magnetoelectroelastic composite consisting of multicoated circular cylinders under anti-plane shear deformation, in-plane electric field, and in-plane mag-netic intensities. Expressions for the effective moduli of the composite are derived. As a practical example, explicit nu-merical calculations for the ME effects of a BTO/CFO/TD coated composite are presented and discussed. These exam-ples show that with a coating appropriate for the inhomoge-neity (fiber), the effective ME effect can be enhanced by one order of magnitude as compared to the noncoated counter-part. While our numerical results are based on piezoelectric BaTiO3 and piezomagnetic CoFe2O4 or Terfenol-D, the
enhancement of the ME effect based on other materials, such as BiFeO3, NiFe2O4, etc., could be possible. Therefore,
dif-ferent material phases and volume fraction ratios are some alterative channels for improving the effective material prop-erties of multiferroic composites.
ACKNOWLEDGMENTS
We are glad to acknowledge financial support from the National Science Council, Taiwan, under Grant No. NSC 99-2221-E-009-053 and from a special program in Henan Province.
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