pyro and electromagnetic effects in ferrite barium titanate composite

Author’s Accepted Manuscript Pyro- and electromagnetic effects in ferrite/barium titanate composite Andrey A Pan'kov www.elsevier.com/locate/moem PII: DOI: Reference: S2452-1779(16)30091-3 http://dx.doi.org/10.1016/j.moem.2016.12.005 MOEM44 To appear in: Modern Electronic Materials Received date: 24 September 2016 Accepted date: 14 December 2016 Cite this article as: Andrey A Pan'kov, Pyro- and electromagnetic effects in ferrite/barium titanate composite, Modern Electronic Materials, http://dx.doi.org/10.1016/j.moem.2016.12.005 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain Pyro- and electromagnetic effects in ferrite/barium titanate composite Andrey A Pan’kov Perm National Research Polytechnical University, 29 Komsomolsky prospekt, Perm 614990, Russia * Corresponding author mkmk_pr@pstu.ru Abstract New solutions for tensors of effective pyroelectromagnetic properties of piezoactive composites on the basis of boundary value problem solution for electromagnetic elasticity have been obtained For the solution of the boundary value problem, new solutions for singular components of the second derivative Green functions for displacements, electric and magnetic potentials in homogeneous transversal isotropic piezoelectromagnetic medium with ellipsoidal grain of heterogeneity have been used Calculation results on the concentration dependences for effective coefficients of pyromagnetic and electromagnetic coherence of ferrite/barium titanate composite with ellipsoidal, spherical and fibrous inclusions for various polydisperse structures and those of a layered structure composite have been presented Considerable influence of the shape of the inclusions, features of relative positioning and inversion of the properties of phases on the effective coefficients of pyromagnetic and electromagnetic coherence of the composite material have been revealed The conclusion is drawn on the preferable use of the pyroelectric phase as spherical inclusions, and ferrite as the composite matrix This allows for more than a fivefold increase in the effective constant of pyromagnetic coherence of the composite material in comparison with its value for the same structure but with inversion of properties of phases for constant volume fractions of the ferrite and pyroelectric phases Keywords Piezocomposite, boundary value problem of electromagnetic elasticity, effective pyroelectric properties Introduction Piezoelectric transducers allow using electrical measuring tools for measuring a wide variety of physical paremeters To convert between magnetic and electric fields one can use magnetoelectrics that induce magnetic field if placed in magnetic field Electric field induction is proportional to magnetic field magnitude Magnetoelectric effect can be observed in a number of materials in anti-ferromagnetic state This effect is produced by specific asymmetry of magnetic moments arrangement in the crystalline lattice To convert the intensity of infrared heat radiation into electric field, pyroelectric materials are used, i.e those in which electric charges are generated on the crystalline surface during heating or cooling Charge generation on pyroelectric surface is attributed to a change in its polarization state due to crystal temperature variation The magnetoelectric and pyromagnetic effects occur in piezocomposites containing pyroelectric and piezomagnetic phases Placing this composite in magnetic field causes deformation of the piezomagnetic phase and hence the piezoelectric phase and the whole composite Piezoelectric phase deformation in the composite produces electric field At the macro level, the electric field induction vector in the composite is related to the magnetic field magnitude vector via the effective magnetoelectric constant tensor Composite heating causes deformation of all its phases In this case, electric field is generated in the pyroelectric phase due to the piezo- and pyroelectric effects in this phase, and magnetic field is generated in the piezomagnetic phase As a result both electric and magnetic field are generated in heated composite the magnitude and orientation of which are determined by the effective pyromagnetoelectric constant tensors The development of new pyroelectromagnetic piezomaterials and the fabrication of devices on their basis are an intensely growing trend in the magnetoelectric materials science [1–3] In composite materials, pyro– and electromagnetic coherence can be absent in a specific phase Its occurrence at the macro level is caused by the microscopic level tnteractions of structural elements [4, 5] Below is the solution for effective pyroelectromagnetic properties of composite with piezo-active phases obtained within the well-known and tested approach of statistical mechanics for composites [6–8] This approach uses new solutions [9] of singular components of second Green function derivatives for homogeneous transverse isotropic piezoelectromagnetic medium with ellipsoidal heterogeneity grain Micro and Macro Levels We consider two-phase piezo active media in the representative volume V that determine the relationships for the phases f = 1, [5, 7, 8]: (f) (f) (f) ij  Cijmn  mn  enij En  hnij H n  ij( f ); ; (f) (f) D i  eimn mn  in En  i( f ); (1) (f) (f) Bi  himn mn  in H n  i( f ), where  is the tension, D and B are the electric and magnetic field inductions, respectively, E and H are the electric and magnetic field magnitudes, respectively,  is the homogeneous external heating temperature, Cf, ef and hf are the elastic, piezoelectric and piezomagnetic tensors for each phase f, respectively, that are considered to be known, f and f are the dielectric and magnetic permeabilities, respectively, f is the thermal coefficient and f and f Ideal contact conditions are met at the phase boundaries: the displacement, magnitude and induction vectors of the electric and magnetic fields are continuous The effective tensors C*, … * are included in the determinant relationships at the composite macro level: * * * *ij  Cijmn *mn  enij En*  hnij H n*  *ij  ; * Di*  eimn *mn  *in En*  *in Hn*  *i  ; (2) * Bi*  himn *mn  *in H n*  *in En*  *i  , and relate the averaged or macroscopic tensions *   , deformations *   , inductions D*  D and B*  B and magnitudes E*  E and H*  H of the electric and magnetic fields, respectively, where is the averaging operator for the volume V of structural fields Generalized Singular Approximation The solution for the tensors of effective elastic C*, dielectric * and magnetic * permeabilities, piezomechanical properties e* and h*, electromagnetic coherence coefficients * and k* and thermal stresses  *, the vector of effective pyroelectric * and pyromagnetic * constants in the determinant relationships at the composite macro level in a generalized singular approximation [6, 8] can be obtained through the corrections to the respective values averaged over the volume V: C , …, π , i.e   * s (1) s (2) s ; Cijmn  Cijmn  v1 1  v1  Cijdb Adbmn  e pij Fpmn  hpij Fpmn       v 1  v    M h D ; e  e  v 1  v   e H  h H C B  ; (3)  h  v 1  v   e M h M C D  ;   v 1  v    M  e D  ;   v 1  v    H h B ;     v 1  v   C T  e T h T ;     v 1  v    T e T ;     v 1  v    T h T , s s *kn   kn  v1 1  v1   kp H (1) pn  ekpq Bpqn ; * kn * nij * hnij nij nij * ij kn 1 1 1 kp (1) s pn pij (1) s pn pij (2) s pn kpq (2) s pn pij pij (2) s pn kp (1) s pn kpq s pqn * kn 1 kp (2) s pn kpq s pqn s ijdb db (1) s pij p * i i 1 (1) s ip p * i i 1 (2) s ip p s pqn ijpq 1 s pqn ijpq * kn ij s pqn (2) s pij p s ipq pq s ipq pq where C  C1  C2 , e  e1  e2 , …, μ  μ1  μ2 are the difference tensors, 1 is the relative volume content of the first phase in the volume V, e.g C = 1C1 + 2C2 where 2 = – 1 Equations (3) reduce the problem of determining the effective pyroelectromagnetoelastic tensors C*, *, …, * of the piezocomposite to the determination of the difference tensors As  A  A , Bs  B  B ,…, T(2) s  T(2)  T(2) averaged over inclusions and composite matrix for the tensor fields A(r), B(r), …, T(2)(r) included in the decompositions ij (r)  Aijmn (r)*mn  Bijn (r) En*  Dijn (r) H n*  Tij (r), (1) Ei (r)  Fimn (r)*mn  Hin(1) (r) En*  M in(1) (r) Hn*  Ti(1) (r), (2) Hi (r)  Fimn (r)*mn  Hin(2) (r) En*  Min(2) (r) Hn*  Ti(2) (r) of the micro level deformation fields (r) and the magnitudes of the electric E(r) and magnetic H(r) fields in the composite volume V via the preset at the composite macro level values *, E* , H* and  The tensor components A s , B s , Ds , F s , H s , M s and T(2)s in Eqs (3) are found by solving the following system of algebraic equations [9]: (1,1) s (1,2) (1) s (1,3) (2) s (1)  aikdb Adbmn  aikd Fdmn  aikd Fdmn  bikmn ;   (2,1) s (2,2) (1) s (2,3) (2) s (2) akdb Adbmn  akd Fdmn  akd Fdmn  bkmn ;  (3,1) s (3,2) (1) s (3,3) (2) s (3) a A  akd Fdmn  akd Fdmn  bkmn ;   kdb dbmn (4) (1,1) s (1,2) (1) s (1,3) (2) s (1)  aikdb Bdbn  aikd H dn  aikd H dn  cikn ;   (2,1) s (2,2) (1) s (2,3) (2) s (2) akdb Bdbn  akd H dn  akd H dn  ckn ;  (3,1) s (3,2) (1) s (3,3) (2) s (3) a B  akd H dn  akd H dn  ckn ;   kdb dbn (5) (1,1) s (1,2) (1) s (1,3) (2) s (1)  aikdb Ddbn  aikd M dn  aikd M dn  dikn ;   (2,1) s (2,2) (1) s (2,3) (2) s (2) akdb Ddbn  akd M dn  akd M dn  d kn ;  (3,1) s (3,2) (1) s (3,3) (2) s (3) a D  akd M dn  akd M dn  d kn ;   kdb dbn (6) (1,1) s (1,2) (1) s (1,3) (2) s  aikdb Tdb  aikd Td  aikd Td  fik(1) ;   (2,1) s (2,2) (1) s (2,3) (2) s  f k(2) ; akdb Tdb  akd Td  akd Td  (3,1) s (3,2) (1) s (3,3) (2) s a T  akd Td  akd Td  f k(3) ,   kdb db (7) where the coefficients a(1,1), a(1,2), …, a(3,3) are calculated as (1,1) aikdb  Iikdb  U (sik ) js C jsdb  1  2v1  C jsdb   s  (2) s    U ((1) ik ) s esdb  1  2v1  esdb   U (ik ) s  hsdb  1  2v1  hsdb  ; (1,2) s   aikd  U(sik ) js edjs  1  2v1  edjs   U ((1) ik ) s  sd  1  2v1   sd  ; (1,3) s   aikd  U (sik ) js hdjs  1  2v1  hdjs   U ((2) ik ) s  sd  1  2v1   sd  ; (2,1) s   akdb  kjs C jsdb  1  2v1  C jsdb   s (2) s    (1) ks esdb  1  2v1  esdb    ks  hsdb  1  2v1  hsdb  ; (8) (2,2) s   akd   kd   kjs edjs  1  2v1  edjs   s    (1) ks  sd  1  2v1   sd  ; (2,3) (2) s  s    akd  kjs hdjs  1  2v1  hdjs   ks  sd  1  2v1   sd  ; (3,1) s   akdb  kjs C jsdb  1  2v1  C jsdb   s (2) s    (1) ks esdb  1  2v1  esdb   ks  hsdb  1  2v1  hsdb  ; (3,2) (1) s  s    akd  kjs edjs  1  2v1  edjs   ks  sd  1  2v1   sd  ; (3,3) s   akd   kd  kjs  hdjs  1  2v1  hdjs   s   (2) ks  sd  1  2v1   sd  The right-hand sides of Eqs (4) have the following form: (1) s (2) s bikmn  U (sik ) js C jsmn  U ((1) ik ) s esmn  U (ik ) s hsmn ; (2) s (2) s s bkmn  kjs C jsmn  (1) ks esmn   ks hsmn ; (9) (3) s (2) s s bkmn  kjs C jsmn  (1) ks esmn  ks hsmn ; For the second and third systems (Eqs (5) and (6)) we can write (1) s cikn  U (sik ) js enjs  U ((1) ik ) s  sn ; (2) s ckn  kjs enjs  (1)s ks  sn ; (10) (3) s ckn  kjs enjs  (1)s ks  sn ; (1) s dikn  U (sik ) js hnjs  U ((2) ik ) s  sn ; (2) s dkn  kjs hnjs  (2)s ks  sn ; (3) s dkn  kjs hnjs  (2)s ks  sn ; And finally for the fourth system (Eq 7)): (11) s (2) s fik(1)  U (sik ) js  js  U ((1) ik ) s  s  U (ik ) s  s ; s (2) s s f k(2)  kjs  js  (1) ks  s   ks  s ; (12) s (2) s s f k(3)  kjs  js  (1) ks  s  ks  s In Eqs (8)–(12), the parenthesized subscripts (ik) denote the separation of the symmetrical component for the respective pair of indices [6] The difference tensors can be written as ~  μ  μ  C  C  C , ~ e  e  e ,…, μ (13) Equations (8)–(12) contain new solutions [9] for the tensors Us, Us(1), …, s(2) of the singular components of the second Green function G derivatives for a homogeneous anisotropic piezoelectromagnetic • • • • • “reference medium” [6] the properties of which are set by the tensors C , e , h , λ and μ (see Eqs (13)):     s U imjn s (1) U imn G r  r(1)  G s  r  r(1) , U ik U i(1) U i(2) G  k  (1)  (2) k (1) (2) (14) s (2) U imn s s (1) s (2) ; G s  imn ,  mn  mn s imn s (1)  mn s (2)  mn where G = G() is the Green function, () is the Dirac delta function,  = r – r1 (the unit volume force, or an electric or a magnetic source acts in the point r1);  is the operator of differentiation over the coordinates of the vector r The components Us, Us(1), …, s(2) of the Gs matrix in Eqs (8)–(12) and (14) are calculated as follows: s s (1) s (2) Uimjn  U ij  ; Uimn  U i(1)  ; Uimn  U i(2)  ; mn mn mn s s (1) s (2)  mjn   j  ;  mn  (1)  ;  mn  (2)  ;   mn   mn mn s (2) s (1) mjn   j  ; mn  (2)    (1)  ; smn   mn mn mn Here the operator 2   mn   4    m n sin dd 0 acts upon the components of the tensors   hi(1) h(1) hi(2) h(2) j j U ij   ij    (1) (2)     j  1 ; U i(1)  U ij hi(1) hi(2)  (2) U ; j  (1) ij   1 (1)  hi(1)U i(1)  (2)  hi(1)U i(2)  (1) (1) h(1) j  (1) ; U i(2)  U ij h(2) j  (2) ; Uij ; ; (1)  hi(2)U i(1)  (2) ; (2)   hi(2)U i(2)  1  (2) ; , where  ij  Cimjn  m n ;   hi(1)  emin  m n ; hi(2)  hmin  m n ;  (1)  mn m n ;  (2)  mn  m n ; 1  (15) 1 sin  cos  ;   sin  sin  ;   cos  ; a1 a2 a3  and  are the polar angles in the spherical coordination system [9] The surface of the ellipsoidal heterogeneity grain [6] is set by the equality x      i 1 (16) i via the values of the main semiaxes in Eqs (9) and xi = r(1)i – ri are the coordinates of the vector x Note that the solutions of Eqs (3)-(16) obtained in a generalized singular approximation correspond to statistical mixture structures having no correlation between the physical and mechanical properties in arbitrary points of the microheterogeneous medium but take into account the shape of the inclusions via the shape of heterogeneity grains (see Eq (16)) Statistical mixtures can be represented as ultimately polydisperse structures [10] By way of example, Figure shows structural fragments in the transverse plane r1r2 for different cases of fibrous polydisperse two-phase structures similarly oriented along r3 The size distribution of the particles (the transverse sections of single-phase (Fig b) and two-phase (Fig a and c) fibers) is wide and includes infinitely small particles This provides the possibility for the particles to fill the entire volume V of the composite without size or type correlation (Fig b) for different structural particles Figure Fragments of (a and c) two and (b) single phase polydisperse structures: () first phase; () second phase Numeric Calculation We calculate the effective coefficients of pyromagnetic *3 and electromagnetic  *33 coherences of ferrite / barium titanate composite with oriented ellipsoidal inclusions the main semiaxes of which a1 are oriented along the respective axes r3 (see Eq 16)) Let the first phase be ferrite [1] the isotropic elastic properties of which are set by the independent components C The transverse isotropic magnetic properties with the symmetry axis r3 are set by the piezomodules h and the magnetic permeabilities  (Table 1) The second phase is barium titanate [5] with independent constant transverse isotropic electric elastic properties C, e, , and (Table 1) Table Calculation parameters of effective coefficients of pyromagnetic and electromagnetic coherence for ferrite/barium titanate composite Phase Elastic properties Parameters Magnetic properties (1) h311 (1) Ferrite 10 С 1111 = 22 · 10 Pa, (1) 10 С 1313 = 5.5 · 10 Pa  (1) h322 Electroelastic properties = -400 Т, (1) (1) (1) h333  h223 = 800 Т, h113 = 200 Т, — (1) (1) = 100 Т, h123  h213 (1) (1) -5 μ 11 = μ 22 = 3.14 · 10 Т · m/А, (1) -5 μ 33 = 2.51 · 10 Т · m/А (2) = 11.6 C/m , e113 (2) = -4.40 C/m , e311 (2) Barium Titanate (2) = 18.6 C/m , e333 10 С 1111 = 16.80 · 10 Pa, (2) 10 С 1122 = 7.82 · 10 Pa, (2) 10 С 1133 = 7.10 · 10 Pa, (2) 10 С 3333 = 18.90 · 10 Pa, (2) 10 С 1313 = 5.46 · 10 Pa — (2) = 112 · 10 11 -10 F/m, (2)  33 -10 F/m, = 126 · 10 (2) 2.18 · 10 Pa/K, 11 (2) = 1.95 · 10 Pa/K, 33 -5 3(2) = 19 · 10 C/Км Figure shows calculation results for the concentration dependences of the effective pyromagnetic *3 (Fig a and c) and electromagnetic  *33 (Fig b and d) coherence coefficients for ferrite / barium titanate composite with differently shaped inclusions for different polydisperse structures: - with ferrite fibers in the pyroelectric (Fig a and Fig curves 5, and 10); - with pyroelectric fibers in the ferrite (Fig b and Fig curves 1, and 8); - with mutually penetrating phases (Fig b and Fig curves 3, and 9) Figure Effective pyromagnetic *3 (a and c) and electromagnetic  *33 (b and d) Coherence Coefficients for Ferrite / barium titanate composite with different polydisperse structures: (1, and 5) ellipsoid Inclusions, (2, and 6) spherical inclusions, (7) layers and (8–10) fibers Note that the solution of Eqs (3) for the effective constants *3 and  *33 (Fig a and b, curves 7) for the transition to the limit q  in the heterogeneity grain (Eq (16)) “layer” was exactly similar to earlier analytical solutions [11] for the constants *3 and  *33 of layered composite (taking into account the axial symmetry of the grain a1 = a2 and the notation q = a1(2)/a3) For the transition to the limit q   for the heterogeneity grain “fiber” the solution of Eqs (3) for the effective constant  *33 (Fig d, curve 8) was exactly similar to the analytical solution by the asympthotic averaging method for the ideal periodic fibrous structure obtained earlier [12] *33  (1) (2) where e311  e311  e311 ; v1 1  v1  e311h311 (1) (2) k12  v1k12  G12 (1) (2) (1) (2) h311  h311  h311 ; k12  k12  k12 ; , k12 = (C1111 + C1122)/2 is the volume planar deformation modulus, G12 = (C1111 + C1122)/2 is the shear modulus in the anisotropy plane r1r2 The calculation results shown in Fig for different polydisperse structures (Fig 1) were obtained in a generalized singular approximation (3)–(16), respectively, for three different properties of the reference medium in Eqs (13) and (15): - for the structure shown in Fig c the properties of the medium are taken to be equal to those of the first phase (ferrite), i.e C• = C2, e• = e2, h• = h(2), λ• = λ2 and μ• = μ2; - for the structure shown in Fig c the properties of the medium are taken to be equal to those of the composite averaged over the volume V, i.e C• = C1, e• = e1, h• = h1, λ• = λ1 and μ• = μ1; - for the structure shown in Fig b the properties of the medium are taken to be equal to those of the composite averaged over the volume V, i.e C  C , e  e , h  h , λ   λ and μ  μ The calculation results shown in Fig suggest that the geometrical shape of the inclusions, and the inversion of the properties of the inclusions and the pyroelectromagnetic composite matrix can lead to a significant increase in the absolute values of the effective pyroelectromagnetic coherence coefficients *3 and  *33 of the composite at constant volume fractions of the ferrite 1 and pyroelectric 2 phases For example, at equal volume fractions (1 = 2 = 0.5) of the ferrite and pyroelectric phases the absolute values of the pyromagnetic constant *3 (Fig 2, curve 2) of the composite with spherical pyroelectric inclusions in the ferrite matrix is more than fivefold higher than *3 (Fig 2, curve 6) for the composite with spherical ferrite inclusions in the pyroelectric matrix, i.e pyroelectric is preferable as inclusions in the composite Summary New solutions for tensors of effective pyroelectromagnetic properties of piezoactive composites on the basis of boundary value problem solution for electromagnetic elasticity have been obtained in a generalized singular approximation For the solution of the boundary value problem, new solutions for singular components of the second derivative Green functions for displacements, electric and magnetic potentials in homogeneous transversal isotropic piezoelectromagnetic medium with ellipsoidal grain of heterogeneity have been used Calculation results on the concentration dependences for effective coefficients of pyromagnetic *3 and electromagnetic  *33 coherence of ferrite/barium titanate composite with ellipsoidal, spherical and fibrous inclusions for various polydisperse structures and those of a layered structure composite have been presented The occurrence of pyromagnetic and electromagnetic coherence at composite macro level are caused by the interaction of piezoactive structural elements at micro level, these phenomena being absent in any specific phase separately Considerable influence of the shape of the inclusions, features of relative positioning and inversion of the properties of phases on the effective coefficients of pyromagnetic and electromagnetic coherence of ferrite / barium titanate composite material have been revealed The conclusion is drawn on the preferable use of the pyroelectric phase as spherical inclusions, and ferrite as the composite matrix This allows for more than a fivefold increase in the effective constant of pyromagnetic coherence *3 of the composite material in comparison with its value for the same structure but with inversion of properties of phases for constant volume fractions of the ferrite and pyroelectric phases Acknowledgements This work was financially supported by Russian Fundamental Research Fund Grant No 14-01-96004 r_ural_a References Korotkih, N I., Matveev N N., Sidorkin A S Pyroelectric properties of polyethylene oxide Physics of the Solid State 2009, vol 51, no 6, pp 1290-1292 DOI: 10.1134/S1063783409060328 Smirnova E P., Aleksandrov S E., Sotnikov K A., Kapralov A A., Sotnikov A V Pyroelectric effect in lead-magnoniobate-based solid solutions Physics of the Solid State 2003, vol 45, no 7, pp 1305-1309 DOI: 10.1134/1.1594247 Yarmarkin V K., Shul'man S G., Lemanov V V., Pankova G A Pyroelectric properties of some compounds based on protein aminoacids Physics of the Solid State 2005, vol 47, no 11, pp 21352137 DOI: 10.1134/1.2131157 Kerimov M K., Kurbanov M A., 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