FEM analysis of a multiferroic nanocomposite: comparison of experimental data and numerical simulation

Abstract

In this contribution, we analyze the properties of two-phase magneto-electric (ME) composites. Such ME composite materials have raised scientific attention in the last decades due to many possible applications in a wide range of technical devices. Since the effective magneto-electric properties significantly depend on the microscopic morphology, we investigate in more detail the changes in the in-plane polarization due to an applied magnetic field. It was shown in previous works that it is possible to grow vertically aligned nanopillars of magnetostrictive cobalt ferrite in a piezoelectric barium titanate matrix by pulsed laser deposition. Based on x-ray linear dichroism, the displacements of titanate ions in the matrix material can be measured due to an applied magnetic field near the boundary of the interface between the matrix and the nanopillars. Here, we focus on (1–3) fiber-induced composites, based on previous experiments, where cobalt ferrite nanopillars are embedded in a barium titanate matrix. In the numerical simulations, we adjusted the boundary value problem to match the experimental setup and compare the results with previously made assumptions of the in-plane polarizations. A further focus is taken on the deformation behavior of the nanopillar over its whole height. Such considerations validate the assumption of the distortion of the nanopillars under an external magnetic field. Furthermore, we analyze the resulting magneto-electric coupling coefficient.

Appendix

The equations for the stress field ${\varvec{\sigma }}$, dielectric displacement ${{\varvec{D}}}$ and magnetic induction ${{\varvec{B}}}$ for piezoelectricity and piezomagnetism in direct and index notation

$$\begin{aligned} {\varvec{\sigma }}= & {} \mathbb {C} : {\varvec{\varepsilon }}- {{\varvec{e}}}^T \cdot {{\varvec{E}}}- {{\varvec{q}}}^T \cdot {{\varvec{H}}}\; , \quad \sigma _{ij} = \mathbb {C}_{ijkl} \varepsilon _{kl} - e_{kij} E_{k} - q_{kij} H_{k} \; , \end{aligned}$$

(14)

$$\begin{aligned} {{\varvec{D}}}= & {} {{\varvec{e}}}: {\varvec{\varepsilon }}+ {\varvec{\epsilon }}\cdot {{\varvec{E}}}+ {\varvec{\alpha }}\cdot {{\varvec{H}}}\;, \quad D_{i} = e_{ijk} \varepsilon _{jk} + \epsilon _{ik} E_k + \alpha _{ik} H_k \;, \end{aligned}$$

(15)

and

$$\begin{aligned} {{\varvec{B}}}= {{\varvec{q}}}: {\varvec{\varepsilon }}+ {\varvec{\alpha }}\cdot {{\varvec{E}}}+ {\varvec{\mu }}\cdot {{\varvec{H}}}\;, \quad B_{i} = q_{ijk} \varepsilon _{jk} + \alpha _{ik} E_k + \mu _{ik} H_k \;. \end{aligned}$$

(16)

They can be reformulated to

$$\begin{aligned} {\varvec{\varepsilon }}= \mathbb {C}^{-1} {\varvec{\sigma }}+ \underbrace{\mathbb {C}^{-1} : {{\varvec{e}}}^{T}}_{{{\varvec{d}}}_e} \cdot {{\varvec{E}}}+ \underbrace{\mathbb {C}^{-1} : {{\varvec{q}}}^{T}}_{{{\varvec{d}}}_q} \cdot {{\varvec{H}}}\,. \end{aligned}$$

(17)

With (17), Eqs. (15) and (16) can be reformulated to

$$\begin{aligned} {{\varvec{D}}}= {{\varvec{d}}}_e \cdot {\varvec{\sigma }}+ ({{\varvec{e}}}: {{\varvec{d}}}_e + {\varvec{\epsilon }}) \cdot {{\varvec{E}}}+ ({{\varvec{e}}}: {{\varvec{d}}}_q + {\varvec{\alpha }}) \cdot {{\varvec{H}}}\; , \end{aligned}$$

(18)

and

$$\begin{aligned} {{\varvec{B}}}= {{\varvec{d}}}_q \cdot {\varvec{\sigma }}+ ({{\varvec{q}}}: {{\varvec{d}}}_e + {\varvec{\alpha }}) \cdot {{\varvec{E}}}+ ({{\varvec{q}}}: {{\varvec{d}}}_q +{\varvec{\mu }}) \cdot {{\varvec{H}}}\; . \end{aligned}$$

(19)

In our approach, we use a coordinate-invariant setting based on the following invariants for the transversal isotropic material law

$$\begin{aligned} I_1= & {} \text {tr}\,{\varvec{\varepsilon }}\; , \quad I_2 = \text {tr}\, {\varvec{\varepsilon }}^2 \; , \quad I_4 = \text {tr}\,({\varvec{\varepsilon }}{{\varvec{m}}})\; , \quad I_5 = \text {tr}({\varvec{\varepsilon }}^2 {{\varvec{m}}})\; , \nonumber \\ J_1^e= & {} \text {tr} \,({{\varvec{E}}}\otimes {{\varvec{E}}}) \; , \quad J_2^e = \text {tr} \,({{\varvec{E}}}\otimes {{\varvec{a}}})\; , \quad K_1^e = \text {tr}\,({\varvec{\varepsilon }}({{\varvec{E}}}\otimes {{\varvec{a}}}))\; , \nonumber \\ J_1^m= & {} \text {tr} \,({{\varvec{H}}}\otimes {{\varvec{H}}}) \; , \quad J_2^m = \text {tr} \,({{\varvec{H}}}\otimes {{\varvec{a}}})\; , \quad K_1^m = \text {tr}\,({\varvec{\varepsilon }}({{\varvec{H}}}\otimes {{\varvec{a}}}))\; , \end{aligned}$$

(20)

with the second-order structural tensor ${{\varvec{m}}}:= {{\varvec{a}}}\otimes {{\varvec{a}}}$, then the quadratic electric enthalpy function $\psi ^{*}$ is given with

$$\begin{aligned} \psi ^{*}= & {} \dfrac{1}{2} \lambda \, I_1^2 \, + \, \mu \, I_2\,+\, \omega _1\, I_5\, + \,\omega _2\, I_4^2\,+\,\omega _3 \, I_1\, I_4\nonumber \\&+\, \beta _1\, I_1\, J_2^e\,+\,\beta _2\, I_4\,J_2^e\,+\,\beta _3\, K_1^e\,+\,\gamma _1\,J_1^e\,+\,\gamma _2\,(J_2^e)^2\nonumber \\&+\, \kappa _1\, I_1\, J_2^m\,+\,\kappa _2\, I_4\,J_2^m\,+\,\kappa _3\, K_1^m\,+\,\xi _1\,J_1^m\,+\,\xi _2\,(J_2^m)^2 \; . \end{aligned}$$

(21)

Then, ${\varvec{\sigma }}= \partial _{{\varvec{\varepsilon }}}\psi ^{*}$, ${{\varvec{D}}}= -\partial _{{{\varvec{E}}}}\psi ^{*}$ and ${{\varvec{B}}}= -\partial _{{{\varvec{H}}}}\psi ^{*}$ appear as

$$\begin{aligned} {\varvec{\sigma }}= & {} \left( 2\,\omega _2\,I_4\,+\,\omega _3\,I_1 + \beta _2 \, J_2^e + \kappa _2 \, J_2^m \right) \,{{\varvec{m}}}+ 2 \, \mu \, {\varvec{\varepsilon }}\nonumber \\&+ \left( \lambda \; I_1 + \omega _3 \, I_4 + \beta _1 J_2^e + \kappa _1 J_2^m \right) \, \mathbf 1 + \omega _1 \left( {{\varvec{a}}}\,\otimes \,{\varvec{\varepsilon }}\,\cdot \,{{\varvec{a}}}+ {{\varvec{a}}}\,\cdot \,{\varvec{\varepsilon }}\,\otimes \,{{\varvec{a}}}\right) \nonumber \\&+ \dfrac{1}{2} \,\beta _3\,\left( {{\varvec{a}}}\,\otimes \,{{\varvec{E}}}\,+\,{{\varvec{E}}}\,\otimes \,{{\varvec{a}}}\right) + \dfrac{1}{2} \,\kappa _3\,\left( {{\varvec{a}}}\,\otimes \,{{\varvec{H}}}\,+\,{{\varvec{H}}}\,\otimes \,{{\varvec{a}}}\right) \; , \end{aligned}$$

(22)

$$\begin{aligned} {{\varvec{D}}}= & {} - (2\,\gamma _2\,J_2^e \,+\,\beta _1\,I_1 \,+\, \beta _2\,I_4)\,{{\varvec{a}}}- 2\,\gamma _1\,{{\varvec{E}}}\,- \beta _3\,{{\varvec{a}}}\,\cdot \, {\varvec{\varepsilon }}\;, \end{aligned}$$

(23)

and

$$\begin{aligned} {{\varvec{B}}}= - (2\,\xi _2\,J_2^m \,+\,\kappa _1\,I_1 \,+\, \kappa _2\,I_4)\,{{\varvec{a}}}- 2\,\xi _1\,{{\varvec{H}}}\,- \kappa _3\,{{\varvec{a}}}\,\cdot \, {\varvec{\varepsilon }}\;. \end{aligned}$$

(24)

With the second derivatives of Eqs. (22), (23) and (24), $\mathbb {C} = \partial _{{\varvec{\varepsilon }}}{\varvec{\sigma }}$, ${{\varvec{e}}}= \partial _{{\varvec{\varepsilon }}}{{\varvec{D}}}$, ${{\varvec{q}}}= \partial _{{\varvec{\varepsilon }}}{{\varvec{B}}}$, ${\varvec{\epsilon }}= \partial _{{{\varvec{E}}}}{{\varvec{D}}}$ and ${\varvec{\mu }}= \partial _{{{\varvec{H}}}}{{\varvec{B}}}$, we get

$$\begin{aligned} \mathbb {C}= & {} \lambda \mathbf 1 \otimes \mathbf 1 \,+\, \omega _3 \,(\mathbf 1 \otimes {{\varvec{m}}}+ {{\varvec{m}}}\otimes \mathbf 1 ) \,+\, 2\,\omega _2\, {{\varvec{m}}}\otimes {{\varvec{m}}}\, + \, 2 \,\mu \mathbb {I} \,+\,\omega _1 \Xi \; , \end{aligned}$$

(25)

$$\begin{aligned} {{\varvec{e}}}= & {} -\beta _1 {{\varvec{a}}}\otimes \mathbf 1 \, - \,\beta _2\,{{\varvec{a}}}\otimes {{\varvec{m}}}\, - \,\beta _3\, {\varvec{\theta }}\;, \end{aligned}$$

(26)

$$\begin{aligned} {{\varvec{q}}}= & {} -\kappa _1 {{\varvec{a}}}\otimes \mathbf 1 \, - \kappa _2 {{\varvec{a}}}\otimes {{\varvec{m}}}- \,\kappa _3\, {\varvec{\theta }}\;, \end{aligned}$$

(27)

$$\begin{aligned} {\varvec{\epsilon }}= & {} -2\,\gamma _1\, \mathbf 1 \, + \,2\,\gamma _2\,{{\varvec{m}}}\;, \end{aligned}$$

(28)

$$\begin{aligned} {\varvec{\mu }}= & {} -2\,\xi _1\, \mathbf 1 \, - \,2\,\xi _2\,{{\varvec{m}}}\;, \end{aligned}$$

(29)

with the fourth-order unit tensor $\mathbb {I}$, $\Xi = {{\varvec{a}}}\otimes \mathbf 1 \otimes {{\varvec{a}}}+ {{\varvec{a}}}\otimes \mathbf 1 \otimes {{\varvec{a}}}$ and ${\varvec{\theta }}= \dfrac{1}{2}({{\varvec{a}}}\otimes \mathbf 1 + {{\varvec{a}}}\otimes \mathbf 1 )$. Taking these equations written in Voigt notation and renaming the parameters which are equal, we get with the preferred direction ${{\varvec{a}}}= {{\varvec{x}}}_3$ of the transversal isotropic material

$$\begin{aligned} \left[ \begin{array}{r} \sigma _{11} \\ \sigma _{22} \\ \sigma _{33} \\ \sigma _{12} \\ \sigma _{23} \\ \sigma _{13} \\ \end{array} \right] = \left[ \begin{array}{cccccc} c_{1111} &{} c_{1122} &{} c_{1133} &{} 0 &{} 0 &{} 0 \\ c_{1122} &{} c_{2222} &{} c_{2233} &{} 0 &{} 0 &{} 0 \\ c_{1133} &{} c_{2233} &{} c_{3333} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} c_{1313} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} c_{1212} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} c_{1212} \\ \end{array} \right] \left[ \begin{array}{c} \varepsilon _{11} \\ \varepsilon _{22} \\ \varepsilon _{33} \\ 2\varepsilon _{12} \\ 2\varepsilon _{23} \\ 2\varepsilon _{13} \\ \end{array} \right] - \left[ \begin{array}{ccc} 0 &{} 0 &{} e_{311} \\ 0 &{} 0 &{} e_{311} \\ 0 &{} 0 &{} e_{333} \\ 0 &{} 0 &{} 0 \\ 0 &{} e_{123} &{} 0 \\ e_{123} &{} 0 &{} 0 \\ \end{array} \right] \left[ \begin{array}{c} E_{1} \\ E_{2} \\ E_{3} \\ \end{array} \right] \, , \end{aligned}$$

(30)

and

$$\begin{aligned} \left[ \begin{array}{c} D_{1} \\ D_{2} \\ D_{3} \\ \end{array} \right] = \left[ \begin{array}{cccccc} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} e_{123}\\ 0 &{} 0 &{} 0 &{} 0 &{} e_{123} &{} 0 \\ e_{311} &{} e_{311} &{} e_{333} &{} 0 &{} 0 &{} 0 \\ \end{array} \right] \left[ \begin{array}{c} \varepsilon _{11} \\ \varepsilon _{22} \\ \varepsilon _{33} \\ 2\varepsilon _{12} \\ 2\varepsilon _{23} \\ 2\varepsilon _{13} \\ \end{array} \right] + \left[ \begin{array}{ccc} \epsilon _{11} &{} 0 &{} 0 \\ 0 &{} \epsilon _{11} &{} 0 \\ 0 &{} 0 &{} \epsilon _{33} \\ \end{array} \right] \left[ \begin{array}{c} E_{1} \\ E_{2} \\ E_{3} \\ \end{array} \right] \;. \end{aligned}$$

(31)

We can identify

$$\begin{aligned} \begin{array}{ccc} \lambda = c_{1122} \;, &{}\mu = \dfrac{1}{2}(c_{1111}-c_{1122})\;, &{}\omega _1 = 2\, c_{1212} + c_{1122} - c_{1111} \;, \\ \omega _2 = \dfrac{1}{2}(c_{1111} + c_{3333}) - 2\,c_{1212} - c_{1133} \;, &{} \omega _3 = c_{1133} - c_{1122} \;, &{}\beta _1 = -e_{311} \;, \\ \beta _2 = e_{311} - e_{333} + 2\,e_{123} \;, &{} \beta _3 = -2\;e_{123}\;, &{}\gamma _1 = -\dfrac{1}{2} \epsilon _{11}\;, \\ \gamma _2 = \dfrac{1}{2}(\epsilon _{11} - \epsilon _{33})\;, &{}\kappa _1 = -q_{311}\;, &{}\kappa _2 = q_{311} - q_{333} + 2\,q_{123}\;, \\ \kappa _3 = -2\;q_{123}\;, &{}\xi _1 = - \dfrac{1}{2} \mu _{11}\;, &{}\xi _2 = \dfrac{1}{2} (\mu _{11} -\mu _{33})\;.\\ \end{array} \end{aligned}$$

(32)