ElectrostrictiveCouplingPolarDerivative

Overview

Computes the residual and jacobian contributions corresponding microforces generated by variations of polarization $\mathbf{P}$ of the electrostrictive free energy density $f_\mathrm{elecstr}$. The governing time-dependent Landau-Ginzburg-Devonshire (TDLGD) equation of relaxation of the ferroelectric order is given by,

$$\begin{aligned} \frac{\partial \mathbf{P}}{\partial t} = - \Gamma_P \frac{\delta f}{\delta \mathbf{P}}, \end{aligned}$$

and we look for variational derivatives of

$$\begin{aligned} f_\mathrm{elecstr} &= q_{11} \left(P_x^2 \varepsilon_{xx} + P_y^2 \varepsilon_{yy} + P_z^2 \varepsilon_{zz}\right) \\ &+ q_{12} \left[ \left(P_x^2 + P_y^2\right) \varepsilon_{zz} + \left(P_y^2 + P_z^2\right) \varepsilon_{xx} + \left(P_x^2 + P_z^2\right) \varepsilon_{yy}\right] \\ &+ q_{44} \left(P_x P_y \varepsilon_{yx} + P_x P_z \varepsilon_{xz} + P_y P_z \varepsilon_{yz}\right), \end{aligned}$$

suitable for cubic parent phase ferroelectric materials (i.e. $\mathrm{PbTiO}_3$ and $\mathrm{BaTiO}_3$). The coefficients $q_{11}, q_{12}$, and $q_{44}$ are the electrostrictive tensor components (in Voight notation). We can write this index notation in general for any symmetry $q_{ijkl}$ tensor,

$$\begin{aligned} f_\mathrm{elecstr} &= q_{ijkl} P_i P_j \varepsilon_{kl}, \end{aligned}$$

Note that for the variational derivative,

$$\begin{aligned} \frac{\delta f_\mathrm{elecstr}}{\delta \mathbf{P}} = \frac{\partial f_\mathrm{elecstr}}{\partial \mathbf{P}} - \frac{\partial}{\partial \mathbf{r}} \cdot \frac{\partial f}{\left(\frac{\partial \mathbf{P}}{\partial \mathbf{r}}\right)} \end{aligned}$$

only the first term is important because there are no explicit gradients in $\mathbf{P}$. Therefore, we have after multiplying the TDLGD equation by a test function $\psi_h$ and integrating on both sides, in index notation we have,

$$\begin{aligned} \left(\psi_h,\frac{\delta f_\mathrm{elecstr}}{\delta P_\beta}\right) = \left(\psi_h,\frac{\partial \left(q_{ijkl} P_i P_j \varepsilon_{kl}\right)}{\partial P_\beta}\right) &= 0 \\ = \left(\psi_h, 2 q_{ijkl} \delta_{i\beta} P_j \varepsilon_{kl}\right) &= 0\\ = \left(\psi_h, 2 q_{\beta jkl} P_j \varepsilon_{kl}\right) &= 0 \end{aligned}$$

Note that we ignore the time derivative as it is handled by another Kernel. This is our residual contribution for $P_\beta$.

$$\begin{aligned} \mathcal{R}_{P_\beta} = \left(\psi_h, 2 q_{\beta jkl} P_j \varepsilon_{kl}\right) \end{aligned}$$

Next we need the on-diagonal jacobian contribution,

$$\begin{aligned} \mathcal{J}_{P_\beta, P_\beta} &= \frac{\partial \mathcal{R}_{P_\beta}}{\partial P_\beta} \\ &= \left(\psi_h, 2 \phi \, q_{\beta \beta kl} \varepsilon_{kl}\right) \end{aligned}$$

where $\phi$ is a shape function of the finite element method. To compute the off-diagonal jacobian contributions, we have

$$\begin{aligned} \mathcal{J}_{P_\beta, P_\gamma} &= \frac{\partial \mathcal{R}_{P_\beta}}{\partial P_\gamma} \\ &= \left(\psi_h, 2 \phi \, q_{\beta \gamma kl} \varepsilon_{kl}\right). \end{aligned}$$

Rewriting the strain in terms of the symmetric and antisymmetric derivatives of the elastic displacement, we have

$$\begin{aligned} \mathcal{J}_{P_\beta, u_\gamma} = \frac{\partial \mathcal{R}_{P_\beta}}{\partial u_\gamma} &= \frac{\partial}{\partial u_\gamma }\left(\psi_h, q_{\beta jkl} P_j \left[\frac{\partial u_k}{\partial x_l} + \frac{\partial u_l}{\partial x_k}\right]\right)\\ &= \left(\psi_h, P_j \left[ q_{\beta jkl} \delta_{k\gamma}\frac{\partial \phi}{\partial x_l} + q_{\beta jkl} \delta_{l\gamma} \frac{\partial \phi}{\partial x_k}\right]\right) \\ &= \left(\psi_h, P_j \left[ q_{\beta j\gamma l} \frac{\partial \phi}{\partial x_l} + q_{\beta jk\gamma } \frac{\partial \phi}{\partial x_k}\right]\right) \end{aligned}$$

with again using the fact that $\partial \frac{\partial u_i}{\partial x_l} / \partial u_j = \delta_{ij} \frac{\partial \phi}{\partial x_l}$ with $\delta_{ij}$ the Kronecker product. This Kernel is hard-coded currently).The residual vector components, in Voight notation for a cubic symmetry parent phase ferroelectric, are,

$$\begin{aligned} \mathcal{R}_{P_x} &=\\ \mathcal{R}_{P_y} &=\\ \mathcal{R}_{P_z} &= \end{aligned}$$

and

$$\begin{aligned} \mathcal{J}_{P_x,P_x} &= \\ \mathcal{J}_{P_x,P_y} &=\\ \mathcal{J}_{P_x,P_z} &=\\ \mathcal{J}_{P_y,P_x} &=\\ \mathcal{J}_{P_y,P_y} &=\\ \mathcal{J}_{P_y,P_z} &=\\ \mathcal{J}_{P_z,P_x} &=\\ \mathcal{J}_{P_z,P_y} &=\\ \mathcal{J}_{P_z,P_z} &= \end{aligned}$$

Example Input File Syntax

Input Parameters

Input Files

• (test/tests/electrooptics/BTO_monodomain_T298K_REF.i)
• (test/tests/msca/BFO_P0A0.i)
• (test/tests/topology/A_skyrm.i)
• (tutorial/ferroelectric_domain_wall.i)
• (tutorial/BFO_dwP1A1_100.i)
• (tutorial/BFO_P111_TO_P111b_switch_m1_a1.i)
• (examples/domain_walls/BTO_wall_T298K.i)
• (examples/monodomain/BTO_monodomain_Tdef.i)
• (test/tests/electrooptics/BTO_monodomain_T298K_REFnoEO.i)
• (test/tests/dispersion/genDomain_PzEz.i)
• (test/tests/dispersion/genDomain_PyEz.i)
• (test/tests/film/PTO_film3_T298K.i)
• (examples/films/PZTfilm_e12_T298K_E0.i)
• (test/tests/auxkernels/surface_charge.i)
• (examples/monodomain/PZT_monodomain_Tdef.i)
• (test/tests/dispersion/perturbBTO_PyEz.i)
• (examples/other/PTO_E0.i)
• (test/tests/pbc/pbc.i)
• (test/tests/film/PTO_film3_T298K_noP.i)
• (examples/domain_walls/BTO_90wall_T298K.i)
• (test/tests/msca/BFO_dwP1A1_100.i)
• (test/tests/domain_wall/test_BTO_domain_wall.i)
• (tutorial/film.i)
• (test/tests/pbc/bcc_pbc.i)
• (test/tests/dispersion/perturbBTO_PzEz.i)
• (examples/monodomain/PTO_monodomain_Tdef.i)
• (test/tests/auxkernels/microforce.i)
• (examples/films/PTOfilm_e12_T298K_E0.i)
• (tutorial/BFO_homogeneous_PA.i)