Large enhancement of magnetic-field-induced strain in two-phase ferromagnetic nanodispersions

A phase-field model is implemented to investigate the microstructure formation and evolution of ferromagnetic alloys consisting of ferromagnetic nanoprecipitates of low-symmetry phase coherently embedded into a ferromagnetic cubic matrix. It is shown that the nanodispersions produced at the early stage of decomposition are structurally in single-domain state because their nanosize makes the formation of multidomain state energetically unfavorable. An application of magnetic fields could rearrange the crystal lattices within the nanosized precipitates through a gradual switching of the crystal orientations in the case of strong elastic anisotropy and magnetocrystalline coupling. The resultant macroscopic strain responses (magnetostrictions) could be largely enhanced and low hysteretic. We also formulate conditions for finding such materials with a combination of excellent mechanical and functional properties but free from the rare-earth elements.

Figure 1

Visualization of the intrinsic free energy,$f_{\varepsilon }^{\beta }\left({\varepsilon }_{ij}^o\right)$, on the plane of $\varepsilon {}_{ii}^{0*}={\varepsilon }_c^{*}+2{\varepsilon }_a^{*}$ for a case with a large elastic anisotropy (${\varepsilon }_{ii}^{00*}=1/3$). Visualizations for cases without (${\varepsilon }_{ii}^{00*}=0$) or with a weaker anisotropy (${\varepsilon }_{ii}^{00*}=1/7$) can be found in Ref. [18]. It is noted that the energy outside of the visualized plane is much higher, and thus not shown for clarity. The topology of energy landscapes and the heights of energy barriers separating different minima are very similar in all three cases, except for positions of minimized states.

Figure 2

Ferromagnetic two-phase microstructure simulated with an initial composition of $c_0=0.5$ for case II. In (a), nanosized tetragonal precipitates of two different orientation variants are shown in red and green. The cubic matrix is the other part of the simulation box, which is set as transparent for clarity. In (b) and (c), the magnetizations are visualized by arrows. (c) is a slice at the location indicated by the blue line in (b), where the black represents the matrix, and the red and green are precipitates of the first and second orientation variants, respectively.

Figure 3

Simulated 3D microstructures (a)–(e) of the frozen nanodispersions shown in Fig. 2 and the corresponding longitudinal strain responses (f) to a cyclic magnetic field applied along the [100] direction. No bias stress is applied during the simulation. Arrows in (a)–(e) show the magnetization directions. In (f), letters of (a)–(e) indicate the corresponding strain responses of (a)–(e), and $M_s$ and N are the saturation magnetization and demagnetization factor, respectively. The magnitudes of H field for (a)–(e) are 0.8, 0.6, 0.2, −0.2, and $-0.6N{M}_s$, respectively.

Figure 4

Simulated 3D microstructures (a)–(e) of the frozen nanodispersions shown in Fig. 2 and the longitudinal strain responses (f) to a cyclic magnetic field applied along the [100] direction. A constant uniaxial bias stress is applied along the [100] direction during the simulation. Arrows in (a)–(e) show the magnetization directions. In (f), the longitudinal strain responses of a cubic sample (green) and a tetragonal sample (blue) are also simulated for comparison; letters of (a)–(e) indicate the corresponding strain responses of (a)–(e); the magnitudes of H field for (a)–(e) are 0.95, 0.6, 0.0, −0.6, and $-0.95N{M}_s$, respectively.

Figure 5

Pseudo-3D simulation results. (a) Tweedlike microstructure with the white and dark regions representing precipitates of different orientation variants, and the gray regions representing the matrix. (b) Zoom-in of the white box region in (a), where precipitates of two different orientation variants are visualized in red and blue, and the matrixes are shown in green. (c) Reduced longitudinal strain response, and (d)–(i) simulated magnetic domains evolution of the nanostructure in (a) under a cyclic H field and a uniaxial constant bias stress both applied along the horizontal direction. Colors in (d)–(i) distinguish magnetizations of different directions, and white segments in (e)–(i) show the actual directions of magnetizations. In (c), letters of (d)–(i) show the corresponding strain responses of (d)–(i). The H fields for (d)–(i) are 0.8, 0.4, 0.0, −0.4, −0.8, and $-1.2N{M}_s$, respectively.

Figure 6

The as-quenched ferromagnetic two-phase microstructures simulated with different initial composition $c_0$ for case II, (a)–(e),and their longitudinal strain responses, (f), to a cyclic H field and a uniaxial constant bias stress both applied along the [100] direction. Nanosized particles in red, green, and blue in (a)–(e) are tetragonal precipitates of three different orientation variants. The cubic matrix is the other part of the simulation box, which is set as transparent for clarity. The reduced simulation time steps for obtaining (a)–(e) are $1.5\times {10}^5$, $1.0\times {10}^5$, $6\times {10}^4$, $3\times {10}^4$, and $2\times {10}^4$, respectively. In (f), the strain responses are labeled by letters of (a)–(e) for the initial nanodispersions in (a)–(e), respectively, where the responses of single-phase materials are also shown for comparison.

Figure 7

Pseudo-3D tweedlike microstructures for case I (a) and case III (b) simulated at $c_0=0.5$ and their longitudinal strain responses, (c), under a cycling H field and a uniaxial constant bias stress both applied along the horizontal direction. In (a) and (b), red and blue show nanoprecipitates of two different orientation variants of the tetragonal phase, and green shows the cubic matrix. White segments are guide to the eye showing a characterizing direction of nanoprecipitates. The time steps for obtaining nanodispersions of (a) and (b) are $2\times {10}^4$ and $6\times {10}^4$, respectively. The responses of case II [Fig. 5] are also shown in (c) as the blue line.

Figure 8

An illustration showing the decrease of energy barrier of a double-well potential by increasing the strength of an externally applied bias field. The calculated potential is $E(x,f)=A{\left(x-0.3\right)}^2{\left(x-0.7\right)}^2-xf$, where A is a parameter set as 100 for simplicity, $x$ and $f$ are the order parameter and the applied field that are coupled to each other. The double-head arrows demonstrate the height of the energy barriers for the cases of $f=0$ and $f=f_1$. The energy barrier even vanished at a high bias field, e.g., $f=f_2$. For better comparison, curves for $f=f_1$ and $f=f_2$ are vertically shifted to show the difference of the barriers.