Landau-Lifshitz-Bloch simulations of the magnetocaloric effect in continuous ferromagnetic-paramagnetic transitions

TL;DR

LLB micromagnetic simulations are used to model the magnetocaloric effect near the Curie transition in ferromagnets, addressing both ferromagnetic and paramagnetic regimes for monocrystalline and polycrystalline samples with varying anisotropy. The method computes isothermal entropy changes via Maxwell relations and analyzes the universal curve and the exponent n to characterize field dependence. Results show close agreement with Brillouin theory for simple cases and reveal how demagnetizing fields, anisotropy, and microstructure influence Delta s_iso and n, validating micromagnetic modeling as a tool for complex magnetocaloric materials. This approach enables systematic exploration of microstructure effects and provides a framework for predicting magnetocaloric performance in new materials.

Abstract

The usefulness of modeling magnetocaloric materials expands from the understanding of their behavior to the prediction of new materials, playing a fundamental role in the optimization of their performance. In contrast with other areas of magnetic materials research, micromagnetic simulations of magnetocaloric materials are scarce due to the difficulty of modeling the material in the vicinity of the transition. To solve this limitation, we propose the use of micromagnetic simulations based on the Landau-Lifshitz-Bloch equation to study the magnetocaloric effect of a ferromagnetic material around its Curie transition. Following our proposed methodology, we obtain reliable isothermal entropy change curves for both monocrystalline and polycrystalline configurations, where we consider different anisotropic contributions. The robustness of the method was evaluated, yielding results that agreed with previous experimental and theoretical observations. Our study shows that micromagnetic simulations are a powerful tool for analyzing magnetocaloric materials with complex microstructures.

Figures (5)

• Figure 1: (a) Temperature dependence of LLB inputs: zero-field reduced magnetization (up) and reduced longitudinal susceptibility (down). (b) Equilibrium magnetization at $\qty{280}{\kelvin}$ and $\qty{0.001}{\tesla}$ for a monocrystal composed of a $128\times128\times128$ cells. (c) Polycrystalline sample with microstructure generated by Voronoi tesselation (each grain region is colored to distinguish it).
• Figure 2: Temperature dependence of the (a) longitudinal magnetization, (b) isothermal entropy change, (c) phenomenological universal curve, and (c) exponent $n$ for different applied fields and field changes. The hollow symbols correspond to the micromagnetic results, while the solid lines corresponds to the Brillouin solutions. Most symbols appear filled as they are superimposed with solid lines.
• Figure 3: (a) Temperature dependence of the magnetization for low applied fields for a sample with cubic geometry. Temperature dependence of the (b) isothermal entropy change, (c) phenomenological universal curve, and (d) exponent $n$ for different samples and orientations (labeled by an effective demagnetizing factor $N$). Panel (d) inset: exponent $n$ at $T_{\mathrm{C}}$ as a function of $N$ for 0.5 T and 1.0 T. The hollow circles are superimposed with symbols and appear as filled circles in the paramagnetic range.
• Figure 4: (a) Temperature dependence of the magnetization for low applied fields along easy and hard directions for a sample with temperature-dependent uniaxial anisotropy (inset). Calculated (b) isothermal entropy change, (c) phenomenological construction, and (d) exponent $n$ for different applied field changes along easy and hard axis. The hollow circles are superimposed with symbols and appear as filled circles in the paramagnetic range.
• Figure 5: Isothermal entropy change (left) and exponent $n$ (right) magnitudes for: 1) polycrystal (pc) with randomly orientated grains and monocrystal (mc) for both easy (e.o.) and hard (h.o.) orientations [(a) and (b)], 2) polycrystal with randomly orientated grains with different exchange coupling between grains, $C$ [(c) and (d)] and 3) oriented polycrystal with different mean grain size, $D$ [(e) and (f)]. Corresponding phenomenological construction of panel (a) data is shown in as an inset (same scale and units as previous plots).