As the energy consumed by cooling systems is expected to increase over the next decades, magnetocaloric cooling systems are of high interest as they are a promising more energy efficient alternative to conventional vapor compression refrigerators. 

Since the discovery of the giant magnetocaloric effect (MCE), several promising material systems, e.g. La-Fe-Si-, Fe$_2$P-based, and Heusler alloys have been studied extensively for potential application in magnetocaloric cooling devices. These materials are of high interest, as their MCE properties like isothermal entropy change $\Delta S_\text{T}$, adiabatic temperature change $\Delta T_\text{ad}$, and Curie temperature $T_C$ are tunable by the elemental composition. However, systematic studies on the effect of microstructure and mangetocrystalline anisotropy are limited.

As tuning of both is difficult experimentally, theoretical studies of the relation between microstructure and anisotropy and the MCE are of high interest. We propose a methodology of calculating the MCE in second-order-type materials, like Fe$_2$P-based alloys, which combines micromagnetic simulations with the Arrott-Noakes equation (ANE).

In this work, the proposed methodology is used to investigate the influence of microstructure and magnetocrystalline anisotropy on the MCE in Co$_2$B nanograins. In detail, we propose a three step calculation of $\Delta S_\text{T}$. Firstly, micromagnetic simulations are performed using the 3D NIST Object Oriented MicroMagnetic Framework (OOMMF) code in which the microstructure and magnetocrystalline anisotropy can be varied. These simulations are performed for temperatures up to $T_C$ to obtain magnetization curves. Secondly, the parameters of the ANE are determined from the simulation results. The ANE is an equation of state, which phenomenologically describes the temperature and field dependence of the magnetization of SOPT materials in the vicinity of $T_C$. It allows us to extrapolate the simulation results to temperatures larger than $T_C$. Finally, the parameterized ANE is used to calculate $\Delta S_\text{T}$ around $T_C$ according to the Maxwell relations.

For investigating the influence of the microstructure on $\Delta S_\text{T}$, we create a simple model geometry composed of 4x4x4 cube-shaped Co$_2$B nanograins with straigth grain boundaries. Grain sizes range from 5 to 100\,nm, while grain boundary thicknesses range from 0 to 10\,nm. $\Delta S_\text{T}$ calculated by our methodology shows good agreement with the experimental results. For the dependence of $\Delta S_\text{T}$ on the microstructure, we found that decreasing grain sizes result in increased $\Delta S_\text{T}$, while the influence of the grain boundary thickness is negligible. We also found that larger magnetocrystalline anisotropies facilitate higher isothermal entropy changes.

Our proposed methodology is supposed to help optimizing the MCE by tuning the microstructure in materials with a second-order phase transition (SOPT). It bridges the gap between micromagnetic simulations, which can only be performed up to $T_C$, and the calculation of \deltaS in the vicinity of $T_C$. It allows for more systematic studies on the relation between microstructural and intrinsic properties and the MCE. 

The authors acknowledge the support from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (Grant agreement No 743116, project CoolInnov) and from the CRC/TRR 270 HoMMage (DFG).