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Local mean field approximation applied to a 3D spin crossover nanoparticles configuration: free energy analysis of the relative stability of the stationary states

Abstract : The local mean field approximation is applied to an inhomogeneous 3D spin crossover (SCO) nanoparticle configuration with a special focus on its systemic effect on molecules which are localized in the bulk, at the corner, at the edge and at the surface. The matrix effect at the surface is introduced through a specific interaction term, L. The partition function for each region allows the determination of the total free-energy F from which the stability of each configuration is analyzed through thermodynamic considerations. 1.Introduction Fe(II) Spin-crossover (SCO) shows a particular first-order phase transition, with thermal hysteresis [1-5] that is mediated between two spin states, Low-Spin (LS) with degeneracy g LS , stable at low temperatures and High-Spin (HS) with degeneracy g HS (> g LS), stable at high temperatures. Between two temperatures, namely termed T down and T up a SCO molecule can be in one of these two states depending on its thermal history. This "bi-stable" character is the result of the competition between the ligand field energy acting on each spin-state and the elastic interactions between the molecules. 2. Model In the framework of the Ising-like model [6-11] each SCO molecule is described by a two-level fictitious spin having two eigenvalues and , respectively associated with the HS and LS states. The total Hamiltonian of a system of molecules, taking into account the short (J)-and long-range (G) interactions, as well as the matrix effect (L), is expressed as: where g = g HS /g LS , is the energy difference between the (HS) and (LS) states, T is the absolute temperature, is the Boltzmann constant, and M is the total number of molecules located at the surface. In this contribution, the local mean-field approximation (LMFA) [12] which consists in replacing the spin state of each neighbor by its mean value is applied. Accordingly, the short (J)-and the long (G) range interactions are replaced by only one coupling interaction Γ. The global Hamiltonian of the system can then be rewritten as follows: (1)
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C. Cazelles, Jörge Linares, Y. Singh, Pierre-Richard Dahoo, K. Boukheddaden. Local mean field approximation applied to a 3D spin crossover nanoparticles configuration: free energy analysis of the relative stability of the stationary states. 9th International Conference on Mathematical Modeling in the Physical Sciences, Sep 2020, Tinos island, Greece. ⟨insu-02956374⟩

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