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Shape, size, pressure and matrix effects on 2D spin crossover nanomaterials studied using density of states obtained by dynamic programming
(Elsevier, 2020-09-29)
In the present work, numerical simulations based on a new algorithm specific for 2D configurational topology of spin crossover nanoparticles embedded in a matrix are presented and discussed in the framework of the Ising-like model taking into account for short- (J) and long-range (G) interactions as for surface effects (L). The new algorithm is applied to calculate the density of states for each macro-state, which is then used to calculate exactly the thermal behavior of spin-crossover nanoparticles under an applied pressure. We find that the pressure plays the role of a conjugate parameter of the temperature. Thus, increasing pressure is somehow equivalent to reducing the temperature.
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Three states and three steps simulated within Ising-like model solved by local mean field approximation in 3D spin crossover nanoparticles
(Elsevier, 2021-03-01)
Coordination iron (II) compounds are studied to simulate switching properties between low spin (LS, S = 0) and high-spin (HS, S = 2) states in spin-crossover materials. These two states are diamagnetic (LS) and paramagnetic (HS) in nature, and the switching between these two states is achieved through external excitations which may be of thermal or of pressure origin. In this contribution, a local mean-field approach is proposed to study SCO nano/micro-particles, for which distinctions among the contributions of molecules localized at the edge, corner, surface or the bulk, as well as for the external coupling that concerns only surface particles have been introduced. In this first attempt, the model is solved using a rough approximation which simplifies its treatment, leading to finding out three steps switching and three states, simulated under temperature effect while two steps transitions are obtained under pressure effect.
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Hexagonal-shaped spin crossover nanoparticles studied by Ising-like model solved by local mean field approximation
(MDPI AG, 2021-05-01)
The properties of spin crossover (SCO) nanoparticles were studied for five 2D hexagonal lattice structures of increasing sizes embedded in a matrix, thus affecting the thermal properties of the SCO region. These effects were modeled using the Ising-like model in the framework of local mean field approximation (LMFA). The systematic combined effect of the different types of couplings, consisting of (i) bulk short-and long-range interactions and (ii) edge and corner interactions at the surface mediated by the matrix environment, were investigated by using parameter values typical of SCO complexes. Gradual two and three hysteretic transition curves from the LS to HS states were obtained. The results were interpreted in terms of the competition between the structure-dependent order and disorder temperatures (TO.D. ) of internal coupling origin and the ligand field-dependent equilibrium temperatures (Teq ) of external origin.
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A first-order phase transition studied by an Ising-like model solved by entropic sampling Monte Carlo method
(MDPI AG, 2021-04-01)
Two-dimensional (2D) square, rectangular and hexagonal lattices and 3D parallelepipedic lattices of spin crossover (SCO) compounds which represent typical examples of first order phase transitions compounds are studied in terms of their size, shape and model through an Ising-like Hamiltonian in which the fictitious spin states are coupled via the respective short and long-range interaction parameters J, and G. Furthermore, an environmental L parameter accounting for surface effects is also introduced. The wealth of SCO transition properties between its bi-stable low spin (LS) and high spin (HS) states are simulated using Monte Carlo Entropic Sampling (MCES) method which favors the scanning of macro states of weak probability occurrences. For given J and G, the focus is on surface effects through parameter L. It is shown that the combined first-order phase transition effects of the parameters of the Hamiltonian can be highlighted through two typical temperatures, TO.D., the critical order-disorder temperature and Teq the equilibrium temperature that is fixed at zero effective ligand field. The relative positions of TO.D. and Teq control the nature of the transition and mediate the width and position of the thermal hysteresis curves with size and shape. When surface effects are negligible (L = 0), the equilibrium transition temperature, Teq. becomes constant, while the thermal hysteresis’ width increases with size. When surface effects are considered, L ≠ 0, Teq. increases with size and the first order transition vanishes in favor of a gradual transition until reaching a threshold size, below which a reentrance phenomenon occurs and the thermal hysteresis reappears again, as shown for hexagonal configuration.
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A generalized Ising-like model for spin crossover nanoparticles
(MDPI, 2022-05-01)
Cooperative spin crossover (SCO) materials exhibit first‐order phase transitions in the solid state, between the high‐spin (HS) and low‐spin (LS) states. Elastic long‐range interactions are the basic mechanism for this particular behavior and are described well by the Ising‐like model, which allows the reproduction of most of the experimental results in the literature. Until now, this model has been applied with an interaction parameter between the molecules, which is considered to be independent of the states. In this contribution, we extend the Ising‐like model to include interaction energy that depends on the spin states and apply it to study SCO nanoparticles. Our research shows that following this new hypothesis, the equilibrium temperature shifts toward higher values.
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An efficient analytical method for determining transition temperatures in core-shell spin crossover nanoparticles described by an extended Ising-like model
(John Wiley and Sons Inc, 2024-12-12)
This contribution is dedicated to the analysis of the size and interaction parameters effects on the thermal properties of square-shaped cross-spin transition core-shell nanoparticles. The present study is carried out by varying the shell size at fixed core size and vice versa. In this problem, the core and shell are active from the spin transition point of view, while having different properties in terms of ligand field (transition temperatures) and interactions. The resulting nanoparticle is described by an Ising-type model including the three interaction parameters JCC, JSS and JCS coupling core-core, shell-shell and core-shell sites, respectively. The thermodynamic investigations have been carried out in 3 different ways: (i) by Monte Carlo Entropic Sampling (MCES) for small size nanoparticles ((Formula presented.)) (ii) by usual Monte Carlo Metropolis for higher dimensions (iii) by an analytical method deducing the equilibrium temperature of the system from the various local environments. We demonstrate that, as long as a clear plateau region occurs in the thermal-dependence of the high-spin fraction, an excellent agreement is obtained, allowing to predict analytically the evolution of the transition temperatures. The limitations of the method are discussed in the case of strong interference between the thermal dependences of the core and shell.
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Surface-Bulk 2D Spin-Crossover Nanoparticles within Ising-Like Model Solved by Using Entropic Sampling Technique
(MDPI, 2023-03-01)
We model the thermal effects in different 2D spin-crossover (SCO) square lattices within the frame of the Ising-like model using Monte Carlo entropic sampling (MCES) method to enhance the scan of macrostates beyond the most probable thermal ones. In fact, MCES allows access to the metastable states, and it is then well adapted to study thermal hysteresis properties. In this contribution, we distinguish, for the first time, the interaction between molecules located in bulk at the surface and those connecting the bulk and surface regions of an SCO lattice. In addition, an extra ligand field contribution is assigned to surface molecules through an interaction parameter (Formula presented.). In the absence of environmental effects on surface nanoparticles, a single thermal hysteresis loop increasing with the lattice size is simulated with a unique bulk and surface equilibrium temperature (Formula presented.). When environmental effects are accounted for, a two-step behavior associated with two hysteresis loops of widths ΔTS (for the surface) and ΔTB (for the bulk) with an intermediate plateau 14 K wide is obtained in the thermal dependence of the high-spin (HS) fraction for the 6 × 6 lattice. The surface and bulk equilibrium temperatures are then different, both decreasing towards lower values, and the (Formula presented.) parameter controls the three states’ behavior as well as the hysteresis loop interval. Size effects show that the equilibrium temperature is governed by the surface atoms for a small lattice size (5 × 5) and by the bulk atoms for a large lattice size (7 × 7). Moreover, a change in the size of the lattice results in a variation of the order–disorder (or Curie) temperature, TO.D., and the surface equilibrium temperature, Teq, while only TO.D. changes in bulk.
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A Comparative Study Between Monte Carlo Entropic Sampling Method and Local Mean Field Investigations of Thermal Properties of Spin-Crossover Nanoparticles Based on Ising-Like Model
(OAE Publishing Inc., 2023-12-01)
The thermally induced transitions between low-spin (LS) and high-spin (HS) configurations of spin-crossover (SCO) nanoparticles are simulated, focusing on the effects of localized surface and bulk interactions on the average magne-tization of 2D square lattices. The thermal behaviors and hysteresis cycles are investigated within the framework of the Ising model Hamiltonian and are conducted following two approaches: local mean field approximation (LMFA) and Monte Carlo entropic sampling (MCES) techniques. The results obtained by these two methods are compared for the two square lattice sizes, 6 × 6 and 7 × 7. Thus, when the bulk-surface interaction term is set to zero, the two approaches lead to identical values of the surface and bulk transition temperatures separated by a long intermediate plateau in both cases. Although hysteresis curves exhibit a similar shape, LMFA shows slightly larger widths ΔT than MCES. On increasing bulk-surface interaction term, the two methods lead to different shifts in equilibrium temperature values for both bulk and surface components, respectively, to lower and higher values by MCES. In general, it is found that LMFA shifts surface equilibrium temperature differently to lower values and enhances the hysteresis effect, particularly for surface molecules. On the other hand, for the 7 × 7 square lattice, the equilibrium temperatures are slightly higher by 1.5% and 3.2% for bulk and surface molecules, respectively, with a narrower hysteresis width in the surface. Moreover, with the MCES method, an abrupt transition instead of a hysteresis transition is calculated for surface molecules (Formula Presented).
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The Role of Substrates in 2D Spin-Crossover Systems: Insights From Monte Carlo Simulations Within the Ising-Like Model
(Wiley, 2025-11-27)
Spin-crossover (SCO) molecular solids are a class of coordination compounds exhibiting hysteretic thermal transitions between low-spin (LS) and high-spin (HS) states, making them capable of collective switching between these two states in response to external stimuli such as temperature, pressure, and electric fields. This bistable behavior directly paves the way for breakthrough technological applications in the field of molecular sensors, molecular switches, and actuators. For thermally induced spin transitions, the transition temperature ( T up ) on heating, at which the system switches from the LS to the HS, is strongly influenced by the ligands coordinating the metal center. In this study, we investigate the effect of the substrate on T up in SCO nanostructures, focusing on how subtle substrate-induced interactions can modulate the transition temperature. To this end, we model substrate effects through an extended Ising-like Hamiltonian, solved using Monte Carlo simulations. The results show that substrate interactions can be used to significantly fine-tune the thermal transition temperature, modify the width of the hysteresis, and induce either abrupt or gradual switching as needed. This groundbreaking control offers a radical new perspective for the design and optimization of next-generation SCO devices, enabling the creation, among other applications, of precision-engineered temperature sensors for complex systems.
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Monte Carlo Simulations of Thermal Behavior in Two-Block Spin-Crossover Structures
(Multidisciplinary Digital Publishing Institute (MDPI), 2026-05-01)
Molecular spin-crossover (SCO) compounds constitute prototypical systems exhibiting first-order phase transitions. These transitions involve an abrupt switch between two well-defined states with distinctly different magnetic, optical, and vibrational properties. One state is diamagnetic (low-spin), while the other is paramagnetic (high-spin). Upon heating, the transition occurs at a characteristic temperature, Tup. Upon cooling, it takes place at a lower temperature, Tdown < Tup, thereby giving rise to thermal hysteresis. Accordingly, each SCO compound is defined by a distinct pair of transition temperatures, Tup and Tdown. The investigation of these molecular solids is of great importance, both for elucidating first-order phase transitions—including the potential emergence of re-entrant phases—and for their broad range of prospective applications. The critical temperatures Tup and Tdown are pivotal in defining their practical utility. We present a strategy to modify and tune the transition temperatures of spin-crossover (SCO) compounds to suit different applications. The approach combines a given SCO material with layers of a second SCO system, enabling precise control of the characteristic temperatures of the resulting heterostructure. We illustrate this method with three case studies that span the 100 K–400 K temperature range. All simulations were performed using Monte Carlo methods within the Metropolis algorithm framework.
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