Monte Carlo Entropic Sampling Research Articles

AN EFFICIENT ANALYTICAL METHOD FOR DETERMINING TRANSITION TEMPERATURES IN CORE‐SHELL SPIN CROSSOVER NANOPARTICLES DESCRIBED BY AN EXTENDED ISING‐LIKE MODEL

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 ([[EQUATION]]) (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.

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

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 magnetization 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 $$ \times $$ 6 and 7 $$ \times $$ 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 $$ \Delta T $$ 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 $$ \times $$ 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 ($$ \Delta T_{surf}^{MCES}=0 $$ K).

Surface-Bulk 2D Spin-Crossover Nanoparticles within Ising-like Model Solved by Using Entropic Sampling Technique

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 L. 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 Teq=Teqbulk= Teqsurf. 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 L 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.

A First Order Phase Transition Studied by an Ising-Like Model Solved by Entropic Sampling Monte Carlo Method

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.

Monte Carlo entropic sampling applied to Ising-like model for 2D and 3D systems

In this paper we present the Monte Carlo entropic sampling (MCES) applied to an Ising-like model for 2D and 3D system in order to show the interaction influence of the edge molecules of the system with their local environment. We show that, as for the 1D and the 2D spin crossover (SCO) systems, the origin of multi steps transition in 3D SCO is the effect of the edge interaction molecules with its local environment together with short and long range interactions. Another important result worth noting is the co-existence of step transitions with hysteresis and without hysteresis. By increasing the value of the edge interaction, L, the transition is shifted to the lower temperatures: it means that the role of edge interaction is equivalent to an applied negative pressure because the edge interaction favours the HS state while the applied pressure favours the LS state. We also analyse, in this contribution, the role of the short- and long-range interaction, J respectively G, with respect to the environment interaction, L.

Multi-Step in 3D Spin Crossover Nanoparticles Simulated by an Ising Model Using Entropic Sampling Monte Carlo Technique

There are currently extended experimental and theoretical developments of spin crossover nanomaterials, in particular based on coordination polymers for the design of smart applications. In this context, we have reproduced a three step thermal transition in a cubic spin crossover nanomaterial with a system dimension of 5 × 5 × 5 metallic centers. For this purpose, we have calculated, using Monte Carlo Entropic Sampling technique, the density of states of all possible system configurations. In order to take into account the local environment, we have included an additional interaction term in the standard Ising like model. We have then analyzed the role of this new interaction as well as the system size effect variation (from 4 × 4 × 4 to 6 × 6 × 6 metallic centers). Comparison with a 2D SCO system shows that the spin transition still proceeds in three steps but is no longer hysteretic.

Analysis of hysteretic spin transition and size effect in 3D spin crossover compounds investigated by Monte Carlo Entropic sampling technique in the framework of the Ising-type model

In spin crossover (SCO) systems, the shape of the hysteresis curves are closely related to the interactions between the molecules, which these play an important role in the response of the system to an external parameter. The effects of short-range interactions on the different shape of the spin transition phenomena were investigated. In this contribution we solve the corresponding Hamiltonian for a three-dimensional SCO system taking into account short-range and long-range interaction using a biased Monte Carlo entropic sampling technique and a semi-analytical method. We discuss the competition between the two interactions which governs the low spin (LS) - high spin (HS) process for a three-dimensional network and the cooperative effects. We demonstrate a strong correlation between the shape of the transition and the strength of short-range interaction between molecules and we identified the role of the size for SCO systems.

Influence of pressure and interactions strength on hysteretic behavior in two-dimensional polymeric spin crossover compounds

A study of thermal behavior for two-dimensional spin-crossover compounds is performed by using an Ising-like model including both short-range and long-range interactions and a Monte Carlo entropic sampling technique for determining the associated density of states. The effects of various factors, such as external pressure, internal interaction strength or system size, on the thermal transition are analyzed and compared to several experimental findings in this area. The study is focused on two-step transitions and the conditions for hysteresis behavior at each step which are especially important for the potential applications of spin-crossover materials in data storage devices or smart sensors.

Role of Edge Atoms in the Hysteretic Behaviour of 3D Spin Crossover Nanoparticles Revealed by an Ising‐Like Model

AbstractWe have applied a phenomenological two‐level Ising‐like model to study the case of 3D coordination polymers showing strongly cooperative but incomplete spin‐transition behaviour. Our numerical results focus on nanoparticles of the cyanide‐bridged 3D network Fe(pyrazine)[Pt(CN)4], considering several factors (i.e. temperature, pressure), the role of short‐ and long‐range interactions and the system size. To be consistent with the experimental studies on a nanometric scale, we used a semiexact method (Monte Carlo entropic sampling) to solve a system that contains molecules on each surface in the high‐spin state (edge atoms). A preliminary analysis of the short‐range antiferromagnetic case has been carried out to illustrate the three‐step transition case.

Analysis of the Hysteretic Behaviour of 3D Spin Crossover Compounds by Using an Ising‐Like Model

AbstractIn this contribution, we use a recently developed Monte Carlo technique to investigate the impact of ferro‐, antiferromagnetic interactions for different thermodynamic parameters in a three‐dimensional Ising‐like model. We present the role of temperature and pressure on the first‐order phase‐transition between a low‐spin and a high‐spin state and identify different response curves. The analysis of one‐ and two‐step spin‐crossover behaviour has been performed by using a self‐consistent approach, because it is known that for a system with both short‐ and long‐range interactions, the Hamiltonian does not have an exact solution. However, by constructing the distributions of the states with Monte Carlo entropic sampling and by taking into account the temperature degeneracy dependence between low‐spin and high‐spin states, we could solve the Hamiltonian. We provide herein a systematic comparison to our previous experimental works and as part of our continuous interest in spin crossover nanosystems, we present a size effect study.

Analysis of long-range interaction effects on phase transitions in two-step spin-crossover chains by using Ising-type systems and Monte Carlo entropic sampling technique

The analysis of two-step spin crossover phenomena in one-dimensional systems is performed in the framework of Ising-type model by using Monte Carlo entropic sampling technique. Both short-range and long-range interactions are considered in the Hamiltonian which also takes into account different degeneracies between the molecular states. The density of states associated to each macrostate is computed by using entropic sampling, via a biased Monte Carlo sampling technique, and the corresponding Ising-type Hamiltonian can thus be solved by using a self-consistent approach. By using this semi-analytical method, the effects of long-range interactions on phase transition driven by temperature variation in spin crossover chains have been investigated. Various types of spin crossover phenomena have been identified from gradual two-step transition to the one accompanied by multiple hysteresis loops, which was also recently observed experimentally. A metastable “plateau” has been observed at a high-spin fraction approximately equal to ½ for intermediate temperature region, but this plateau is not part of the low-to-high spin transition in strong cooperative systems and might be reached only by using special reversal curves or other external factors, such as light or pressure. In addition, a standard Monte-Carlo algorithm with Arrhenius dynamics is applied to this system in order to better understand its molecular behavior during the intermediate phase.

Determination of the physical parameters distribution in spin transition compounds using experimental FORC diagram

Spin transitions materials are characterized with an innovative experimental method, i.e. first-order reversal curve (FORC) diagram. The interpretation of the results is performed in the framework of two different Ising-like models: a mean-field approach and the exact solution done by the Monte Carlo entropic sampling (MCES) method.

Quasi-realistic distribution of interaction fields leading to a variant of Ising spin glass model

The distribution of interaction fields of an Ising-like system, obtained by Monte Carlo entropic sampling is used for modeling the hysteretic behavior of patterned media made of magnetic particles with a common anisotropy axis; a variant of the canonical Edwards–Anderson Ising spin glass model is introduced.

Monte Carlo entropic sampling applied to spin crossover solids: the squareness of the thermal hysteresis loop

The Monte Carlo entropic sampling method previously presented in [I. Shteto, J. Linares, F. Varret, Phys. Rev. E 56 (1997) 5128] is adapted here to an Ising-like system with short- and long-range interactions. Such model is suited to spin crossover solids [J. Linares, H. Spiering, F. Varret, Eur. J. Phys. B 10 (1999) 271; K. Boukheddaden, J. Linares, H. Spiering, F. Varret, Eur. Phys. J. B 15 (2000) 317] where the long interaction is due to elastic coupling mediated by the lattice, while the short-range interaction originates from the bonding between the spin crossover units [J. Linares, H. Spiering, F. Varret, Eur. J. Phys. B 10 (1999) 271]. Taking into account the different degeneracies gHS for high-spin (HS) and gLS for low-spin (LS) states, the Ising Hamiltonian associated with fictitious spins is written:H=−h∑σi−J∑σiσjwithh=−Δ2+kBTlngHSgLS2+G〈σ〉where J and G are the short- and long-range interactions, respectively, and Δ the energy gap of ligand field such that the LS state is the ground state. The numerical method has been tested successfully by comparison to the exact solution for a 1D system: [Fe(Htrz)2(trz)](BF4)2 [J. Linares, H. Spiering, F. Varret, Eur. J. Phys. B 10 (1999) 271; J. Krober, J.P. Audière, R. Claude, O. Kahn, J. Hassnoot, F. Grolière, C. Jay, A. Bousseksou, J. Linares, F. Varret, A. Gonthier-Vassal, Chem. Mater. 6 (1994) 1404]. We describe here the results obtained for 2D systems, and show that the squareness of the thermal hystersis loop, associated with the spin-transition, can be correlated to the strength of short-range interactions.

Metastable states of an Ising-like thermally bistable system.

The lifetimes of the metastable states are investigated in an Ising-like model associated with thermally bistable systems. A discrete mesoscopic Markovian dynamic is established using an optimized version of the previously presented Monte Carlo entropic sampling method. This is well suited to an extensive study of the role of the physical parameters: temperature, interaction parameter, electronic energy gap. By combining a discrete Markovian mesoscopic dynamic and the absorbing Markov chain technique, we obtain an analytical access to the average lifetime of the metastable state. One-variable and two-variable approximations for the original microscopic master equation are presented and discussed. A typical difference in the thermal dependence of the lifetime of the low- and the high-temperature metastable states is found, and explained as a consequence of the temperature-dependent field associated with the Ising-like model. The validity, the advantages, and the limits of the method are discussed, as well as the possible consequences on the behavior of spin transition systems. A prospective for a possible phenomenological finite-size scaling is presented.

Monte Carlo entropic sampling for the study of metastable states and relaxation paths

We present a continuous extension of the recent Monte Carlo entropic method for sampling a density of states restricted in dimensionless macroscopic parameters. The method performs a random walk through a two-dimensional macrostate space and provides complete information in the form of continuous functions of the system's coupling constants. For the example of an Ising system, we project relaxation paths from Monte Carlo Metropolis dynamic over the two-dimensional state space and compare them with a ``most probable path'' associated with the equilibrium distribution, derived from the density of states. We observe a close agreement between them in the stochastic regime, i.e., before the system escapes from the metastable state. We establish a Markovian macroscopic dynamic over the two macroscopic parameters and we discuss it with respect to the Metropolis microscopic dynamic.