## Article

# 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

^{*}Correspondence to: Profs. Jorge Linares, Kamel Boukheddaden, Groupe d'Etudes de la Matière Condensée, Université ParisSaclay, UVSQ, CNRS, 45 Avenue des Etats Unis, Versailles 78035, France. E-mail:

**Received:**26 Apr 2023 |

**First Decision:**19 Jul 2023 |

**Revised:**2 Oct 2023 |

**Accepted:**20 Oct 2023 |

**Published:**18 Nov 2023

**Academic Editor:**Yann Garcia, Bao-Lian Su, Jiang Liu |

**Copy Editor:**Dong-Li Li |

**Production Editor:**Dong-Li Li

**Open Access**This article is licensed under a Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, sharing, adaptation, distribution and reproduction in any medium or format, for any purpose, even commercially, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Abstract

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

## Keywords

*,*local mean-field approach

*,*spin crossover

*,*phase transitions

*,*nanoparticles

*,*bulk-surface interactions

## INTRODUCTION

Spin-crossover (SCO) materials^{[1–6]} are composed of transition metals with an electronic configuration ranging between ^{[7–20]}.

In Fe-based SCO materials, the spin transition between LS and HS is accompanied by an increase of ^{[21–24]}, and gases, notably of hazardous volatile organic compounds, as recently demonstrated in several reports^{[25,26]}. They can also serve as switches of light emission in the electroluminescent devices or photoluminescent^{[27,28]} containing SCO complexes or to enhance plasmonic resonances^{[29]}.

To try to explain and reproduce these switching phenomena from the LS to HS states, researchers have proposed different models, some of which are based on Ising-like^{[30–35]}, mechano-elastic^{[36]}, and electroelastic^{[37]} models, on which switchable core-shell SCO nanoparticles have been investigated in details^{[38,39]}.

In the present study, we focus on the modeling of thermal effects in 2D square SCO nanoparticles. The Ising-like model is used here and solved in the framework of local mean field approximation (LMFA) and within the Monte Carlo entropic sampling (MCES) technique in an attempt to reveal the role of interactions with the external environment (matrix). In this work, we discuss and compare the results obtained by these two methods. Experimental results on the effect of the surface or the substrate have been reported^{[35,40,41]}.

Although the MCES technique is a good choice to solve the Hamiltonian associated with the extended Ising-like model, it is limited in terms of computing capacity for samples containing a small number of molecules. As we would like to analyze the cases of nanoparticles with large numbers of molecules and even thin films in contact with a substrate, which cannot be studied by MCES, we plan to use an alternative formalism based on LMFA. With this contribution, the purpose of this contribution is to determine the reliability, limitations, and drawbacks of the LMFA method when applied to very small systems by comparing it with the MCES technique.

The manuscript is organized as follows. Section 2 introduces the Model and some details of the parameters used in the model. In section 3, the MCES technique is presented, while the LMFA is introduced in section 4. In section 5, we present the simulated thermal behavior of order parameters and the discussion of the obtained results. Finally, we conclude and outline the possible extensions of this work in section 6.

## THE MODEL

In condensed matter physics, magnetic phenomena observed in a regular lattice are described with the well-known Ising Model based on interactions between quantum spin variables that were first introduced phenomenologically by Pauli^{[42]}. Later on, in 1925, Ising^{[43]} used the concept of spins to describe the thermal behavior of the magnetization of magnetic materials using a 1D model whose statistics were treated in the frame of a transfer matrix method. Pointing quickly the exact resolution of this model (with zero applied field) performed with a remarkable mathematical tour de force of Onsager in 1944^{[44]}, it is also important to notice the considerable efforts of several authors in the realization of approximative methods based on variational treatments of the Ising model, which led to mean-field theory^{[45]}, Bethe-Peierls^{[46]}, *etc*., which are today widely used to analytically describe a large panel of cooperative phenomena in solid state physics and beyond.

In the field of molecular switchable materials, the Ising model^{[30–35]} has been adapted to study SCO materials in which two different magnetic states, HS and LS, interact to display at the macroscopic scale various physical properties that give rise to a wide variety of transition behaviors. Among them, one can quote: (ⅰ) thermally-induced hysteretic first-order phase transitions^{[15]}, gradual or continuous^{[47]} conversions, incomplete^{[48]}, and multi-step transformations^{[49]}.

To model SCO nanoparticles, a spin fictitious operator with two eigen-values +1 and -1 is associated with each molecule describing the HS and LS states, respectively. Then, the Ising-like Hamiltonian is used to model the energy operator connected to SCO. In this respect, the Hamiltonian

with

(ⅰ) The single molecule contribution:

where

(ⅱ) The interaction term:

where

(ⅲ) The interaction between the surface molecules and their environment:

where

In the following, we give a detailed description of each of these terms.

### Molecular contribution $$ \boldsymbol{H_1} $$

The two states, HS and LS, have different spin and orbital degeneracies, and the spectrum of intramolecular vibrations of the molecule strongly depends on the spin state of the central metal.

And, as it has been reported^{[1,2]}, the density of states becomes higher in the HS state than in the LS state, which is then equivalent to considering these two states as having different effective degeneracies, denoted here by

Where

At this stage, we see that from the point of view of a single site Hamiltonian, as far as

Simple Boltzmann statistics performed on Hamiltonian (1) leads to the following relations,

and

from which the curves of Figure 1 showing thermal dependence of the HS magnetization and fraction are easily derived.

### Interaction term $$ \boldsymbol{H_2} $$

The interaction term can be split into two contributions: (ⅰ) those arising from nearest neighbors, which are parametrized using a short-range interaction term

*} \sigma_i \sigma_j-G\sum\limits_{i=1}^N \sigma_i $$*

Considering solely this interaction term

Figure 2. Thermal evolution of the average magnetization

The region on the left-hand side below the upper curve (respectively above the lower curve) with respect to this

However, for ^{[14]}.

### Short- and long-range interaction contributions $$ \boldsymbol{H_{1} + H_{2}} $$

When the first two terms are combined to model SCO physical behavior with temperature, the Hamiltonian writes as:

The symmetry in the configurations consisting of positive and negative sigma values about

In this case, negative solutions of

Therefore, if

### Surface contribution $$ \boldsymbol{H_3} $$

This

with

In order to follow how the equilibrium temperature of the surface molecules

However, it must be noted that the long-range interaction

### The total Hamiltonian $$ \boldsymbol{H_T} $$

Now, as one considers the total Hamiltonian, one gets:

which can be written as:

In the present work, the Hamiltonian

## RESOLUTION BY MONTE CARLO ENTROPIC SAMPLING METHOD

In this section, we will distinguish three kinds of short-range interaction terms, namely,

with

where the respective total and surface magnetizations,

Where

The thermal average value of the fictitious magnetization,

Where

and

In Equations (24) and (25), ^{[48–50]}, and Equation (24) is solved by numerical techniques such as bisection. From the thermal average values

Let us recall the basic principles of MCES^{[50–52]}. It consists in introducing the detailed balance equation of the Monte Carlo (MC) procedure, as shown in:

This suited biased distribution

In this case, the balance equation can be written as:

Using

By construction,

The resulting restricted density of states is calculated after the correction for the bias was applied:

The flat character of the histogram

## LOCAL MEAN-FIELD APPROACH

In the present work, three types of sites related to three lattice regions are considered: the atoms located in the bulk (

The case of a system comprising 36 molecules is represented in Figure 3.

Figure 3. Schematic representation of a 2D 6

Note that in the framework of a classical mean-field approximation^{[53]} and the case of a 2D square-shaped lattice, the coordination number is q = 4 [Figure 4].

Figure 4. Mean-field approximation in a 2D square-shaped lattice. The

The consideration of three types of sites leads us to distinguish three local average order parameters

For each region,

with

where

The ligand field contributions and the number of short-range interactions as a function of various localizations in the square-shaped lattice

Site | Bulk | Edge | Corner |

4 | 3 | 2 | |

0 | 1 | 1 | |

Ligand-field |

Here, for simplicity of calculations, we have taken *z* = 1 (the same coordination number) everywhere in the surface (edge and corner), meaning that we are considering the same connectivity for all molecules at the surface.

The short-range interaction term

These two interaction terms,

Let

In a similar way, for the surface sites, Equation (35) is written as:

where

The average spin state can be written as follows:

So, for each region

which is equivalent to:

The weighted average of the three local order parameters associated with the three regions, bulk, edge, and corner, leads to the average fictitious spin of the system expressed as:

and the high-spin fraction, Nhs, which is the probability that the HS state is occupied and which is written as Equation (6):

The purpose is to solve a system of three equations

The calculation of the three points "close to the solution" is performed in two steps: Let X be the space in which values of (

Solving the system amounts to calculating

which allows to write in the vicinity of points

The following system of three equations with three unknowns is thus obtained as:

And where

In a general way, the

With the

## RESULTS AND DISCUSSION

We are interested in the effect of the coupling between surface and bulk molecules, and for that, we perform numerical simulations by MCES and by LMFA. To proceed, we choose a set of realistic SCO thermodynamic parameters, and for this purpose, we chose the data derived from experimental data of one of these SCO compounds: [^{[54]}, whose molecular structure is shown in Figure 5. The energy gap

Figure 5. Schematic structure of the SCO compound iron (II) [

### The case $$ \boldsymbol{J_{bs}^{MCES} = 0} $$

Figure 6 displays the results obtained in the case of a 6

Figure 6. Comparison of the thermal evolution of the HS fraction in a SCO system for a 6

Correspondence between the values of short-range interactions

60 | 60 | 0 | 36 | 33.33 |

Figure 6 illustrates a qualitative and quantitative agreement between the calculations obtained by MCES and LMFA. The hysteresis loops, observable for each method, at lower temperatures,

It can be noticed that the hysteresis widths

Values of the thermal hysteresis width for molecules located in the bulk and in the surface obtained by MCES,

The simulation parameters are: | |||

4.8 | 7 | 7.6 | 10.5 |

### The case $$ \boldsymbol{J_{bs}^{MCES} \neq} $$ 0

The value of the short-range interaction parameter

Figure 7. Comparison of the thermal evolution of the HS fraction in a SCO system for a 6

Figure 8. Comparison of the thermal evolution of the HS fraction in a SCO system for a 6

Figure 9. Comparison of the thermal evolution of the HS fraction in a SCO system for a 6

Correspondence between the values of short-range interactions

60 | 60 | 20 | 44 | 42.22 |

60 | 60 | 40 | 52 | 51.11 |

60 | 60 | 60 | 60 | 60 |

The phase diagrams of the 6 *vs*.

Figure 10. Phase diagram T *vs*.

Figure 11. Phase diagram T *vs*.

By MCES, as can be seen in Figure 10 and Figure 11, when the interaction parameter

In LMFA, as shown in Figure 10 and Figure 11, an opposite behavior is observed for surface molecules. First of all, when the interaction parameter

Values of the thermal hysteresis width for molecules located in the bulk and in the surface in MCES,

The computational parameters are: | ||||

0 | 4.8 | 7.6 | 7 | 10.5 |

20 | 6.4 | 6.2 | 17.5 | 20.36 |

40 | 8.0 | 5.0 | 30.5 | 29 |

60 | 8.8 | 4.0 | 45.0 | 38.5 |

### Size effect

The purpose of this part is to compare the two simulation techniques in the case of a larger system of size 7

Correspondence between the values of short-range interactions

60 | 60 | 0 | 40 | 32.72 |

60 | 60 | 20 | 46.66 | 41.81 |

60 | 60 | 40 | 52.33 | 50.90 |

60 | 60 | 60 | 60 | 60 |

### The case $$ \boldsymbol{J_{bs}^{MCES} = 0} $$

The results obtained and presented in Figure 12, highlight three important points. First of all, as previously observed in the case of the 6

Figure 12. Comparison of the thermal evolution of the HS fraction in a SCO system with a size 7

Values of the equilibrium temperatures of the bulk and surface molecules obtained in MCES and in LMFA, as a function of the system size (6

The computational parameters are: | ||||

6 |
185.0 | 185.0 | 144.0 | 144.0 |

7 |
187.75 | 187.75 | 148.6 | 148.6 |

Values of the thermal hysteresis width for molecules located in the bulk and in the surface obtained by MCES,

The computational parameters are: | ||||

6 |
4.8 | 7 | 7.6 | 10.5 |

7 |
10.4 | 14 | 2.8 | 5 |

Finally, in a larger particle of size 7

Values of the thermal hysteresis width for molecules located in the bulk and in the surface by MCES,

The computational parameters are: | ||||

6 |
8.8 | 45 | 4 | 38.5 |

7 |
23.2 | 48 | 0 | 32 |

### The case $$ \boldsymbol{J_{bs}^{MCES} \neq 0} $$

The phase diagrams of the 7 *vs*.

Figure 13. Phase diagram T *vs*. (

Figure 14. Phase diagram T versus (

When the value of the interaction parameter

First, for each method, the curves presented in Figure 12 and Figure 13 reveal that the equilibrium temperatures of the surface and of the bulk in the 7

In addition, as already observed in the case of a 6

Simulations using the LMFA were carried for very large lattices, 20

Figure 15. Thermal dependence of the HS fraction studied with the LMFA for square-shaped particles of different sizes: 20

## CONCLUSION

The present contribution, a simulation work, with the purpose of giving some hints to colleagues who synthesize spin-transition nanoparticles, is focused on meticulous investigations of the thermal properties of 2D-SCO nanoparticles using an adapted two-state Ising-like Hamiltonian. At this end, three types of interaction terms are considered: long and short-range interactions originating from electronic and elastic molecular couplings inside the lattice, to which we added a specific energetic contribution accounting for the interactions between surface molecules with their surrounding environment (inside and outside the lattice). The model is solved in two distinct ways, which involve different levels of sophistication: (ⅰ) MCES techniques, which are exact methods and (ⅱ) LMFA based on mean-field theory but taking into account the lattice shape. In the former case, three types of two-body short-range interaction terms are involved between bulk-bulk (

Our comparison of the two techniques (MCES and LMFA) shows that when dealing with small samples, LMFA yields results that are in perfect agreement with MCES in the case where the interaction between the surface and the bulk is zero (

Finally, it is worth mentioning that the surface properties have been limited here to the outer layer of the lattice, while the concept of the surface may integrate some inner layers close to the interface material/air, which also have specific properties different from those of the bulk. Extensions of the present work to include the above-cited aspects are in progress, as well as other packing clusters such as rhombohedral or faced centered cubic (FCC). The results of this simulation work can also be extended to explain the behavior of polycrystalline SCO films on the ^{[40]} and the influence of the substrate on the functionality of SCO molecular materials^{[41]}.

## DECLARATIONS

### Acknowledgments

This project has received financial support from the CNRS through the MITI interdisciplinary programs through its exploratory research program, as well as from the ANR project Mol-CoSM No. ANR-20-CE07-0028-02, the Universities of Ver-sailles and Paris-Saclay-UPSAY. We thank all of them for their strong support.

### Authors' contributions

Proposed this contribution: Linares J, Cazelles C, Boukheddaden K

Wrote all the equations for this research work: Linares J, Boukheddaden K, Cazelles C

Calculated the density of states and made the simulations: Cazelles C, Linares J

Analyzed the results: Linares J, Boukheddaden K, Cazelles C, Belmouri N, Dahoo PR

All the authors participated in writing the paper.

All authors have read and agreed to the published version of the manuscript.

### Availability of data and materials

Not applicable.

### Financial support and sponsorship

This project has received financial support from the CNRS through the MITI interdisciplinary programs through its exploratory research program, as well as from the ANR project Mol-CoSM No. ANR-20-CE07-0028-02, the Universities of Ver-sailles and Paris-Saclay-UPSAY.

### Conflicts of interest

All authors declared that there are no conflicts of interest.

### Ethical approval and consent to participate

Not applicable.

### Consent for publication

Not applicable.

### Copyright

© The Author(s) 2023.

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## Cite This Article

**OAE Style**

Linares J, Cazelles C, Richard Dahoo P, Belmouri Ne, Boukheddaden K. 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. *Chem Synth* 2023;3:48. http://dx.doi.org/10.20517/cs.2023.23

**AMA Style**

Linares J, Cazelles C, Richard Dahoo P, Belmouri Ne, Boukheddaden K. 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. *Chemical Synthesis*. 2023; 3(4): 48. http://dx.doi.org/10.20517/cs.2023.23

**Chicago/Turabian Style**

Linares, Jorge, Catherine Cazelles, Pierre Richard Dahoo, Nour-el-islam Belmouri, Kamel Boukheddaden. 2023. "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" *Chemical Synthesis*. 3, no.4: 48. http://dx.doi.org/10.20517/cs.2023.23

**ACS Style**

Linares, J.; Cazelles C.; Richard Dahoo P.; Belmouri N.e.; Boukheddaden K. 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. *Chem. Synth.* **2023**, *3*, 48. http://dx.doi.org/10.20517/cs.2023.23

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