Thermodynamics of Small Magnetic Particles

Eugenio Vogel,Patricio Vargas,Paulo Centres,Julio Valdes,Antonio Ramirez-Pastor,Gonzalo Saravia

doi:10.3390/e19090499

Eugenio Vogel, Patricio Vargas + Show 4 more

Open Access

https://doi.org/10.3390/e19090499

Copy DOI

Abstract

In the present paper, we discuss the interpretation of some of the results of the thermodynamics in the case of very small systems. Most of the usual statistical physics is done for systems with a huge number of elements in what is called the thermodynamic limit, but not all of the approximations done for those conditions can be extended to all properties in the case of objects with less than a thousand elements. The starting point is the Ising model in two dimensions (2D) where an analytic solution exits, which allows validating the numerical techniques used in the present article. From there on, we introduce several variations bearing in mind the small systems such as the nanoscopic or even subnanoscopic particles, which are nowadays produced for several applications. Magnetization is the main property investigated aimed for two singular possible devices. The size of the systems (number of magnetic sites) is decreased so as to appreciate the departure from the results valid in the thermodynamic limit; periodic boundary conditions are eliminated to approach the reality of small particles; 1D, 2D and 3D systems are examined to appreciate the differences established by dimensionality is this small world; upon diluting the lattices, the effect of coordination number (bonding) is also explored; since the 2D Ising model is equivalent to the clock model with q = 2 degrees of freedom, we combine previous results with the supplementary degrees of freedom coming from the variation of q up to q = 20 . Most of the previous results are numeric; however, for the case of a very small system, we obtain the exact partition function to compare with the conclusions coming from our numerical results. Conclusions can be summarized in the following way: the laws of thermodynamics remain the same, but the interpretation of the results, averages and numerical treatments need special care for systems with less than about a thousand constituents, and this might need to be adapted for different properties or devices.

Highlights

Recent developments in experimental techniques and refinements of synthesis methods allow for direct measurements of very small systems [1,2,3,4,5,6]
This broad picture calls attention to the extension of the mathematical treatments valid in the thermodynamic limit (TL) to systems formed by a small number of elements
As magnetic particles get very tiny, comprising a low number of elements, especially at the nanoscopic level, the ergodic separation is no longer possible in the way that it is handled in the TL

Summary

Introduction

Recent developments in experimental techniques and refinements of synthesis methods allow for direct measurements of very small systems [1,2,3,4,5,6]. In a recent work [26], it has been shown that for tiny systems, surface sites are special, and some inner sites might not be equivalent to each other This fact can have important consequences on the thermodynamic properties of small nanoparticles (up to sizes of the order of ≈10 nm). We increase the internal degrees of freedom at each site upon generalizing the Ising model to the discretized XY model in what is called the clock model which is a well-known example of the famous Berezinskii–Kosterlitz–Thouless (BKT) transition [47,48] In this way, we will be able to appreciate what is the role of the increase of the configuration space due to the characteristics of the constituents of small systems and the fact that they interact among themselves.

General Definitions

Exact Theoretical Approach for a Small System

Numerical Simulations

The Clock Model

Results and Discussion

Conclusions