The Eﬀect of Defects on Magnetic Droplet Nucleation Federico Ettori1 Timothy J. Sluckin12 and Paolo Biscari1 1Department of Physics Politecnico di Milano Piazza Leonardo da Vinci 32 20133 Milan Italy

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The Eﬀect of Defects on Magnetic Droplet Nucleation

Federico Ettori ∗1, Timothy J. Sluckin †1,2, and Paolo Biscari ‡1

1Department of Physics, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milan, Italy

2School of Mathematical Sciences, University of Southampton, University Road, Highﬁeld,

Southampton, SO17 1BJ, UK

Abstract

Defects and impurities strongly aﬀect the timing and the character of the (re)ordering or

disordering transitions of thermodynamic systems captured in metastable states. In this

paper we analyze the case of two-dimensional magnetic systems. We adapt the classical

JMAK theory to account for the eﬀects of defects on the free energy barriers, the critical

droplet area and the associated metastable time. The resulting predictions are successfully

tested against the Monte-Carlo simulations performed by adopting Glauber dynamics, to

obtain reliable time-dependent results during the out-of-equilibrium transformations. We

also focus on ﬁnite-size eﬀects, and study how the spinodal line (separating the single-

droplet from the multi-droplet regime) depends on the system size, the defect fraction,

and the external ﬁeld.

1 Introduction

Macroscopic systems exhibit a large variety of phase transitions when subject to continuous

variations of external control parameters, such as temperature, magnetic ﬁeld, pressure, or

chemical potential. The nature and timing of the nucleation and growth processes associated

with the transition strongly depend on a number of factors, including the nature of the transi-

tion and the presence of defects and/or impurities in the system. Ehrenfest [1,2] ﬁrst classiﬁed

phase transitions. In ﬁrst-order transitions the system jumps discontinuously to a diﬀerent

free energy branch, with a consequent discontinuous jump in the free energy gradient. Higher

order transitions were then identiﬁed through the order of the discontinuous derivatives of

the free energy, though now we know that this is an over-simpliﬁed cartoon, and we talk of

ﬁrst-order and continuous transitions.

In the original work it was thought that there was no singularity in the free energy at

a ﬁrst-order transition and that the free energy curve could be analytically continued into a

metastable region until well beyond the point at which the transition should have taken place.

Eventually the system reaches a point at which a susceptibility diverges, and the metastable

state become unstable with respect to ﬂuctuations of any sort. This is known as the spinodal

line (line, because one can draw a set of such points in the full phase space). The ﬂuctuations

then grow, and the process is known as spinodal decomposition. In a liquid-gas system, the

signature of the spinodal line is the instability with respect to density ﬂuctuations. The system

then develops spontaneous density ﬂuctuations and decays into regions of higher and lower

density, a process which stops when the densities become equal to the relevant equilibrium

∗federico.ettori@polimi.it

†t.j.sluckin@soton.ac.uk

‡paolo.biscari@polimi.it

arXiv:2210.02138v2 [cond-mat.stat-mech] 30 Dec 2022

liquid and gas densities. Droplets and/or bubbles form; at later times the droplets coagulate,

eventually triggering a full phase separation in which the liquid sits at the bottom of the

sample and the vapour at the top.

This theoretical picture of metastability is matched experimentally, even though we now

know that actually there is an essential singularity in the gradient at the phase transition. All

mean ﬁeld theories, computer simulations and real systems agree in that this metastable phase

has a real existence, incidentally in the process demonstrating that too rigid an insistence on a

knowledge of equilibrium properties can sometimes be self-defeating in statistical mechanical

studies. However, usually thermodynamic systems do not reach the spinodal line and some

other process intervenes beforehand causing phase separation and sending the system toward

a phase coexistence of true equilibrium phases. What is required in such cases is so-called

nucleation of the new phase.

The process of nucleation and subsequent growth has thus been the focus of much study

over the years. It is also the main focus of the present study, in which we are particularly

concerned with the eﬀects of defects and impurities on nucleation and droplet growth. The

nucleation involves the formation of a nucleus of the new phase, which then grows (depending

on various conservation laws) to invade the whole of the rest of the system, or alternatively

enforces phase separation into two coexisting equilibria. Examples might be a new solid phase

(say, martensite) invading another which had previously been stable (e.g. austenite), a new

magnetic phase (say, spin up) invading a formerly stable spin-down phase, or a homogeneous

metallic A-B alloy separating into two coexisting alloys, one A-rich and the other B-rich. How

this nucleation occurs is itself a subject of considerable study. Textbooks usually draw a

distinction between homogeneous and heterogeneous nucleation.

In homogeneous nucleation droplets of the new phase form (and then usually decay) as ﬂuc-

tuations around the original phase. Droplets in this condition are called subcritical. To reach

the size at which they would grow spontaneously (supercritical droplets) involves a spontaneous

ﬂuctuation with a free energy large enough to overcome the energy barrier ∆Fbarr. When this

is the case the classical Johnson-Mehl-Avrami-Kolmogorov (JMAK) theory [3–6] is expected

to capture the main features of the transition. If the system is kept at constant temperature T,

the probability of ﬂuctuations of the proper size is proportional to exp(−∆Fbarr/(kBT)), with

kBthe Boltzmann constant. As a consequence, the process of developing such a ﬂuctuation is

Poisson-like, with the expected time thus proportional to exp(+∆Fbarr/(kBT)), possibly with

a complex prefactor in front.

By contrast, heterogeneous nucleation involves requires either large (i.e., of a dimension

much larger than molecular dimensions) or many impurities to seed the transition process.

The homo/heterogenous character of a phase transition deeply inﬂuences its character as well

as the metastable lifetime, that is, the mean decay time of a metastable state. When nucleation

is originated by suﬃciently many seeds, the phase transition process is much more rapid and

predictable, in the sense that the standard deviation of the metastable lifetime is reduced.

Clearly, the number of nucleation seeds depends on the size of the system itself: the larger the

system the easier to nucleate transition droplets. For this reason, the term spinodal line has

also been used (see e.g. [7]), and will here be used, to identify the minimum (ﬁnite) size of a

system which exhibits heterogeneous nucleation.

The full phase transformation thus involves several distinct stages. Firstly there is an

early-stage stochastic nucleation process. This is followed by a deterministic phase during

which the critical droplet is growing. Finally there is a late stage stochastic phase, during

which the diﬀerent droplets amalgamate. The dynamics might be expected to be diﬀerent

depending on conservation laws which dictate the ﬁnal state equilibrium (is it phase transfor-

mation, or merely phase separation). Until the advent of computer simulation, these processes

were diﬃcult to examine in detail. Even with computer simulations, the sizes involved in

the case of heterogeneous nucleation, not to mention the inﬂuence of the ﬁnite size of the

simulation box, present signiﬁcant diﬃculties. The present study seeks to alleviate some of

the simulation diﬃculties by studying a very simple model. We seek to extend understand-

ing of nucleation processes by considering a case which encompasses both homogeneous and

heterogeneous nucleation. In our study, the impurities are molecule-sized rather than, as in

the case of classical heterogeneous nucleation, colloid-particle-sized. Our simulation uses a

two-dimensional Ising model, which, of course, possesses a distinguished lineage in the history

of Statistical Mechanics [8,9]. Spins may be either up or down, are coupled to their neighbours

on a lattice.

The precise aim of the present study is to analyze how the presence of impurities in-

ﬂuences the relevant phase transitions, with particular focus on nucleation and growth of

critical droplets, and the homo/heterogenous character of the transition itself. Common ap-

proaches for the modelling of imperfections include the classical Random-Bond [10,11] and

Random-Field Ising Models, where typically Gaussian-distributed bonds and/or quenched lo-

cal ﬁelds [12–14] embed the randomness, and diluted Ising lattices [15] where non magnetic

impurities are mixed with magnetic structure. The presence of quenched disorder requires a

proper treatment of ﬁnite-size eﬀects, including sample-to-sample ﬂuctuations and the possible

lack of self-averaging [16]. In our study we do this in two ways. First, we choose carefully

the number of diﬀerent samples with the same disorder statistics – replicas – over which any

physical quantity must be averaged. Second, we focus on the diﬀerent regimes that might be

present is systems of diﬀerent sizes (see section 4.4).

The present paper is organized as follows. In the next section we specify the modelling

of defects, and describe the Monte-Carlo algorithm used in the numerical simulations. In

Section 3 we focus on the nucleation process. We derive a theoretical prediction of the in-

ﬂuence of impurities on the critical droplet size, and the corresponding free energy barrier,

and test the prediction against the outcomes of the simulations. In Section 4we focus on the

homo/heterogenous character of the transition in the presence of defects, and draw conclusions

about the spinodal line. A careful analysis of the simulations allows us to analyze also the

ﬁnite-size eﬀects. A concluding section summarizes and discusses the main outcomes of the

present study.

2 Model and algorithm

We consider a ferromagnetic Ising Model [17] occupying a L×Ltwo-dimensional square

lattice, in which periodic boundary conditions are enforced. The system Hamiltonian takes

the standard form

H[ s ] = −JX

hi,ji

sisj−hext X

si,(1)

in which Jrepresents the coupling interaction between neighboring spins and hext the magnetic

ﬁeld applied to the system. All the simulations discussed in the present paper adapt the n-

fold way algorithm ﬁrst introduced by Bortz et al. [18]. In its original 2D version, each of

the N=L2spins is assigned to one among n= 10 classes, based on their orientation and

the number of positively oriented neighbors. This allows us to easily monitor the spins which

are most/less likely to modify their state, and therefore to build a rejection-free Monte-Carlo

algorithm.

Defects are modelled as ﬁxed spins which are not allowed to modify their orientation during

the evolution of the system. In a ﬁnite temperature Monte-Carlo simulation the defect-ﬂipping

probability cannot be ruled out. Therefore here we are in fact assuming that the characteristic

defect-ﬂipping time is larger than the longest simulation time considered (the metastable

lifetime analyzed in Sect. 4). The presence of defects does not inﬂuence the eﬃciency of the

Monte-Carlo simulation. Ergodicity is ensured provided we restrict the phase space to the free

spins. Similarly, the detailed-balance condition is not aﬀected by the inclusion of defects. In

the implementation of the algorithm, the defects are all assigned to an additional 11-th class,

whose transition probability is held ﬁxed at zero.

In order to understand how the quenched disorder inﬂuences the physical properties of the

system we study in parallel several diﬀerent realizations of the system with similar quenched-

disorder characteristics. In all realizations of the same system we introduce the same number

of defects of positive and negative defects, so to study neutral samples and therefore reduce the

sample-to-sample variation [19]. We parameterize the number of defects through their total

fraction fin the system. Therefore, when a fraction fof defects is reported, it means that

there are precisely bfL2/2cquenched defects of each sign distributed at random throughout

the system.

2.1 Defects as random ﬁelds

It is known that in 2D the perfect Ising model sustains a low-temperature ferromagnetic phase,

characterized by long-range order [8,20]. The addition of defect sites introduces quenched

randomness and possible frustration. Although randomness and frustration are known to be

two key ingredients leading to spin-glass phases [21], we now show that in the thermodynamic

limit no such behavior should be expected.

The presence of defects can be interpreted as the eﬀect of a peculiar type random-ﬁeld

distribution. To be more precise, let D+(resp. D−) denote the set of defects with ﬁxed

positive (negative) orientation, and consider the following random-ﬁeld distribution on the

entire system

hRF,i =









+hRF if i∈D+,

−hRF if i∈D−,

0otherwise.

(2)

In the hRF Jregime, any reversal of the selected spins is prevented at any ﬁnite (non-zero

and non-inﬁnite) temperature, so those spins will eﬀectively behave as defects. How large

should hRF be taken depends on the chosen temperature. It is well know that a random-ﬁeld

Ising model in thermal equilibrium and in the thermodynamical limit, no spin glass phase can

be observed [22,23]. Moreover, no ordered phase can survive in the two-dimensional random-

ﬁeld Ising model [12,24]. As a result, in the thermodynamic limit, no ferromagnetic nor spin-

glass phases are to be expected. What we can and we do observe instead are pseudo-phases [25]

in ﬁnite systems, where domain clusterization and ﬁnite-size eﬀects generate pseudo-ferro or

pseudo-glassy phases, depending on the temperature and defect density. We postpone to a

later study the report of the characterization of such pseudo-phases.

2.2 Dynamic Monte-Carlo algorithm

The simulations presented in the present work were performed by using the so-called n-fold

Monte-Carlo algorithm [18], subject to Glauber dynamics [26]. We now discuss how this

algorithm is adapted so as to describe the out-of-equilibrium response of magnetic systems in

the presence of defects.

The primary quantities of interest in our simulations concern magnitudes and time scales

associated with droplet formation and growth processes. More generally, it is thus necessary

to characterise the out-of-equilibrium and dynamic response of the system to perturbation.

To implement the Glauber dynamics, we associate with each possible single spin ﬂip si→ −si

the transition probability rate

wi[ s ] = 1

2α1−sitanh β hi[ s ],(3)

where αis a microscopic characteristic time, and hi=hext +PjJij sjis the local ﬁeld acting

on the i-th spin.

We recall that in our model the transition probabilities associated with the defects are set

equal to zero. In an underlying real physical system, this will not in general be rigorously true.

Such a system would rather be composed of two species with two very diﬀerent microscopic

characteristic times ααdef. Our key approximation involves taking the limit αdef/α →+∞.

This condition is rather strong and may be weakened in future studies. With this choice, we

ensure that at any speciﬁc time it is much more probable that a normal spin ﬂips rather than

a defect. As noted above in our discussion of the basic model, our simulations also rely on yet

another asymptotic limit, αdef/τ → ∞. Here τrepresents the longest time in our simulations.

In Section 4this is labeled the metastable lifetime. This second limit ensures that no defect

can possibly ﬂip during the numerical experiments.

The sum wT=Piwiprovides the global transition rate for the entire system. The

interaction energy Jand the characteristic time αare chosen as units for energy and time,

respectively. Then for each Monte-Carlo step, two operations are required. These are:

1. Identify the associated time interval ∆t. This involves extracting a random number from

a Poisson distribution with parameter w−1

2. Perform the move which occurs over this time interval ∆t. A speciﬁc spin ito be ﬂipped

is chosen with probability proportional to wi. To enable this choice to be made, we use

the Bortz n-fold algorithm [18].

3 Droplets and defects

We now turn our attention to droplet formation and growth in the presence of defects. We

ﬁrst adapt some theoretical predictions to account for the presence of defects. This part of our

analysis relies on the Droplet Theory, adapted to a defect-free magnetic system as in [7] and

then report and discuss the results of a number of simulations which help us understanding

the proper behaviour of a magnetic system out of equilibrium.

3.1 Domains and free energy balance

We consider an initially magnetized system, in which all spins, with the exception of the

negative defects, are aligned parallel, and equal to +1. We apply a negative ﬁeld hext =−h,

with h > 0, and study the reversal transition in which droplets of negative spins are expected

to form and grow. We identify a droplet as a connected domain, possibly including n+positive

and n−negative defects, in which all the free spins have already reversed their sign. Positive

defecs are counted as part of a negative droplet only if internal to it, that is if it is surrounded

by four already reversed spins. We notice that the positive defects are aligned with the

initial orientation of all remaining spins, while the negative ones are aligned with the external

magnetic ﬁeld, and act as seeds for droplet formation.

To characterize a droplet we introduce the following parameters:

•The area Acounts all the spins in the droplet, including the defects.

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TheEectofDefectsonMagneticDropletNucleationFedericoEttori*1,TimothyJ.Sluckin1,2,andPaoloBiscari11DepartmentofPhysics,PolitecnicodiMilano,PiazzaLeonardodaVinci32,20133Milan,Italy2SchoolofMathematicalSciences,UniversityofSouthampton,UniversityRoad,Higheld,Southampton,SO171BJ,UKAbstractDefectsandim...

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The Eﬀect of Defects on Magnetic Droplet Nucleation Federico Ettori1 Timothy J. Sluckin12 and Paolo Biscari1 1Department of Physics Politecnico di Milano Piazza Leonardo da Vinci 32 20133 Milan Italy

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