DOCTORAL THESIS

Jan Kulveit

Nucleation in complex systems

Institute of Physics of the Czech Academy of Sciences

Supervisor of the doctoral thesis: prof. Pavel Demo

Study programme: Physics

Study branch: Physics of Nanostructures

Prague 2019

I declare that I carried out this doctoral thesis independently, and only with thecited sources, literature and other professional sources.

I understand that my work relates to the rights and obligations under the ActNo. 121/2000 Sb., the Copyright Act, as amended, in particular the fact that theCharles University has the right to conclude a license agreement on the use ofthis work as a school work pursuant to Section 60 subsection 1 of the CopyrightAct.

In Prague date 15.8.2019 Jan Kulveit

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I would like to thank my supervisor prof. Pavel Demo for patience and help withthis work, and for many interesting discussions about physics in general. Thanksalso to my collaborators, in particular Tomas Gavenciak, for help with software.

I’m grateful to my family - my late father Frantisek Kulveit and my motherJaroslava, for their love and support. Last but not least I would like to thankVerca.

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Title: Nucleation in complex systems

Author: Jan Kulveit

Institute: Institute of Physics of the Czech Academy of Sciences

Supervisor: prof. Pavel Demo, Institute of Physics of the Czech Academy ofSciences, Department of Optical Materials

Abstract: We studied nucleation in progressively more abstract contexts andsystems, starting from classical nucleation theory and ending with nucleationin complex networks. The cases studied include impurity nucleation in a solidmatrix on several alkali halide crystals, where we determined formation energiesfor clusters, treated as defects, starting from single impurity-vacancy dipole andsmall aggregates to possible configurations of larger clusters. In the next part, weturn to the study of heterogeneous nucleation. While in the usual treatment ofheterogeneous nucleation the surface energy is assumed to be homogenous, we askthe question what happens if we consider the surface energy to be heterogeneous.Utilizing umbrella sampling computer simulations we find the nucleation barriercan be significantly lowered in the presence of surface heterogeneity, even if theaverage surface energy is kept constant. In the last part we study influence ofclustering coefficient on phase transitions in scale-free networks, using forwardflux sampling (FFS).

Keywords: nucleation, complex systems, phase transitions, Ising model, networks

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Contents

1 Introduction 31.1 Conceptual metaphors . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 The metaphor of potential and landscape . . . . . . . . . . 31.2 Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Classical theory of nucleation . . . . . . . . . . . . . . . . . . . . 5

1.3.1 Master equation . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Heterogeneous nucleation . . . . . . . . . . . . . . . . . . . . . . . 81.5 Heterogeneous substrates . . . . . . . . . . . . . . . . . . . . . . . 91.6 Nucleation on networks . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Classical nucleation theory 112.1 Homogeneous nucleation without strain . . . . . . . . . . . . . . . 11

2.1.1 Critical parameters - simple case . . . . . . . . . . . . . . 132.2 Homogeneous nucleation with strain . . . . . . . . . . . . . . . . . 13

2.2.1 Incoherent interfaces . . . . . . . . . . . . . . . . . . . . . 132.2.2 Interface development during nucleation and growth . . . . 14

2.3 Nucleation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.1 Classical nucleation rate . . . . . . . . . . . . . . . . . . . 152.3.2 Nucleation rate in a quasi-steady-state . . . . . . . . . . . 162.3.3 Non-steady-state nucleation rate . . . . . . . . . . . . . . . 192.3.4 Nucleation and growth . . . . . . . . . . . . . . . . . . . . 22

3 Nucleation with formation of nanophases 243.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4.1 KCl-PbCl2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4.2 NaCl-PbCl2 . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Ising model as a testbed of nucleation 344.1 Modern statistical sampling methods . . . . . . . . . . . . . . . . 344.2 Testbed systems and results . . . . . . . . . . . . . . . . . . . . . 35

4.2.1 Hard spheres . . . . . . . . . . . . . . . . . . . . . . . . . 354.2.2 Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3 Conclusions and prospects for future research . . . . . . . . . . . 37

5 Heterogenous nucleation on heterogenous surface 385.1 Classical heterogeneous nucleation . . . . . . . . . . . . . . . . . . 385.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2.1 Energy landscape, sampling technique and reaction coordi-nates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2.2 Model of surface heterogeneity . . . . . . . . . . . . . . . . 415.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.3.1 Regular stripes . . . . . . . . . . . . . . . . . . . . . . . . 42

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5.3.2 Random pattern . . . . . . . . . . . . . . . . . . . . . . . 425.3.3 Random surface . . . . . . . . . . . . . . . . . . . . . . . . 42

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6 Nucleation on complex networks 456.1 Graphs and networks . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.1.1 Network statistics . . . . . . . . . . . . . . . . . . . . . . . 466.1.2 Network models . . . . . . . . . . . . . . . . . . . . . . . . 476.1.3 Generative and descriptive models . . . . . . . . . . . . . . 48

6.2 Dynamics on networks . . . . . . . . . . . . . . . . . . . . . . . . 536.3 Ising model on networks . . . . . . . . . . . . . . . . . . . . . . . 53

6.3.1 Mean-field description . . . . . . . . . . . . . . . . . . . . 546.3.2 Tree-like approximation . . . . . . . . . . . . . . . . . . . 556.3.3 Nucleation on random graphs and scale-free networks . . . 55

6.4 Simulation study of nucleation in Ising model on a Barabasi-Albertnetwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.4.1 Core degree distribution throughout a transition . . . . . . 58

6.5 Ising model on clustered scale-free networks . . . . . . . . . . . . 60

Conclusion 68

A Numerical simulations implementation 69A.1 Forward-flux sampling algorithm . . . . . . . . . . . . . . . . . . 69

B Implementation notes 72B.1 Execution speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72B.2 GPU implementation and drawbacks . . . . . . . . . . . . . . . . 72

B.2.1 Parallel spin updates . . . . . . . . . . . . . . . . . . . . . 73B.2.2 Connected component search . . . . . . . . . . . . . . . . 73

Bibliography 74

List of Figures 80

List of publications 83

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1. Introduction

The first chapter of this work is somewhat unusual: non-technical and, philosoph-ical. Its purpose is twofold: first, to put the theory of nucleation into the widercontext of physics. Second, to explain concepts of nucleation in a way accessibleto non-specialists - I think some ideas and models from this theory may be ofinterest to a wider audience, including e.g. researchers interested in theory ofsocial change. Some motivation also comes from a question “what do you workon and what is it good for?” which the author of this work is sometimes asked.

Rest of the work is in part based on articles Formation of structured nanophasesin halide crystals [48], Ising model simulations as a testsbed of nucleation the-ory [49], Heterogeneous nucleation on a surface with heterogeneous surface en-ergy (in review)[50]. The parts about on nucleation on networks are new (to besubmitted).

1.1 Conceptual metaphors

As we will try to avoid the use of equations in the Introduction, instead, we willintroduce several conceptual metaphors [51]. What is a conceptual metaphor?Basically, a mapping from one area of human knowledge to another. Other namesfor the same concept are cognitive metaphor or conceptual analogy. For example,we often think about linear variables using metaphors of distance or height. Thestock price is falling, rising, high, suddenly falling, slowly rising.

The fact such language is very natural and common conceals that it is ametaphor - there is no a priori reason why prices could not be mapped to otherspatial dimensions (going left of right), or why they should be mapped to spatialdimensions at all. According to conceptual metaphor theory, metaphors do notend with language, but they are deeply related to how we think. The concept ofpricing thus activates the same parts of our brain that we would use for thinkingabout direction. The claim that when thinking about numbers on the abstractmathematical level we use the same neural circuits as for space representation,currently got support from neuro-imaging methods [13].

It is plausible that part of the intuitive understanding of mathematics andphysics that each mathematician or physicist gradually acquires, is precisely theacquisition of new conceptual metaphors, which map concepts from more abstractto “more natural” areas. A beautiful example is the mapping from the “systemwith potential energy” domain to the domain of understanding the movement innatural landscape.

1.1.1 The metaphor of potential and landscape

Everyone can imagine an undulating landscape. Peaks, ridges, hills, saddles, val-leys. Steep slopes or, on the contrary, the slightly curved, mostly flat, bottom ofthe valley. One of the most frequently used metaphors connects this mental imagewith the concept of potential or a scalar field. In a more technical description,(scalar) potential is a property of fields where the energy of an object in differentplaces of space depends only on the position and does not depend on the path

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Figure 1.1: Landscape metaphor: physical landscape and abstract potential land-scape. Photo by Jay Huang.

that the object has reached there. This allows a number to be assigned to eachpoint of space, proportional to how much energy it would take to move a unitof “charge” of the field to such point. In the landscape metaphor, this numberis translated into altitude. High numbers correspond to hills, low numbers tovalleys.

As an example, let’s take the problem of a particle in a potential well. Themetaphor makes it is very easy to imagine and reason about the situation - asthe description suggest, the visual image is of a well and an object somewhere atits bottom, which needs energy to escape from the pit.

An important feature of this conceptual analogy of “potential as a landscape”is that it is physically correct. Gravitational potential is such a conservative field,and the intuitive concepts of motion in the landscape can be very accuratelymapped on formal concepts described by the language of mathematics.

local maximum peak of a hilllocal minimum bottom of the valley

gradient gradient slopeisolines coutour

In addition to mapping a potential energy, the same landscape metaphor canbe used for a great number of other domains. For example in optimization, whenlooking for minima of some function, we speak about gradient descent algorithm,valleys and ridges, steep walls, etc. In evolutionary biology we imagine ”fitnesslandscape”.

In many cases, the mapping is less much accurate - for example, organizationsand groups of people are metaphorically mapped to organisms or bodies (leadingto words like a “head of a company” ). Transportation network or arteries inthe body are metaphor for general network systems. War a physical fight is ametaphor for discussion.

Metaphors can not replace mathematical mathematical models. Yet they areuseful for intuitive understanding of the meaning of formal models, and also for

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the ability to create models, and transfer concepts from one domain of knowledgeto another.

In spite of their importance, metaphors are rarely the subject of a formaldescription in physics: their understanding is rather implicit knowledge, acquiredintuitively from informal explanations.

Equipped with the concept of conceptual metaphor, we may continue towardthe main topic of this work, nucleation. This introduction is attempting to de-scribe the rest of this work informally using metaphors.

1.2 Nucleation

What is a nucleation? In physics, the term is used to describe the first phaseof the process in which a new phase or a self-organizing structure emerges. Wecan imagine bubbles, emerging in a glass of sparkling wine. How does suchbubble arise? In the beginning, we have isolated molecules of CO2 dissolvedin the wine. Molecules randomly meet, forming tiny nuclei (also called clustersor embryos). Such clusters can grow by attaching other molecules. Or, on thecontrary, decreasing by detaching molecules.

Nucleation is in fact much more general than formation of bubbles in wine.For physicists, it is sufficient to state that nucleation is usually a part of firs-order phase transition, and therefore occurs in as diverse contexts and systems asphase transitions in general. Informally, we can find nucleation in the formationof objects as big galaxies and as small as nanocrystals. And also as common asraindrops or steam bubbles in a pot of boiling water. The controlled nucleationprocess is one of the ways to create quantum dots or to crystallize active sub-stances in pharmacology, and in a number of other interesting or technologicallyimportant applications.

The nucleation theory typically tries to answer questions such as: how manyclusters of the new phase in a given volume arises? (How many bubbles?) Howlong will it take? How is the process affected conditions such as temperature,pressure, presence of impurities and defects, or flow and turbulence?

We can try to apply the same models and similar mathematical language alsoto many other complex systems - biological, economical and social. In such cases,nucleation-like dynamic can describe phenomena such as innovation spreading,formation of new scientific fields, or also difficulty of societies to move away frominadequate social equilibria.

1.3 Classical theory of nucleation

How does the classical theory describe nucleation? Informally, by looking at thetwo basic features that a new phase cluster has - surface and volume. When acluster of the new phase forms, it means creation of a volume of the new phase,and also a creation of an interface between the new and the old phase.

The basic condition where the nucleation model is applicable is when thecreation of the interface costs energy. At the microscopic level, this surface energyhas the nature of broken bonds of molecules at the interface. On the mesoscopiclevel of the description, we assume that this energy can be expressed as a function

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Figure 1.2: Competition of surface and volume terms, creating the nucleationbarrier.

of the surface size and shape of the new particle. In abstract complex systems,the bonds mean arbitrary relationships which can have some positive or negativequality - for example cooperative and uncooperative moves in a game.

The other condition for nucleation is that is a volume of the cluster, consist-ing of the new phase, which is energetically more favourable. For example themolecules are arranged in a different order, and the difference between the energyof the particle in the old and new phases is negative. Existence of this ’volumeterm’ is the driving force of the phase transition.

We can try to extend the model to social systems. Imagine a situation wheresome scientists start working in a completely new field. If more researchers workon a topic, their productivity is usually higher than if the topic is interesting justfor one or a very small number of people. For an individual researcher, whileworking in the new, un-established field would be potentially more impactful,switching from established to new fields has also costs, often in interrupted linksto co-workers and limited opportunities for collaboration. These broken links havean associated costs, and can be mapped to “surface energy” , having oppositesign from the “volume” .

From the basic dimensional analysis of the problem, in three-dimensionalspace the surface and the volume of the cluster are scaling differently - when weincrease the size of the cluster, with linear growth in diameter the volume growswith the third power, while the surface only with the second power. Roughlyspeaking, the difference in energy terms has a maximum at a certain size of thecluster. Such a core is energetically “the least profitable”. The energy needed

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a.

b.

c.

d.

Figure 1.3: Movement of clusters in the space of sizes. a. one step correspondsto addition of one particle. c. critical cluster on top of the hill

for its formation is called nucleation barrier in the nucleation theory, and its sizeis the basic ingredient used by theory to predict observable quantities, such asthe number of nuclei emerging. Usually such parameters are named critical, asin critical size or critical nucleus.

1.3.1 Master equation

The technical approach used in the classical nucleation theory to describe theproblem is mapping: representation of the real state of the system (a very richstate space, the description of which required a huge amount of information: forexample, the positions of all the particles in the system) into the ”cluster size”space. The described map feels natural for a physicist, but we should notice itis in fact conceptually and mathematically non-trivial. From the very rich statespace we map the state of the system to a very simple representation, whereeach nuclei is described by just one parameter, its size. Moreover, in this spacewe map all the complex interactions to a simple ”energy landscape”, where theposition in the landscape corresponds to the size of the emerging nucleus, and theenergy depends just on this position. Movement in the landscape corresponds tochanges in size of the nucleus, for example, movement to the left to its shrinking,movement to the right to growth.

What does the “landscape” like? The energy barrier is a hill. The clustersbegin their journey in an energy valley (an isolated particle is at the bottom).With the addition of other particles, they “climb uphill” . At the top of thebarrier, the terrain flat. When even more to the right, the terrain starts to slopedown again, and once the clusters are here, they tend to grow.

How do the clusters move in such a size-space? The simplest processes areadding one particle to the cluster, and breaking of one particle from the cluster.This corresponds to one step left, and one step right. Other processes that wemight consider are two or more clusters uniting, or splitting of a cluster in twoor more parts - in practice while these processes happen, their contribution tonucleation is in the classical case negligible.

We can also intuitively visualize how the usual development of a cluster pro-ceeds in such a space. Near to the beginning, small clusters rapidly emerge anddisappear. Rarely, a cluster “climbs” up the barrier in a series of “uphill” steps,with considerable difficulty. Since the top of the barrier is almost flat, the clustersmove here in a close approximation of random walk - and the probability thatthe cluster dissolves or on the contrary starts to grow and leaves the “nucleation”stage are exactly same on the top of the hill. Further to the right, the randomwalk leads some of the clusters toward “the other side of the hill”, where they

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start descending, in this case progressively faster. Further movement is usuallycalled growth phase and is no longer within the scope of of nucleation theory.

The crucial insight of nucleation theory is that the height of the barrier isthe main factor affecting the nucleation rate: the number of clusters that reachthe peak depends on this parameter very dramatically, that is, exponentially.Nucleation as a process is therefore extremely sensitive to conditions that affectthe height of the energy barrier. For example, one practical insight is, when thereare several alternatives, nucleation will mostly proceed via the the path where thebarrier is smallest. We will see this on examples of heterogeneous nucleation or theformation of metastable nanoparticles. Often it helps the intuitive understandingof nucleation to imagine the processes as some kind of “agent”, seeking the bestopportunities to cross the the ridges of of some unpleasant mountains, on a wayto better optima.

1.4 Heterogeneous nucleation

From the simple case of creation of a new phase in an environment that is basicallythe same everywhere, we can move to more complex scenario.

Virtually the most significant case is heterogeneous nucleation, a commonname for situations where nucleation occurs on irregularities in the environment -for example at boundaries, on impurities, foreign particles, but also on “dynamic”irregularities, such as swirling. Examples of a boundary could be the wall ofchampagne glass, or walls of a reaction chamber, or a surface of a magneticmaterial. Examples of irregularities may be the small particles of dust floating inthe atmosphere, or defects in material. Or, if we speculate on nucleation in socialsystems, the “foreign particles” may be the special and charismatic founders ofnew social groups. In most practical cases of physical systems, heterogeneousnucleation is many times faster than homogeneous nucleation so, for example,rain drops are much more likely to form on particles of dust or small ice crystalsthan from pure water vapor. Bubbles in a glass of champagne are more quicklyformed on the walls, and when looking on social systems, we might speculate thatnew sects or social movements are more likely to arise around leaders who areexceptional, not really fitting into the larger society.

The reason why heterogeneous nucleation is often much faster is the energyefficiency - a new phase cluster, such as droplet, attached to a boundary, such asglass surface, has two different interfaces. If the glass surface is ”more similar”to liquid water than air, and less bonds are frustrated on that interface, theformation of the water-glass interface is “ cheaper ” than the water-air interface.Of course this depends on the degree - in the case of droplets the different surfaceenergies are the result of different wetting angles, and some surfaces could beactually unfavourable.

In the case of heterogeneous nucleation on impurities, the formation of a newdroplet causes the formation of a water-air interface, and water-solid interfacee.g. on dust particle boundary, but on the other hand, the air-dust interface willdisappear.

The practical importance of heterogeneous nucleation is difficult to overstate:in terms of how new phases occur in systems studied in natural science or engi-neering, homogenous nucleation is rather unusual, and heterogeneous nucleation

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dominates.

1.5 Heterogeneous substrates

The usual classical description of heterogeneous nucleation on the surface is thatthe surface on which nucleation occurs is itself homogeneous. The fourth chapterof this paper deals with the case where we drop this simplifying assumption, andwe assume that the surface could be inhomogeneous. Our results indicate thatif we keep surface energy the same on average, by patterning the surface we cancause changes in the nucleation rate in several orders of magnitude, even if thesurface structures are smaller than critical clusters. This is in part similar toresults of previous studies where nucleation was influenced by the presence ofpores.

1.6 Nucleation on networks

While for many systems the natural way how to think about them is as aboutbeing embedded in two or three dimensional space, or on a lattice, we can uncovera rich field of models by thinking about different underlying structure: networks.What can be thought of as a network? Social networks, transportation networkor computer networks are some of the central examples of practical interest,but we are thinking about networks in more general. Any complex system inwhich we can distinguish some individual nodes, and some connections, can bestudied as a network. This map could be quite abstract: we can think aboutindividual protein configurations as nodes, and allowed transition between themas links. Or, we can take recipe ingredients as nodes, and the frequency withwhich they appear in recipes together as the strength of the links. Many artificialnetworks are of immense importance –both the power grid and the the networkof international financial transactions are in some ways indispensable for theway how society functions. Given all that realisations of networks, one may besceptical if some useful research can be done at this level of generality – yes, bothneurons in the brain and humans and many other systems have some networkstructure, but are there actually more commonalities? Somewhat surprisinglylate, several physicists, mathematicians and social scientists realized that yes,many real-world networks have surprisingly similar statistical properties, can bestudied as a class, and this regularity could be explained by some simple localgenerative principles. This effort now developed into a whole field of networkscience, with interdisciplinary applications in many other fields. One way howto think about networks is about a whole mathematical language on a similarlevel as the classical calculus: once you learn derivatives, you start to see themeverywhere. Similarly, with networks.

If we hope to apply physics reasoning and models to complex systems such asin social phenomena, network approach usually provides the right topology.

In Chapter 6 we examine effects of structural properties on the global phasetransition behaviour. Example of such question could be:

• For making the phase transition faster, is it better to have one undifferenti-ated large network, or is it better to have several densely connected clusters

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with smaller number of connections between them?

• What is the role of ”hubs”, that is, nodes with many links?

• How to increase or decrease the probability of a cascade failure?

The specific question we are trying to answer in this work is about influenceof clustering, where informally the coefficient measures “how densely the nodesare connected”. Our results show that higher clustering actually makes it easierfor networks to undergo phase transition.

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2. Classical nucleation theory

In the broader view, nucleation is a process almost as omnipresent as phase tran-sitions. It was studied in different transitions including condensation, cavitation,solidification, crystallization and precipitation. And in different fields of physicsand technology ranging from atmospheric physics concerned with condensationof water vapor to the study of damage in neutron-irradiated materials importantfor reactor technology applications.

In most of these situations, some general properties are the same. Discontin-uous phase transitions usually proceed in three steps. First, some of the smallclusters of the new phase, “embryos”, appear due to stochastic fluctuations. Ifthey reach a certain critical size, embryos become “nuclei“. This stage of thetransition is called nucleation. In the second stage, particles grow. Finally, inclosed systems the growth is limited by supply of the untransformed remainingphase.

The formation of nuclei is associated with an energy barrier limiting the pro-cess and allowing persistence of metastable phases over long periods of time. Thebarrier may be lowered, if the cluster forms on proper site of an existing impurity,which leads to heterogeneous nucleation. The barrier may also be lowered if thenucleus is of some intermittent phase, different in structure or composition fromthe stable one [64, p. 93].

In this chapter, we will give a short review of classical nucleation theory andsome of the modern developments and alternatives.

2.1 Homogeneous nucleation without strain

We will start with the simplest case: a single component system without strainenergy, such as a liquid phase condensing from gas, or the precipitation from aliquid solution. In classical nucleation theory, the capillarity approximation isfrequently used, where the values of the parameters used in the model are takento be the same as in macroscopic objects.

Initially, the system is in some α-phase, which is metastable with regard tothe phase β. In order to change the β-phase, first some small cluster of β-phasemust be formed.

The energy balance for the formation of a small cluster consisting of N par-ticles (atoms, ions, etc.) is thermodynamically given as

∆GN = N(µβ − µα) + ∆Ginterface, (2.1)

where µβ (respectively µα) are chemical potentials in the β-phase (resp. α-phase)and ∆Ginterface is the energy of the newly formed interface.

The first term is always negative and represents the driving force of the pro-cess. The surface term is positive and competes with the first (volume) term. Forsmall radii, the ratio of surface to volume is large and the surface term dominatesin effect creating a barrier for nucleation. The nature of the dependence becomesclear if we rewrite (2.1) to be

∆GN = N(µβ − µα) + ηN23σ, (2.2)

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Figure 2.1: Free energy ∆G(N) as a function of cluster size N in a nucleationregime

where η is a shape-factor surface/N23 (constant for a given shape) and σ

denotes the interfacial energy per unit area (see Fig 2.1). The capillarity approx-imation implies the interfacial energy is assumed to be the same as for a largeflat interface.

For small radii, the ratio of surface to volume is large and the surface termdominates creating a barrier for nucleation of the height ∆Gc. The cluster of thecorresponding size is known as a critical nucleus Nc, with the critical radius rc,etc. While clusters smaller than Nc tend to go down the energy slope and shrink,clusters larger than Nc grow further and form stable particles of the new phase.

In the simplest case, where the interfacial energy is isotropic and the formationof the cluster doesn’t cause any strains, the cluster will take a spherical form. Thecorresponding shape factor is then

η = (36π)13V

23A , (2.3)

where VA is the atomic volume of the cluster ”building unit“ (atom, ion, etc.).

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2.1.1 Critical parameters - simple case

Critical parameters in this case may be easily found explicitly from the extremumcondition ∂∆G/∂N = 0:

Nc =

(2ση

3∆µ

)3

(2.4)

and

∆Gc =4 (ση)3

27∆µ2. (2.5)

We can see that the height of the energy barrier depends on the third powerof the surface energy σ and and on the second power of ∆µ. As the change in theenvironment leading to nucleation is often induced by change of temperature, it isinteresting to estimate the influence of temperature on the process. For example,in the case of vapor condensation, we can use ∆µ = kT lnS , where S = pv/peqis the supersaturation (ratio of the actual vapor pressure, pv, to the saturatedvapor pressure peq). Using the Clausius-Clapeyron equation

∆µ ≈ k∆hcon

R

(∆T

Tcon

)(2.6)

where ∆T (respectively Tcon) is undercooling (respectively condensation) temper-ature, ∆hcon is the enthalpy of vaporization and R is the gas constant.

Hence, the driving force of nucleation is linearly dependent on temperature.The lower the temperature, the greater the driving force.

2.2 Homogeneous nucleation with strain

To generalize our conclusions, let us consider the effects of misfit of shape and sizeof the new nucleus within the matrix. While this effect is absent in liquid-solidnucleation, it is generally present in solid-solid transitions. Strains act both inthe nucleating particle and in the surrounding matrix, thus the elastic energy ofthe deformation have to be accounted for in the nucleation energy balance (wedenote the corresponding term ∆Gel). Furthermore, this energy also generallydepends on the number of building units involved within cluster, on its shape andorientation and together with the surface energy unambiguously determines theproperties of the formed nuclei.

2.2.1 Incoherent interfaces

We start with the simplest case of acting of the strain, where the particle interfaceis incoherent[6, p. 597]. This means that only stress is transmitted across theinterface. Strains are relaxed. As a further simplification, we can assume thatthere are no pre-existing stresses in the matrix and, moreover, properties of bothphases are isotropic.

Inclusion in the shape of the general ellipsoid of revolution was treated byNabarro [54] within the context of classical elasticity. The ellipsoid with semiaxesa,a,c also includes important limit cases of thin needles (c ≫ a) and flat discs(a ≫ c). Without any calculations, we can conclude that the strain energy,

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Figure 2.2: Shape dependence of strain energy. (For c/a→∞, E = 3/4).

∆Gstrain, influences the nucleus shape in a different way than the surface energy.While in the limiting case of a very thin disk, the strain energy would go to zero.This shape would maximize the surface energy.

Nabarro achieved a general expression for the strain energy per volume

∆Gstrain = 6µϵ2E( ca

), (2.7)

where the shape-dependent function E(c/a) is of the form, as shown in Fig 2.2and ϵ is strain. The important conclusion is that this contribution, at least fordilute solutions (where particles are not close to each other), is volume dependent.

2.2.2 Interface development during nucleation and growth

The above considerations lead to a picture which can be generally varied. On oneside, the interfacial energy is low, if the particle keeps coherency with the matrix(if it is possible). On the other side if there is any misfit in lattice parameters,this generates strain that eventually has to be released and the particle ”breaksaway“. Hence, the interface l ++ oses its coherency and becomes semicoherentor incoherent.

14

2.3 Nucleation rate

2.3.1 Classical nucleation rate

The nucleation rate is defined as the rate at which stable nuclei are formed withinthe unit volume in unit time.

Various descriptions of the process exist. Early calculations were done byFarkas [31], and steady-state calculations by Becker and Doring [10]. Here, weuse the cluster dynamics approach to allow the derivation of both classical theoryand its flavors as well as some of the recent models. These naturally showsthe links between them. Conceptually, we can understand the approach as aprojection of the huge state space of the system onto a single variable, where wewant the projection to keep the Markov process properties of the original system.

In the non-nucleation regime, the new phase β is not stable, ∆GN is alwayspositive, no stable nuclei form and the nucleation rate is zero. The equilibriumdistribution of the clusters, minimizing the free energy of the systems, is

XN = exp(−∆GN

kT), (2.8)

where XN is a fraction of the clusters of size N to all the clusters.In a nucleation regime, the system is out of equilibrium and the clusters are

larger than Nc, which grow to stable sizes.The growth can be described as a flux of clusters in a size-space. If the

coalescence rate is small (which is most often true at least at the early stageof nucleation), it can be assumed that the growth is governed by single particleprocesses (addition or lose of one particle, the so-called step-by-step process).The cluster flux rate at one particular size can then be written as

J(N)(t) = β(N)F(N)− α(N + 1)F(N + 1), (2.9)

where β(N) (respectively α(N)) are transition probabilities of a particle joining(respectively leaving) a cluster of size N and F(N) is a number of clusters ofsize N . The rate is, in general, time-dependent and the whole set of equationsdescribing the time-dependent problem reads (using the condition of a constantnumber of particles):

∂F(N)

∂t= J(N − 1)− J(N), (2.10)

∂F(1)∂t

= −2J(1)−∑M>2

J(M). (2.11)

To solve equation (2.9), we define recursive quantity Z as

Z(N + 1) = Z(N)α(N + 1)

β(N + 1), Z(1) = 1 (2.12)

summarizing information from the β(N) and α(N) coefficients. Z can be ex-pressed explicitly as follows:

Z(N) =N∏

M=1

α(M)

β(M), Z(1) = 1. (2.13)

15

For the ratio α(N)/β(N), we generally have:

α(N)/β(N) > 1 ∀ N < Nc, (2.14)

α(N)/β(N) = 1 ↔ N = Nc, (2.15)

α(N)/β(N) < 1 ∀ N > Nc, (2.16)

limN→∞ Z(N) = 0.Multiplying (2.9) by Z(N) and summing up N , one obtains:

N∑1

J(N)Z(N) =N∑1

β(N)F(N)Z(N)− α(N + 1)F(N + 1)β(N + 1)

α(N + 1)Z(N + 1).

(2.17)The first and second terms on the right side of the summation differ only in theindex and cancel, except at the bounds of summation. Hence,

N∑1

J(N)Z(N) = β(1)F(1)− β(N + 1)F(N + 1)Z(N + 1). (2.18)

2.3.2 Nucleation rate in a quasi-steady-state

For large N , the second term in (2.18) can be neglected. Further, losing generalitywe assume the system has reached quasi-equilibrium and the cluster flux steady-state value, J(N)(t) = J . (It is not apparent whether it is good approximationof some real systems. Whether it is actually a good approximation or how long isthe “relaxation“ period before steady flux is achieved shall be discussed in Section2.3.3.

Using the above assumptions, a teady nucleation rate can be expressed as:

J =β(1)F(1)

1 +∑N

M=2

∏MN=2

α(N)β(N)

. (2.19)

To proceed further, some estimates of α(N) and β(N) are necessary. Onepossible way is to take the attachment probabilities directly from some model.This leads to the kinematic nucleation theory by Katz and Weidersich [42], ifthe attachment probabilities for gas molecules and droplets are used. But fromthis point, we can also derive the classical nucleation equation by taking theprobabilities from the equilibrium distribution (2.8) in true equilibrium withoutnucleation, or in a slightly different approach, by constructing an artificial con-strained equilibrium, in which cluster sizes are limited by the upper boundaryNmax, F(N) = 0∀N > Nmax.

In a real equilibrium (which naturally forms a boundary condition for thenucleation), J(N)(t) = 0 and (2.9) becomes:

0 = β(N)F(N)− α(N + 1)F(N + 1). (2.20)

By substituting the equilibrium distribution (2.8), we get a relation for the tran-sition probabilities

0 = N(β(N)e−∆GNkT − α(N + 1)e−

∆GN+1kT ), (2.21)

16

leading toα(N + 1) = β(N)e

1kT

(∆GN+1−∆GN ). (2.22)

We use this expression to solve the summation from (2.19). The inner term is

M∏N=2

α(N)

β(N)=

β(M − 1). . . . β(1)

β(M).β(M − 1) . . . β(2). (2.23)

. exp1

kT((∆GM −∆GM−1) + (∆GM−1 −∆GM−2). (2.24)

. . . (∆G2 −∆G1)) (2.25)

=1

β(M)exp

1

kT(∆GM −∆G1)β(1). (2.26)

If we approximate the summation in (2.19) by an integration, the denominatorbecomes: ∫ N

2

1

β(M)e

1kT

(∆GM−∆G1)β(1)dM. (2.27)

This integral depends on the given ∆GM . We use the classical expression(2.1) based on the capillarity assumption. Then, as the value of the integrand issignificant only near Nc, we can approximate β(M) by β(Nc) and we can extendthe integration range to (−∞;∞):∫ ∞

−∞

1

β(M)e

1kT

(∆GM−∆G1)β(1)dM. (2.28)

In this approximation, we can also expand ∆GM around Nc to the second order:

∆GM = ∆GNc +(M −Nc)

2

2

(∂2∆GM

∂M2

)M=Nc

(2.29)

= ∆GNc −1

3

∆Gc

N2c

(M −Nc)2 (2.30)

and by solving the resulting Gaussian integral we obtain:

β(1)

β(Nc)e

1kT

(∆Gc−∆G1)

√3πN2

c kT

∆Gc

. (2.31)

Thus, quasi-steady-state equation for the nucleation rate (2.19) in classical theoryleads to

J =

(∆Gc

3πN2c kT

) 12

βcFe1kT

(−∆Gc+∆G1). (2.32)

.The first dimensionless term is called the Zeldovich factor and its magnitude

is typically 10−1 [6, p. 466].The ∆G1 term in the exponent requires some explanation. It seems natural

to expect the energy of ”formation“ of ”clusters” of size 1 to be 0. But theexpression for ∆G(N), (2.2) gives generally a non-zero value for N = 1, whichleads to apparently self-contradictory predictions for clusters of size 1 even in anequilibrium distribution (2.8) and in all consecutive calculations. This can be

17

Figure 2.3: Schematic representation of the energies of clusters of sizes N , re-spective of N + 1 and the intermediate state of higher energy.

understood as a result of stretching the capillarity approximation used in (2.2)clearly beyond it’s limits (to a single monomer). One alternative is to use anatural ∆G(N) = 0. Other proposed alternative [35] is to include the term,which can be thought of as a correction to (2.2)

∆GNSCT = (N − 1)(µβ − µα) + ηN23σ − ησ. (2.33)

This leads to an internally consistent nucleation theory.

Connection to Zeldovich-Frenkel and Turnbull-Fisher treatment

While the above described expressions are quite generic, in various contextsslightly different pictures of the process are most commonly used. In conden-sation of vapors, the Zeldovich-Frenkel (or the Becker-Doring) picture seems tobe the most common version of the classical nucleation theory. The main differ-ence of the Zeldovich-Frenkel picture is in the use of the continuum size-spacefrom the beginning, leading to the immediate use of differential quantities andequations.

In the context of crystallization and solid solutions, references to the Turnbull-Fisher [75] picture are common, so we should make a quick connection to the T-Fpicture.

The T-F nucleation theory can be understood as a branch of the above givendescription in the point, where estimations of α(N) and β(N) are given. T-Fobtains the probabilities α(N) and β(N) from the classical reaction rate theory.An intermediate configuration (known as an “activated complex”) is assumedbetween stable clusters of size N and (N + 1)-cluster.

The intermediate state creates an additional energy barrier of the height ∆gabove the mean of ∆GN and ∆GN+1, as shown in Fig 2.3. Transition probabilitiesread:

β(N) = a(N)+kT

he−

1kT

(∆g+ 12(∆G(N+1)−∆G(N)), (2.34)

respectively

α(N + 1) = a(N + 1)−kT

he−

1kT

(∆g− 12(∆G(N+1)−∆G(N)), (2.35)

18

where a(N)+ and a(N+1)− is a number of α atoms “in contact” with a cluster ora number of cluster atoms in contact with α phase and the difference between thetwo is considered negligible for all but the smallest nuclei, a(N)+ ∼= a(N+1)− ∼= a.Then, it is possible to relate the probabilities in a way completely analogous to(2.22):

α(N + 1) = β(N)e1kT

−(∆G(N+1)−∆G(N)) (2.36)

and onward proceed in the same way as from (2.22) - (4.2.1). As β is given moreexplicitly, it is possible to express J as:

J =

(∆Gc

3πN2c kT

) 12

F(kT

h

)ac exp

1

kT(−∆g −∆Gc +∆G1), (2.37)

where ac is the number of surface atoms belonging to the critical nucleus. Theterm −∆g in the exponent may be understood to be the activation energy of theattachment of monomers.

2.3.3 Non-steady-state nucleation rate

To what extent is the quasi-steady-state with the J constant described above isa good approximation of a real physical system?

First, we can see the stationary state has to be preceded by some transi-tion period in which the initial size distribution of the clusters F(N)0 from thenon-nucleation conditions evolves to a semi-steady distribution F(N)steady. Thisperiod, referred sometimes slightly ambiguously as an “incubation time”, hasbeen subject of a lot of study, where both numerical and analytical solutionshave been obtained.

Second, the stationary of the state is dependent both on the constant drivingforce µβ − µα and the constant boundary condition F(1)(t) = F(1)(0). This canbe true only in an open system, where the number of monomers is always keptconstant and is approximately true in a closed system, only when the ratio ofmonomers transformed to nuclei to all monomers is negligible [44]. Otherwise,depletion of the monomers lowers the driving force and eventually stops the nu-cleation. While it is shown in numerical simulations that in some situations themaximum nucleation rate achieved in the system is well below the “steady“ ratecalculated from initial supersaturation [45], this problem is less studied.

Also, other parameters of the system, such as temperature and pressure, maybe varied.

Time-scale analysis of transient processes

When describing real systems, often a big part of the successful description is anassessment of which simplifications are acceptable and which are not. A simpleand useful tool for this task is a time-scale analysis. So for every non-stationaryconsidered, we will try to obtain some time scaling constant characterizing theprocess.

Transient nucleation rate

The time dependence of a number of nuclei in a system (where steady-state wasachieved) is shown in Fig 2.4. Convenient parameters to describe the evolution

19

Figure 2.4: Definition of the incubation time tinc.

are Jss and incubation time tinc, defined as a time lag to crossing point of thetangents of the growth curve in a steady-state with a time axis (see Fig 2.4).

Analytical treatments of the transient period have most often been based ona Z-F continuum approximation. We substitute equilibrium rates (2.22) to (2.9)dividing by the equilibrium distribution (2.8):

J = β(N)Feq(N)

(F(N)(t)

Feq(N)− F(N + 1)(t)

Feq(N + 1)

). (2.38)

We express the flux in differential terms together with (2.10):

J = −β(N)Feq(N)∂

∂N

[F(N)(t)

Neq(N)

](2.39)

∂F(n)∂t

= −∂J(n)

∂n. (2.40)

(2.41)

Using a classical expression for the nucleation barrier (2.2), suitable boundaryconditions

F(N)(t = 0+) = Feq(1)Θ(N − 1), (2.42)[F(N)(t)

Feq(N)

]N→1

= 1, (2.43)[F(N)(t)

Feq(N)

]N→+∞

= 0, (2.44)

(2.45)

20

where Θ is the Heavyside function. Approximate solutions in good agreementwith the exact numerical solutions were found by Demo and Kozisek [25].

F(N)(t) =1

2Feq(N)ercf

⎛⎝3ZNc

√π

[(N/Nc)

13 − 1

]+ (1−N

− 13

c )e−t/τ

√1− e−2t/τ

⎞⎠ (2.46)

and the corresponding cluster flux at critical size

J(Nc)(t) = Js1√

1− e−2t/τexp

(−3ZNc

√π(1−N

− 13

c )e−t/τ

√1− e−2t/τ

)(2.47)

where Js is the steady-state flux (4.2.1). Z is the Zeldovich factor and τ =7/(10πZ2β(1)). Often older approximate solutions by Trinkaus and Yoo [74] andKashchiev [40] are also used.

We can observe that (2.47) behaves as expected, for long time periods J(Nc)(t→∞) = Js and for very short time periods J(Nc) is exponentially small. Unfor-tunately, the relation of the time scale constant τ to the incubation time tinc isdefined as above is not easy. We need to to calculate the integrated cluster fluxat Nc (the cluster flux in a transient regime depends on cluster size) to be

I(Nc)(t) =

∫ t

0

J(Nc)(t)dt, (2.48)

which is an important measurable quantity on its own. Then tinc can be deter-mined from the linear part of this dependence.

While the above described solutions are much more accurate, the simple phys-ical picture and the simpler approximations can lead to a qualitative understand-ing of the process. Close to the critical size NC , the ∆GN becomes more and moreflat. Hence, the transition probabilities are almost the same and the relation (2.9)can be approximated by

J(N)(t) = β(N)(F(N)−F(N + 1)) (2.49)

In the continuum approximation,

J(N)(t) = −β(N)F(N)

N + 1. (2.50)

This is equivalent to a simple diffusion equation in the presence of only a con-centration gradient, so the movement of the cluster in size-space in the vicinityof NC exhibits the character of a random walk. Further, the movement may beunderstood to be a diffusion in the presence of a certain potential. So, the timelag consists of the time necessary to reach the critical region in the vicinity ofNC plus the time to random walk the distance where the potential is flat. Thenatural boundary of the critical region is such that the potential deviates fromflatness less than kT as shown in Fig 2.5.

The distance of the random walk is denoted by δ. As expected, the absolutevalue displacement of the random walk after n jumps is ∼

√n, the necessary time

is then τ ≈ δ2/2β(Nc). Using the expansion of ∆GN from (2.29) we easily get

δ2 =kTN2

c

3∆Gc

. (2.51)

21

Figure 2.5: Variation of free energy in the critical region for sizes close to Nc

Hence,

τ ≈ kTN2c

6∆Gcβ(Nc)=

1

18πZ2β(Nc), (2.52)

which has the same form as the time scale from (2.47) and as several scalesused in approximate solutions reviewed by Kelton [43]. As various authors usedifferent definitions of transition rates, the direct comparison is somewhat diffi-cult.

2.3.4 Nucleation and growth

In closed systems, the process of rapid nucleation changes the untransformed partof the system, this often provides negative feedback for the process.

If the process is driven by supersaturation and the mobility of monomers islow, the composition in the vicinity of nuclei changes generating concentrationgradients. Consequently, the growth becomes limited by the drift rate.

If the process is driven by supersaturation and the mobility of monomersis sufficiently high (as is true in the case of condensation), the concentrationgradients quickly relax and supersaturation is lowered in whole system.

In both cases, the greater the initial supersaturation, the more pronouncedthe changes are.

Growth and nucleation processes compete so as smaller particles may becomeunstable under decreasing supersaturation then bigger particles may grow at theirexpense (”coarsening”).

Transition from nucleation to growth

In the cluster dynamics approach we follow, the growth region corresponds tocondition N > Nc, where movement in cluster space is driven by drift. We canaverage over the random fluctuations and rewrite the master equation (2.9) as

J(N)(t) = γ(N)F(N), (2.53)

22

where γ = β(N)− α(N + 1) is the growth rate. Now, if we describe not the fluxof the cluster at specific N but rather an average growth of a single large clusterof size n(t), the descriptions are related simply by

dn

dt= γ(N). (2.54)

Some classical models (such as T-F (2.34)) describe certain systems up to anarbitrary size - in large clusters the T-F model predicts [24]

γ(N →∞) = a(N)kT

he−

1kT

∆g sinh (∆µ/2kT ) . (2.55)

. Translated to a single-cluster, the T-F model describes surface limited growth.As a(N) ∼ N2/3 ,

dn

dt∼ n2/3, (2.56)

and, asymptotically,n ∼ t3. (2.57)

By similar reasoning, we can account for any “monomer supply” limiting effectin cluster dynamics given the model of growth of big clusters, where γ(N) (forlarge N).

23

3. Nucleation with formation ofnanophases

3.1 Introduction

In this section, we study the nucleation pathway in a system of alkali halides.It was known at least since the observations of Burstein et al. [15] that optical

absorption and emission spectra of alcali halide crystals doped with small amountsof divalent impurities exhibit interesting structures, varying with the growth andthermal history of a crystal. Formation of small aggregates of the impurity is thereason for this and despite a simple structure of halide matrices, the aggregationprocesses vary considerably between different matrix-impurity systems.

The systems were experimentally studied by X-ray diffraction. Of those stud-ies, the most important were by Suzuki [53][71] in 1954 and 1955 on NaCl-CaCl2mixed crystals. Suzuki observed larger impurity precipitates and proposed amodel for their structure explaining observed X-ray diffractions, consisting ofthe formation of thin plate-zones in {111} and 310 planes. These consist ofthinner ”platelets“, nanophase structures coherent with surrounding lattices yetresembling the structure of CaCl2. Both the observed patterns and their in-terpretation is analogous to patterns observed in age hardened alloys (so-calledGuinier-Preston zones) [6, p. 557]. In later pioneering work [72], Suzuki stud-ied NaCl-CdCl2 systems and explained observed patterns by the emergence ofa new metastable phase (now called the Suzuki phase), with the primitive unitcell edge twice the size of the NaCl primitive cell edge, and of the CdCl2.6NaClstochiometry. The Suzuki phase maintains consistency with the NaCl lattice.

Careful dielectric, UV absorption and emission studies including the study ofthe evolution of the spectra with time were made mostly by Dryden [20][21][29]in NaCl-CaCl2, NaCl-MnCl2, KCl-Sr2 and KCl-CaCl2 systems. Similar measure-ments have been made by Capelletti and Benedetti [16] in NaCl-CdCl2.

In the initial state, the impurity is present in the form of the impurity-vacancypairs (dipoles). The next stage,formation of small dipole aggregates, is less clear

Figure 3.1: Suzuki phase unit cell in an NaCl crystal in the 100 plane. Cl ions inviolet and Pb ions in yellow. Vacancies indicated by letter V.

24

and various models have been developed. Dryden suggested [21] the third-orderreaction of dipoles to form trimer of a hexagonal structure, and, subsequently,by addition of two dipoles a time to form a pentamer, heptamer, etc. Differingmodels by Strutt and Lilley [70] supposed the precipitation of the Suzuki phase.Crawford [23] improved the model by Dryden by assuming the intermediate dimerformation, which was supposed to be more plausible than the reaction of the thirdorder (direct formation of the trimer from three dipoles) originally proposed byDryden.

In 1980, theoretical studies of the problem were done by Corish et al. [22]and Bannon et al. [7] using the lattice model based on interionic potentials.The most comprehensive studies until now were computations of defect-clusterenergies in NaCl:Mg2, KCl:Mg2 and KBr:Mg2 by Corish et al. [22] and of Mn,Cd and Pb incorporated in NaCl, KCl and KBr structures by Bannon et. al.[7]. In calculations of Bannon et al. [7], most of the stable small clusters forboth systems are the hexagonal trimers described by Dryden, but for KCl thestabilization energy of this trimer was smaller than then stabilization energy ofthe bulk Suzuki phase. Bannon concluded that aggregation of the nn dipole leadsto this exceptionally stable hexagonal trimer and for some systems the processends there. For other systems, where the stabilization energy of Suzuki phasewould be higher, the dipoles are likely ”converted“ in the dimer stage to the nnnform more suitable for formation of the Suzuki phase. Aggregation of such dipolesthen leads to the aggregation of the Suzuki phase and, eventually, the growingSuzuki phase causes the redissolving of trimers. This aggregation path remainedsomewhat unexplained by results of the calculations.

Later experiments confirmed nucleation of the Suzuki phase aggregates insome systems [32] or the calculated energy of Suzuki phases by high quality abinitio computations [18], but these results were often limited to a single alkalihalide impurity system. While such results may be high in precision, they arealso mostly of limited use in explaining the nucleation behavior and differencesbetween various halide impurity combinations.

Surprisingly, while the Suzuki phase remained a topic of sustained attentionsince its proposal, his other model of ”platelets“ occurring in NaCl-CaCl2 wasnot considered in most of the later studies. Recently Polak et.al [38] measuredPb in NaCl and KCl nucleation and suggested differences in the behavior of thesystems is related to the intermediate formation of some complex lead halide (e.g.KPbCl3) which does not exist in the case of Na-Pb-Cl system, and modeled theaggregation of PbCl2 in NaCl by the classical nucleation theory.

In the present work, we use lattice modeling to evaluate the relative stability ofthe various small aggregates in NaCl and KCl crystals doped with PbCl2 and usethe results to explain the marked difference in their nucleation behavior observedin experiment.

3.2 Model

Generally, we treated small aggregates as defects embedded in cluster containing1229 ions. Then, the classical lattice relaxation based on Ewald summation andpair interionic potentials was used for the optimization of energy of the structure.

Computations were done using the program GULP [34] and some auxiliary

25

nn D2 T5

Q7 Q8 Q9

Figure 3.2: Selected configurations of clusters in KCl system

original code for the generation of aggregate configurations. Empirical interionicpotentials (IOPs) by Catlow [17] were used for the description of mother phaseinteractions and an original potential for Pb-Cl interaction was created by fittingtheempirical data of PbCl2 structure in GULP. IOP parameters are listed in ....

We started with dipoles, examining all the possible configurations up tosome impurity-vacancy distance. For dimers (aggregates of two impurity-vacancypairs), trimers, and higher aggregates an exhaustive search in configurationalspace would be computationally prohibitive, so we decided to “follow“ the growthand examine only clusters which can be formed from (N − 1)-particle clusters byaddition of a single dipole to most stable configurations(s), with some arbitrarymanual pruning of the configuration space and some additions. All configura-tions present in the previous studies of Corish et al. and Bannon et.al. were alsopresent.

For the next step, modelling of the larger clusters, we changed the treatmentfrom “defects in a cluster“ boundary conditions to periodic boundary conditions.

General quantity unifying both steps are energy differences ∆U between acluster and its constituent dipoles (nn or nnn) expressed per Pb atom.

3.3 Results

KCl

In case of KCl, the most stable dimer is quadrupole D2, most stable trimeris T2, and generally, the initial steps of the aggregation seem straightforward.The emerging structure is a phase of the Suzuki phase cell. Calculated energy

26

Structure U(L)

T5 -0.30Suzuki phase -0.84PbCl2 -0.94

Table 3.1: Energies of stabilization of selected structures in eV, per Pb atom, inKCl.

differences between simple planar growth Q7 and three dimensional configurationidentical to quarter the Suzuki phase unit cell Q8 are negligible.

Stabilization energies of the formation per Pb atom U(L) of various metastableobjects and the final PbCl2 phase are located in Table 3.1. In this case, thedifference between the metastable Suzuki phase and the stable PbCl2, which isthe driving force of further transformation, is only 0.1 eV.

Hence, further transition to full PbCl2 structure is difficult.

NaCl

The NaCl-Pb system exhibited very different patterns. Most stable dimer con-figurations, labeled D4, have lead atoms in the 2nd next nearest neighbor sitesand vacancies located between them. Addition of an impurity and a vacancyto this dimer leads to many configurations with small energy differences, butgenerally configurations with lower energy stay planar in the {111} plane andhave distances between Pb atoms greater than between vacancies. The emergingstructure is a two-dimensional arrangement of impurities and vacancies in the{111}-Na-plane. The most stable trimer seems to be a symmetrical hexagonalstructure T3, though there is only a small difference between the symmetrical T3and the crescent-shaped T2. Both in energy and structure, the structures can beconverted by one vacancy jump. Further addition of a dipole to this trimer leadsto an S shaped chain Q2 or to a hexagon with a dimer attached to Q4, againwith little energy difference and convertible by one vacancy jump.

As results for small cluster indicate that the clusters first grow in one plane,in the next step, we calculated energies of various arrangements of Na, respectiveof Pb ions and vacancies in the {111} Na-layer. A supercell was constructed fromthe planar arrangements in the {111} plane and supplemented by several Cl andNa layers forming an ordinary NaCl lattice, extending n layers in the directionperpendicular to the plane of the growing cluster structure (see Fig 3.4).

All arrangements with 2 × 2 and 3 × 2 2D-periodic structure unit cells wereconsidered, along with some additional ones, inspired by energetically favorablecluster configurations, up to 4× 4 2D unit cell.

Precisely speaking, the calculated system with this choice of supercell is aninfinite ensemble of infinite planar layers of NaPb separated by n layers of NaCl.As coherency of the layer (respective of its interfaces) is imposed by the definitionof the supercell, interfacial energy and coherency strain created on the interfaceare mixed. We can partially overcome this problem by analysing cases witha different number of attached NaCl layers. The case of individual layers isequivalent to n → ∞, so it is in theory possible to split the interfacial and

27

Figure 3.3: Energies of Pb-vacancy clusters in the {111} Na plane of NaCl crystal.Relative positions of the clusters represent energies of formation from their closestconstituent parts.

28

Figure 3.4: Example of a supercell used for layer energy calculations. Pb atomsare yellow and Na atoms are blue.

Layer energy s U(L)

L1 0.00 -0.31L2 0.50 -0.23L3 0.88 -0.16L4 0.38 -0.24L6 0.67 -0.14L7 0.67 *-0.37

Table 3.2: Energies of stabilization of selected arrangements in eV, per Pb atom,relative to Pb in nn dipoles. Most stable configuration is marked *.

coherency strain by analysing the behaviour of the dependence of layer formationenergy U(L) on n (defined later). In the numerical results, this dependencewas observed, but was generally insignificant for relative comparison of differentarrangements. It must be noted, however, that energies of structures obtained inthis way are not exactly comparable to defect energies calculated from embeddingin clusters.

Examples of several such arrangements are given in Fig 3.5. Average energiesof formation of arrangement layers per Pb atom were calculated. For layer L,supplemented by the Cl layer and n NaCl layers the energy is

∆U(L) = U(cell)− ((n+ 1)U(NaCl) + U(nn)), (3.1)

The most stable arrangement found in this way is the one marked by L7, whichcan be thought of as a tiling of a plane with hexagonal pattern based on the moststable trimer T3. It is possible in a real crystal that the real arrangements will bedifferent, or more complicated. Limited classes of arrangements were examinedand the effects of the influence of dislocations were not considered. Nevertheless,even if this configuration minimizing energy is not the global minimum in thewhole energy landscape, some observations can be made: the comparison of theenergy of formation per Pb atom U(L) of the L7 layer and respective quantityfor small clusters (U(c)/n − U(nn)) shows the layer is more stable than small

29

L1 L2

L3 L4

L6 L7

Figure 3.5: Examples of periodic arrangements in the {111} plane. Na ions inblue and Pb ions in yellow.

30

Figure 3.6: PbCl2 crystal in the 001 plane

clusters, or in other words, energy is released by the growth from small to largeclusters.

Multiple layers and transition to PbCl2 in NaCl-matrix

Further growth can occur in the direction perpendicular to the planar structure,which is a case analogous to surface growth. An attempt was made to explorethis likely next stage and energies of various double-layer and triple-layer ar-rangements were calculated in a way similar to the single-layer case. Resultingenergy differences between the various structures based on the L7 configurationwere of the order 0.01 eV per Pb atom. Also the energy difference betweene.g. triple-layer L7 structure and triple-layer L1 structure shrank to 0.02 eVper Pb atom. So it is clearly beyond the precision of the presented calculations.Furthermore, for large clusters, probably the long-range coherency stresses (asso-ciated with the large planar coherent interface) will be relaxed by the introductionof anticoherency dislocations and the interface would lose coherency. Similarly,roughening of the surface will likely occur because of the free energy contributionof configurational entropy.

It seems most probable that at some point, the existing aggregates undergoanother transition, while staying in the shape of thin plates in the {111} plane,the structure further deforms to full PbCl2 structure and loses further coherencywith the surrounding matrix. From comparison of the PbCl2 crystal in the 001plane (Fig 3.6) and NaCl crystal in the {111} plane (Fig 3.7) it seems likely thatthe loss of coherency still will be only partial.

A scaling analysis of interfacial and strain energies suggests the decisive pa-rameter for this transition is plate thickness. For thin plates consisting of nmetastable structure layers, interfacial energy is proportional to the area S of theplatelet, while bulk lattice energy depends on volume S.n.

Stabilization energies given in Table 3.3 indicate the difference between theintermittent planar phase and stable PbCl2 is approximately −0.4 eV, muchgreater than in case of KCl.

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Figure 3.7: NaCl crystal in the {111} plane, Cl ions in violet

Structure U(L)

D4 -0.17L7 -0.37Suzuki phase -0.34PbCl2 -0.74

Table 3.3: Energies of stabilization of the selected arrangements in eV, per Pbatom, in NaCl.

3.4 Discussion

In summary, in a lightly doped KCl-PbCl2 system, the Suzuki phase easily nucle-ates from the first aggregation steps, and further transition to PbCl2 is hinderedby the high stability of this phase. This is in good agreement with experimen-tal data. In comparison, NaCl-PbCl2 systems exhibit complex behavior, wherethe first metastable 2D phase is completely coherent with host lattice nucleates.Transition to the PbCl2 structure most likely occurs after the plate-like parti-cle reaches critical thickness. This result is different from some of the previoustheoretical studies, but seems to be compatible with experimental measurements[38].

3.4.1 KCl-PbCl2

Results of the presented calculations agree with the general understanding (e.g.[69]) that when ratio of dopant ionic radius to cation ionic radius is relativelysmall, formation of the Suzuki phase is likely. Bannon et al. [7] conclude thatthe Suzuki phase is stable with respect to small clusters once formed. However,in these calculations, the hexagonal cluster T3 appeared to be more stable thanthe natural precursors to the Suzuki phase (e.g. T5). Hence, the initial steps ofthe process remained somewhat unexplained. As results of this work show, thenatural pathway for the Suzuki phase nucleation from direct precursors seemslikely by the limited accuracy of the calculations (this was noted in [7] as possibleexplanation).

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3.4.2 NaCl-PbCl2

Presented results for NaCl-PbCl2 system are in agreement with early suggestionsby Dryden that the first stage of nucleation takes place in the {1, 1, 1} plane viahexagonal structures. At variance with the original proposal, structures consist-ing of an even number of dipoles (dimers,tetramers,heptamers,...) are not signif-icantly more stable than odd-numbered clusters (trimers, pentamers,...). Hence,the nucleation proceeds by a single particle processes, as is usual in many othersystems.

Again, direct comparison is possible with results of Bannon et al. [7]. Whiletrends of stabilization energies calculated here are the same, the dimer andtetramer Pb-vacancy clusters which are most stable (D4, Q2) are missing in Ban-non’s study. Also, conclusions that trimer T3 is likely a stable end point of theaggregation sequence in these systems is different from our results, which indicatesaggregation proceeds further, eventually leading to large particles.

The nature of the expected planar metastable phase is very similar to platezones described by Suzuki in a NaCl-CaCl2 system.

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4. Ising model as a testbed ofnucleation

In the development of nucleation theory, there is obviously a huge space for ex-tensions and improvements of the classical theory described in previous chapters.

The first big class results from lifting some of the simplifying assumptions (e.g.not assuming the steady state) we can examine the time lag to nucleation [26]or we can study the theory of nucleation in closed systems [46]. Another bigclass consists of attempts to improve the core of the theory, for example byincluding some seemingly neglected entropy contributions, or by imposing someformal requirements on consistency (e.g. note that in the above derivation the“formation energy” of a size 1 “cluster” is nonzero, which seems unnatural).

Until recently, such modifications were typically very hard to test. Individ-ual nuclei are usually too small to be directly observed, particularly “in vivo”,when the nucleation process is happening. The quantity accessible to experimentwith is often only the total nucleation rate partially obscured by the subsequentgrowth processes. Due to the exponential dependence of the nucleation rate onparameters, including temperature and the energy barrier, it is often very hardto distinguish experimentally whether some proposal is really an improvementto the theory, or if it just happens to push the predicted nucleation rate in the“correct” direction, compensating for often large errors of the experimental dataor the parameter control. The difficulty with experimental tests is also relatedto the fact that predictions for a relatively small space of experimental data (e.g.the dependence of the nucleation rate on a single parameter such as tempera-ture) are based on a much bigger space of model parameters and assumptions(e.g. chemical potentials, surface energies taken from macroscopic systems, as-sumptions such as the insignificance of the time scale with which the system istempered in relation to the nucleation time scale, etc.).

This is a situation where computer simulations can be of enormous use, allow-ing precise control over the big parameter space and allowing individual aspectsof the theory to be tested.

4.1 Modern statistical sampling methods

The difficulty with computer simulations of nucleation lies in the rarity of nucle-ation events. For example, in the case described below of nucleation in the latticeIsing model, the typical time until one nucleation event occurs is 105 simulationsteps. Straightforward simulations may wander endlessly in the initial phase,then the nucleation event proceeds very fast in a few steps, and then the systemsremains in the final phase. To obtain a meaningful statistical sample, or anysample at all, it is therefore necessary to employ algorithms which enhance theprobability of rare events and lead to a detailed exploration of the phase spaceclose to the transition point. A detailed description or a comparison of thesemethods is beyond the scope of this paper. For a comprehensive review includingpractical comparisons, see Van Erp [76].

From several typically used methods to study nucleation, in this work we

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implement Forward Flux Sampling (FFS) [3, 1] and Umbrella sampling [73, 33] -see A

4.2 Testbed systems and results

Even in a simulation, when modeling real systems using molecular dynamics,nucleation theory gets tested along with various other simulation properties (e.g.a description of interatomic forces). For a systematic improvement of nucleationtheory itself, the ideal case is a system with as few as possible arbitrary parametersof both the system and the simulation. Two such model systems are particularlyimportant. The system of hard spheres, often used as a reference model of aliquid, is also used for studies of nucleation. The second system is the latticeIsing model, one of the simplest statistical systems exhibiting phase transitions.

4.2.1 Hard spheres

In 2001, Auer and Frenkel [5] used a model of hard spheres to predict absolutecrystal nucleation rates without any adjustable parameters and most of the as-sumptions of CNT. In their comparison of the results with CNT, their conclusionwas that the CNT predictions for the height of the nucleation barrier ∆G are notaccurate (30 − 50% is too low), but the data from the simulation can be fittedto the functional form given by CNT, except for very small clusters. Auer andFrenkel also studied the nucleation pathway, the sequence of structures of smallclusters. This topic was later also studied by O’Malley and Snook [57] and others.

Prestipino et al. [60] used the hard sphere model to systematically test theassumptions of CNT, giving particular attention to the definition of clusters andrelated problems with cluster shape and interfacial energy, leading to correctionsto the first part of the theory (determining the nucleation barrier and capillarityapproximation).

Heterogeneous nucleation of hard spheres on walls was examined by Auer andFrenkel [4]. Drastic lowering of the nucleation barrier was observed, as would beexpected from classical heterogeneous nucleation theory. An interesting observa-tion was that the nucleation barrier was dominated by line tension. Xu et al. [79]also studied heterogeneous nucleation of hard spheres on patterned substrates(consisting of patterns of the same spheres in fixed positions). They noted thatthe time required for crystallization can be greatly reduced on a suitable substrateand the crystallizing phase can to a large extent be influenced by the substrate.Even if in some of the studies no explicit comparison with classical theory wasmade, or the results are mostly qualitative, there seemed to be at least qualita-tive agreement, and not surprisingly, a problem of CNT with correct interfaceenergies.

Sandomirski et al. [63] used hard and soft sphere models to study heteroge-neous crystallization on flat and curved interfaces.

4.2.2 Ising model

Detailed comparisons of classical nucleation theory with simulations of nucle-ation in the 2D and 3D lattice Ising model were made by Ryu and Cai [62, 61].

35

A particularly interesting aspect of this study was the independent testing of the“nucleation barrier” part and the “nucleation rate” part of CNT. The two parts infact rest on different sets of assumptions, and their validity is relatively indepen-dent. In the case of the 2D lattice Ising model, Ryu and Cai demonstrated a goodagreement of CNT with the simulation with no adjustable parameters. The CNTmodel in this case included two important improvements to classic theory. Theseare the Langer field theory correction [52] to nucleus energy, and the correctedtemperature dependent interfacial energy taking into account anisotropy of thesurface energy in the Ising model and changes in the shape of the equilibriumnucleus with temperature. The results of these studies establish the Ising modelas an extremely useful reference point for testing various fundamental improve-ments to nucleation theory, and also for testing changes and additions to CNTthat are necessary in different scenarios.

Brendel et al. [14] studied the nucleation times in the two-dimensional Isingmodel, using cluster energies and transition rates directly obtained from simula-tion. With the input of these parameters, the nucleation times predicted by CNTwere in reasonable agreement with the simulation.

Page and Sear [58] studied the influence of pores and surface patterning on theheterogeneous nucleation rate and energy barriers, finding a significant change inthe nucleation rate caused by the presence of the pores, and satisfactory agree-ment with CNT if different nucleation rates are assigned to nucleation in and outof pores. Building on this work, Hedges and Whitelam [36] asked how to patternthe surface in order to maximally speed up nucleation, and studied nucleationin the presence of pores with various dimensions. An interesting and potentiallypractically useful result is that the maximum nucleation rate is achieved if onedimension of the pore has a critical length.

Kuipers and Barkema [47] focused on memory effects (non-Markovian dy-namics) in the Ising model with local spin-exchange dynamics, which introducesdiffusion-like properties to the model. In such circumstances, events of particleattachment and detachment from the cluster are often strongly correlated, andthe moves in cluster size space are no longer Markovian. Accounting for this byintroducing new events, such as “particle leaving to infinity”, “particle leavingto return”, Kuipers and Barkema demonstrated an influence of memory effect ondynamics. Effectively, the outcome was increased fluctuations around the criticalsize, leading to a smaller time spent on the “energy plateau” of the nucleationbarrier, and hence an increased nucleation rate. An analytical description of thesituation remains an open topic.

Allen et al. [2] focused on another important scenario, ie. nucleation in thepresence of shear, in the 2D Ising system. They observed a peak in the nucleationrate in the intermediate nucleation rates, and suppression of nucleation in highshear rates. It seems to remain an open question whether concepts from CNT,especially cluster size as a reaction coordinate, are suitable simplification.

Recently, Schmitz et al. [65] carefully examined the definitions of the clustersused in most of the studies, showing that the most commonly used “geometri-cal” definition is unsuitable for defining clusters at higher temperatures. On theother hand, at low temperatures the cluster energy and the shape are grosslyanisotropical. A big part of the previous results need to be reconsidered in lightof the more physical definitions of clusters.

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4.3 Conclusions and prospects for future research

The results described above clearly show that agreement of CNT with simulationsin simple cases is a great starting point for understanding nucleation in morecomplex scenarios.

From a comparison with earlier studies of CNT and hard spheres, we canpropose several directions in which a comparison of the theory with numericalsimulations in the Ising model can be made, and the required additions to thetheory can be tested. One big relatively sparsely explored topic is the field ofnon-stationary systems and conditions. We can vary not only the external drivingforce, but also the temperature may be varied, as is often the case in experimentalscenarios. In the case of surface nucleation, the surface energy may also be non-stationary. Other interesting cases may be generated by lifting the condition ofspatial homogeneity. For example, we can introduce a temperature gradient, orwe can form a more complicated and more realistic surface, which may exhibitheterogeneous surface energy, roughness, or curvature.

Of these options, we selected the case of heterogeneous nucleation on an het-erogeneous surface as a topic of further study.

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5. Heterogenous nucleation onheterogenous surface

In the standard treatment of nucleation[41], the energy of the surface is takento be homogeneous. Often the surface energy is obtained from the macroscopicmeasurement, as determined e.g. by the contact angle of a liquid drop. Obviously,this is an idealization.

We ask the question what happens if we consider the surface energy to beheterogeneous. This is a straightforward generalization and may realistically beimportant in a number of scenarios, e.g. when the phase forming the surface isa binary alloy, solid solution, in presence of self-organized ordered patterns, ordue to intentional modification of the surface. It should be noted that it is adistinct case from more commonly studied problem of heterogeneous nucleationin presence of perturbations of the surface geometry, e.g. surface roughness,pores[58] or morphological instabilities. To answer this question we examine theeffects of surface heterogeneity in a few scenarios in a model system. While ourinitial motivation was based on an attempt to explain differences of nucleationrates of diamond on chemically identical substrate in one case in the form of ananofiber and in the other of planar layer, due to the complexity of such systemand signifficant problems with observing nanoscale nuclei in situ, we resorted totheoretical study and computer simulations of a much simpler system, specificallyIsing model in 3-dimensional cubic lattice.

Our answer for “wettable” surface is that the heterogeneity of surface energycan affect the height of the nucleation barrier and strong heterogeneity can sig-nificantly lower the barrier leading to much more rapid nucleation. (See Figs.and following sections)

We believe this can be useful both for preparation of functional surfaces andalso for understanding of some discrepancies between predicted and observednucleation rates in cases where surface energy homogeneity assumption cannotbe not satisfied.

The structure of the article is as follows: First, due to use of nucleation theoryin very diverse fields and resulting variance in terminology, we briefly mentionsome concepts from the classical nucleation theory. In further section we explainthe choice of the model used and describe applied model parameters. Finally, theobtained results are discussed and summarized.

5.1 Classical heterogeneous nucleation

Initially, the system is in some α-phase, which is metastable with regard to thestable β-phase. In order to change to the β-phase, first some small cluster ofβ-phase must be formed. Small clusters constantly appear because of sufficientlymassive thermal fluctuations, but too small clusters tend to dissolve, as the gainin free energy proportional to size of the cluster is more than compensated bythe interfacial energy of newly formed surface. Thus, there is an energy barrier,separating two attractors - the dissolution of the cluster at small size, and itsgrowth for large sizes. We will label the height of the barrier ∆Gc. Clusters at

38

the transition state on the top of the barrier are conventionally called “criticalclusters” and their size “critical size” nc.

During homogeneous nucleation, the process takes place in the whole α-phase.In a case of heterogeneous nucleation, the clusters are formed at heterogeneities.In this paper we consider the case of planar heterogeneous surface (labeled as γ),with β-phase - γ surface interfacial energy lower than on α-β interface, creatingfavorable conditions for nucleation.

5.2 Model

The model system which we use is a 3-dimensional rectangular lattice of spins.For each lattice site k, there is a variable σk taking values −1, 1. Spins in twoadjacent sites j, k interact with energy Jσjσj, where J is an interaction strengthenergy, same for all neighboring pairs. There is also an external field h inter-acting with each spin with energy hσj, and surface energy term slσl, for spinsneighboring surface site l. Consequently, the energy of the system is describedby the hamiltonian

H =∑j,k

Jσjσk +∑j

hσj +∑l

slσl (5.1)

when the first sum is over all pairs of neighboring spins, the second one is overall the spins, and the third term is summation over surface spins.

We can eliminate unnecessary parameters by using the coupling constant Jas a unit of energy. The temperature is measured in units of J/kB where kB isthe Boltzmann constant, and the strength of magnetic field h also in units of J .

We use rectangular simulation cell with surfaces on two opposing sites, andperiodic boundary conditions in the remaining two directions parallel to the sur-faces. (See fig.) All the non-surface spins have 6 nearest neighbors, the surfacespins have only 5 neighbors.

The Ising model exhibits the famous ferromagnetic phase transition[55]. In3D the critical temperature of the model is Tc = 4.51J/kB [59]. We explore thesystem at temperature T = 0.6Tc = 2.71J/kB which is well below the transitionto disordered phase.

The temperature is above the roughening temperature Tr = 0.57Tc of the spinphase interfaces[78] / if the system was bellow the roughening temperature, wewould expect cubical nuclei with flat walls (with some noise). Above the rough-ening temperature, expected nuclei are more spherical with irregular interfaces.

The applied magnetic field is h = 0.57J in direction opposite to the initialorientation of the spins.

5.2.1 Energy landscape, sampling technique and reactioncoordinates

The evolution of the system is simulated using Monte Carlo approach with thespin-flip dynamics (see, e.g. [12]).

In a straightforward simulation, the system would spend most of the timeclose to the two attractors - the initial state with most spins in α phase and

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the final state, with almost all spins in β phase, minus fluctuations. However,for understanding nucleation, these states close to the two attractors are quiteuninteresting, and contrarily, the transition states “on the top” of the nucleationbarrier are the most important. This leads to necessity to use some advanced sta-tistical sampling technique, which collects more information about the nucleationbarrier.

We use the umbrella sampling approach[73, 33]. The idea of the method isas follows: In Metropolis Monte Carlo, in every step of the simulation, flip ofone spin is attempted. If the new state is energetically favorable, it is accepted.Otherwise, a random number is compared to the Boltzmann probability of theflip, and if the random number is larger, the flip is still accepted, otherwise, thespin returns to previous state. In umbrella sampling, the standard Boltzmannprobability is replaced by a factor adding a bias potential V (σ) to the energy ofthe system. The potential is chosen to keep the system in a region of the energylandscape, which would normally be undersampled. A series of such umbrellasamplings may be used to explore the whole nucleation transition, and from theresults we can recover the original energy profile. For this estimate we use Benettacceptance ratio (BAR)[11] as implemented in the PyMBAR code[66, 67].

Sometimes misunderstood or neglected in nucleation studies is the importantrole of the reaction coordinate. In fact the classical theory approach basicallyis coarse-graining the system into one dimmensional Markov chain, where somemeasure of the largest cluster size is used as the coordinate. In capillarity ap-proximation, the number of cluster particles, its radius, surface, and surface en-ergy are usually taken to be tied by some simple relations (e.g. r(n) = a.n1/3,S(n) = b.n2/3 where r is radius, S surface, n the number of monomers formingthe cluster), any of such variables may be used as the reaction coordinate. How-ever, in realistic situations, and also in the simple Ising model, it is clearly notobvious that this mapping of a huge configuration space into one coordinate isenough. E.g. two nuclei consisting of the same number of monomers but withwidely different surface areas may be one more likely to dissolve, the other togrow. Fortunately, in the case of Ising model it was demonstrated the size of thecluster seems to be a good reaction coordinate[47].

An important pitfall in this projection is a correct counting of clusters: oftenused is geometrical cluster counting, in which case a spin is considered a memberof a cluster if any of its neighboring spins is also a member of the cluster. Thisis straightforward, fast to compute, and unfortunately inappropriate way how toproject the space - geometrical clusters are unphysical, and not reflecting correctlythe thermodynamics of the system[65]. We use adjusted cluster counting, in whichmembership of a spin in a physical cluster is determined by following procedure:we consider each bond between neighboring spins with the same orientation activeonly with probability

p(T ) = 1− exp (−2J/kBT ) (5.2)

where J is the interaction strength energy constant, and T is temperature. Then,when we discover the clusters by following bonds, we test if the probability p isgreater than a random number a ∈ (0, 1), and depending on the result extend thecluster only when the bond is ”active“. A geometrical cluster hence can containmore than one physical clusters, and an element of randomness is introduced intothe cluster counting procedure.

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Figure 5.1: Schematics of the simulation cell: the lower heterogeneous surfacehas favorable conditions for nucleation. The upper surface has surface energypreventing nucleation. The boundary conditions are periodic in remaining direc-tions.

Figure 5.2: Three examined cases of surface heterogeneity - shade of gray repre-sents the difference from a homogeneous surface, which would be uniform gray.From the left: 1. regular stripes 2. random pattern of species 3. correlatedrandom variable

5.2.2 Model of surface heterogeneity

.As mentioned above, we model the surface by including a surface term

∑l slσl

where energy of the bond between surface surface site l and the attached spinmay differ from site to site. The surface energy sl may be split into homogeneouspart s0 and the variable, heterogeneous part a.p(l), where a is an amplitude (or“strength”) of the heterogeneity, and p(n) is a pattern.

We examine three patterns of heterogeneity, and for each pattern we runa series of simulations where the amplitude of the heterogeneity is graduallyincreased. First we use regular stripes of sites with lower and higher surfaceenergy (p(n) ∈ {−1, 1}). Second random pattern of sites with lower and highersurface energy (p(n) ∈ {−1, 1}). and finally, we applied surface where its energyis a random variable with binomial distribution (p(n) ∈ (−1, 1)).

5.3 Results

The model was run in a cubic cell consisting of 243 spins. One boundary of thecell is generated by the surface with the above described heterogeneous energy.On the opposite side of the cell we created a “non-wetting” surface with fieldorientation reversed, strongly unfavorable for nucleation.

In all of three surface patterns, we observed the same pattern: clusters form byheterogeneous nucleation, and the nucleation barrier height depends on amplitude

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Figure 5.3: Decrease of nucleation energy barrier ∆G with increasing “contrast”of surface pattern in case of surface patterned with regular stripes. Heterogeneityamplitudes from the top: 0.2, 0.4, 0.8 in units of h. Size of the cluster N is simplythe number member spins.

of the heterogeneities and also on a characteristic length scale of the pattern.Heterogeneity of the surface can cause marked decrease in the nucleation barrier.

5.3.1 Regular stripes

In this case the inhomogeneity consists of regular stripes 2 lattice constant wide.When the inhomogeneity is introduced, first we observe small increase of nucle-ation barrier, but with increasing amplitude the trend is soon reversed and thenucleation barrier decreases. See Fig 5.3. In the presence of strong inhomogene-ity, the barrier is almost halved. Due to the exponential dependence of nucleationrates on barrier height, this means nucleation rate can be increased by severalorders of magnitude.

5.3.2 Random pattern

This heterogeneity is a random pattern of sites with two different energies. (SeeFig 5.4). The change of the nucleation barrier is observeable, but less pronouncedthan in previous case. Again with increasing inhomogeneity the nucleation barrieris lowered.

5.3.3 Random surface

Here the introduced inhomogeneity is a pattern generated by addition of noiseat length scales of 1,2,4 lattice units, normalized so the mean value of the noiseacross the surface is zero. See Fig 5.2. Again, with increasing amplitude of theinhomogeneity, we observe decrease in the nucleation barrier. See Fig 5.5.

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Figure 5.4: Decrease of nucleation energy barrier ∆G with increasing “contrast”of surface pattern in case of random surface. Heterogeneity amplitudes from thetop: 0.2, 0.4, 0.8 in units of h. Size of the cluster N is simply the number memberspins.

Figure 5.5: Decrease of nucleation energy barrier ∆G with increasing “contrast”of surface pattern in case of random surface. Heterogeneity amplitudes from thetop: 0.2, 0.4, 0.8 in units of h. Size of the cluster N is simply the number memberspins.

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5.4 Discussion

In this chapter we have studied heterogeneous nucleation in Ising model on simplecubic lattice, with a planar wall. While usually such walls are considered likehomogeneous, we examined several cases of surface with heterogeneous energy.

The simulation results indicate the nucleation barrier can be substantiallyreduced by the inhomogeneities of the surface energy on the surface where thenucleation takes place. While our model is realively simple, we expect this conclu-sion is true also for more realistic nucleation scenarios. It shows that on surfaceswith nanoscale heterogeneities it is insufficient to use the average surface energy,obtainable from macroscopic measurement.

Intuitively, this can be understood as an ability of the nucleation process totake advantage of sites with lower surface energy, even when the average surfaceenergy remains stable.

From a more abstract viewpoint we can ask how a distribution of surfaceenergies on a heterogeneous surface will influence nucleation, and if the classicalmodel which uses average of the energy can be improved by addition of somesimple term describing heterogeneity.

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6. Nucleation on complexnetworks

While previous chapters dealt with physical systems on lattices, here we switchto systems with different underlying topology, networks (or graphs) .

We will start with a brief definition of terms, then describe some motivationswhy networks are an interesting model of many real systems, review some of theexisting results on phase transitions on various kinds of networks, and describethe problem of nucleation on networks.

The questions motivating our research are similar to the case of heterogeneoussurfaces - we are interested in how local changes (in this case the graph structure)can make phase transition easier of harder. One of the more interesting questionsin this domain are scaling properties of the critical cluster size and energy. Or ina terminology more commonly used in the field - we are interested in the influenceof local structural properties on hysteresis behaviour.

On one side, in two and three dimensional lattices, the critical cluster size is afunction of intensive parameters, and not of the size of the system (in the limit oflarge systems). On the other side, asymptotic behaviour of nucleation in randomgraphs can be solved by mean field theory, and the critical cluster sizes scaleslinearly with the size of the graph. As we will see, neither of these models hasthe same statistical properties as many real networks of practical interest, andexisting research shows that the models in some way interpolating between theseclasses exhibit more complex behaviour[] Also the systems with more realisticproperties were so far much less studied, perhaps because they are not solvableby elegant analytic methods.

When investigating such, we move from systems which are well described bymean field approximation to cases which are less obvious - finite systems andnetworks with clusters. As in the case of the previous complex settings, wherean analytical model is not solved, we will resort to MC numerical simulations.

The main result of our investigation is demonstration of a strong influence ofclustering on the phase transition, and observed critical cluster scaling which isdifferent from the mean-field case.

While we limit ourselves to studying Ising model dynamic, we expect quali-tatively similar effect would occur for many other models of dynamics.

6.1 Graphs and networks

Mathematical abstraction of the network is a graph. Graph is a structure con-sisting of a set of vertices V , and edges E. Edges are 2-element subsets of a setof vertices (the edges connect vertices that have a mutual relationship). Sucha type of graph would be more precisely labeled as non-oriented (the edges arenot directional) and simple (there is at most one edge between the two vertices).Network science often uses different terminology - instead of (graph, vertex, edge)the triplet (network, node, link). We will use the network science convention.

If there is an edge between two nodes, we say they are adjacent. If there is apath of adjacent nodes between two nodes, we say these nodes are connected. A

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Figure 6.1: Degree distributions for four real world networks, from [9]: internetconnectivity at the router level, protein-protein interaction network, email net-work and a citation network. The green lines shows Poisson distribution withthe same average degree as in the data, illustrating the fact that many real-worldnetworks have a degree distribution which can not be accounted for by randomgraph mode.

component is part of the network in which any two nodes are connected to eachother by one or more paths, and which is connected to no additional nodes in thenetwork.

Another concept we need to introduce is the node degree (vertex degree). Itis the number of links from the node. We will use the ki for the number of linksfrom i-th peak. Trivially, in a non-oriented simple graph, the total number ofedges

L =1

2

∑i

ki

An important feature of the network is the average node degree < K >=1N

∑i ki where N is the number of vertices.

6.1.1 Network statistics

Degree distribution

If we select a random network node, pk is probability that its level is k. Thedistribution of the degrees is one of the basic characteristics of the network - the

46

v v v

Cv = 46 = 2

3 Cv = 06 = 0 Cv = 6

6 = 1

0 0

0

1

13

16

23

hCi = 1n

Pv∈V Cv = 13

42

Figure 6.2: Local clustering coefficient. Values of Ci are indicated next to nodes.

importance of this characteristic for real networks was discovered by Barabasi andAlbert [8], showing that many real-world networks have approximately power-lawdistribution of degrees - see Fig 6.1 As we will see later, degree distribution isalso important for phase transitions.

Clustering coefficient

Another important graph statistics, is the clustering coefficient[27]. Clusteringcoefficient is a measure of “ how densely the nodes are connected ”. The com-monly used formal definition is that the local cluster coefficient Ci = 2Li

ki(ki−1)

where Li is the number of links between nodes adjacent to node i, and ki is thenumber of these nodes. The average clustering coefficient for the whole networkis given by C = 1

N

∑iCi. See Fig 6.2 for illustration of the calculation.

The common 3D cubic grid, which we studied in the chapter ... has the samedegree of all nodes. Also, all peaks have the same cluster coefficient, Ci = 0.

6.1.2 Network models

Simple models

Grids

We can view regular d-dimensional grids as a special kind of network, and list it’smain properties when condsidered as a network.

• All vertices have the same degree.

• Clustering coeficint depends on the exact grid format (square, triangular,...)

Random networks

Random graphs,that is, graphs where the nodes are linked ranomly, are someof the most studied subjects in discrete mathematics since Erdos and Renyi[30].Their classical definition starts with a set of nodes N and L randomly placed links

47

among them. An alternative older definition, which is often more useful for realproblems, is by Solomonoff and Rapoport[68]: in the so called Gilbert model, wedo not to fix the number of links, but rather a probability p that two randomlychosen nodes are connected. While in this model the total number of links isno longer fixed, it is easier to calculate many of the graph’s properties. Whatis usually studied are not individual instances of such net, but rather statisticalensembles of all graphs on N vertices with given properties.

The two options (Erdos-Renyi and Gilbert) are essentially a canonical andgrandcanonical ensemble where the number of links plays the role of the numberof particles.

The average degree of a node is < k >= p(N − 1) (where N is number ofnodes of the network). It easily follows the expected total number of links in the

network is < L >= pN(N−1)2

. The number of links follows a binomic distribution.In the limit where the number of nodes gets to infinity the distribution can beapproximated by Poisson one, and the models are equivalent.

We will consider other classes of graphs defined in a similar way - by speci-fying some constrains or or properties of the graph, and considering a statisticalensemble were all the members satisfying the restrictions are realized with thesame probability.

Several important properties of random graphs can be calculated using mean-field approximation.

Topological phase transition - percolation on random graphs

A famous result in from Erdos and Reny describes percolation behaviour on thesegraphs. Let’s increase the probability p of connection from 0 to 1. Close to 0,there are few links, and the graph is essentially disconnected. Close to 1, thegraph is connected - there is a path between almost any two nodes. We will callparts of the graph which are connected - that is, there is a path between anytwo nodes of the part, components. Then, we may ask what is the behaviour ofthe size of the largest component in dependence on p, in the large graph limit.Famously, a percolation transition is at p = 1/N where for p < 1/N the size ofthe largest connected component is O(logN). Above the threshold, so calledgiant component appears.

Local structure of random graphs

[ todo ]

6.1.3 Generative and descriptive models

Moving beyond the simplest models, a sizeable field of network models canbe broadly divided into two imprecise categories: generative and descriptivemodels[80]. Generative models usually describe some process by which the re-sulting network structure can emerge, often using some rules by which links inthe network are formed when a new node is attached. This way of modellingcan help understand how statistical properties of networks emerge, but are notparticularly suitable for fitting for data. Examples which belong to this cate-gory are Barabasi-Albert preferential attachment model, models depending on

48

Figure 6.3: Cayley tree.

link redirection, Wattz-Strogatz small worlds, and various models based on nodeembedding in latent space.

The other way of modeling is more rooted in descriptive statistics, and usuallyproceeds this way: First, infer some statistics about the real world network, forexample, degree distribution. Second, study a statistical ensemble of graphsmaximizing entropy with the observed characteristic as a constrain. This way ofmodeling does not require additional assumptions about the system, and oftenit is possible to calculate some parameters on the whole ensembles. Examplesfrom this category would include scale-free networks, stochastic block models,clustered networks.

It is often very fruitful to connect these two ways of description - for exam-ple, part of the success of the influential Barabasi-Albert studies on scale-freenetworks can be attributed to linking a property observed in some real worldnetworks, namely scale-free degree distribution, with a generative model (prefer-ential attachment), which may explain how can such degree distribution emerge.unfortunately, the relation of such models is sometimes misunderstood - in factthere are other processes which may generate power-law degree distributions, anymany more processes which lead to degree distributions which are not actuallypower law, but can be approximated as such.

Wattz-Strogatz networks

Wattz-Strogatz model [77] is one of the earliest network models trying to replicatea small-world property of many real world networks - that is, average shortest pathlength between nodes scaling slower than log(N), and high clustering coefficient.Also, the model can be understood as interpolating between lattices and randomgraphs. Wattz-Strogatz network with N nodes, mean degree K and rewiringparameter β is constructed by this procedure:

1. Construct a regular ring lattice with N nodes, each connected to K neigh-bors.

2. For every node i = 1 . . . N , take every edge connecting i to its rightmostneighbors, that is every edge (i, jmodN) with i < j ≤ i + K/2 , and

49

Regular graph Small world graph Random

p = 0 p = 1

Randomness (p)

Figure 6.4: Wattz-Strogatz model.

rewire it with probability β, by replacing it with (i, k) where k is chosenuniformly at random from all possible nodes while avoiding self-loops andlink duplication [ref wiki]

In the β = 0 limit, the average path lenght scales linearly with the systemsize - l = N/2K. In the β → 1 limit, average path l ∼ logN

logK. In intermediate

values, the path lenghts sharply decreases even at small values of β - which is alsoone of the insights historically learned from the model relatively small amount ofrandom long-distance links is enough to significantly decrease average distancesin the network.

The degree distribution for ring lattice is just a delta function centered onK, and with increasing β broadens, but still has pronounced peak at k = K anddecays exponentially P (k) ∼ e−|K−k|.

Clustering coefficient for the ring lattice is

C =3(K − 2)

4(K − 1)(6.1)

and for β → 1 approaches the clustering coefficient of a random graph, whilein intermediate values of β staying relatively large. See Fig 6.5

All these parameters lead to a model which, at intermediate values of β,has small-world properties, has large clustering coefficient and relatively narrowdegree distribution. Compared to many observed systems, the disadvantage ofthe Wattz-Strogatz model is the degree distribution is very different from mostsystems.

Barabasi-Albert model

Barabasi-Albert model is a generative model of a growing network, describingprobability of link creation when a new node is attached [9]. The network startswith m0 connected nodes. Nodes are then added one by one. When new node isattached, new m < N links are formed with existing nodes with probabilities

pi =Ki∑j Kj

(6.2)

This leads to a pattern where nodes with higher degree get more new con-nection - that is, nodes prefer to connect to highly connected nodes - preferential

50

Figure 6.5: Evolution of a Barabasi-Albert network. In each step, the newlyadded node is indicated by a white circle, the old nodes by grey circles

attachment. This pattern leads to emergence of highly connected hubs, whichare commonly observed feature in many real world networks. See Fig 6.5.

After t steps, the model produces network with N = m0+t nodes and m0+mtlinks. We can analyze the dynamics of node degrees - the rate at which numberof degree ki node i grows is, switching to continuous approximation,

dkidt

= mpih = mki∑j Kj

(6.3)

As the sum is over all nodes except the newly added node, and the number oflinks is twice the sum of node degrees, we can arrive at

dkidt

= mki

2mt−m+m0

(6.4)

For large t this leads to

dkiki

=dt

2t(6.5)

By integrating and using the fact that newly added node had m links in step ti,we have the expected degree of node i at time t

ki = (t

ti)

12

(6.6)

This means all nodes degrees grow in the same way. We can use this to deriveasymptotic degree distribution. Let i(k, t) be the node having degree k at timet. We can express i as

i(k, t) = m2k−2t (6.7)

Given that we introduce one node per time step, the fraction of nodes of degreek at time t is the difference between i(k)− i(k + 1) divided by the total numberof nodes, which grows as t - substituting, we get a power-law degree distribution

p(k) ≈ m2k−3 (6.8)

Same result could be obtained by master equation approach.

51

General preferential attachment

The attachment pattern in Barabasi-Albert model be considered a special caseof attachment process where the probability

pi =Ki∑j Kj

+ a (6.9)

- so the attachment probability is not proportional, but a linear function ofthe node degree. In the limit case a → ∞ the result is a random graph. Fora > −1 the results are scale-free graphs with degree distribution

p(k) ≈ k−3−a (6.10)

Clustering coefficient approaches 0 in the N →∞ limit.As we see, the properties of B-A and similar model are somewhat comple-

mentary to Wattz-Strogatz networks - both models have small-world property,but while Wattz-Strogatz networks have narrow degree distribution and non-vanishing clustering, Barabasi-Albert model leads to power-law degree distribu-tion, and low clustering.

As many practically important networks have small world properties, widedegree distributions often resembling power law, and high clustering coefficient,these models still leave a lot of room for developement of other models whichwould have all these properties at once.

Clustered networks

A simple generalization of the B-A model allowing to generate clustered scale-free networks is the Holme-Kim model [37]. The model additionally depends ona parameter 0 ≤ p△ ≤ 1, a probability that, after adding an edge from newnode v to existing u, the next edge is added from v to a (prefferentially selected)neighbor of u. p△ exactly corresponds to the Barabasi-Albert model.

Stochastic block models

Another feature often observed in real-world networks is community structure- where by communities we mean parts of the network which are more closelyconnected. Networks with this kinds of global structure are called modular, andare often encountered in networks created for some functional purpose, both man-made such as transport network, and evolved, such as protein interaction network,and also in all kinds of social systems. One of the most popular models of thiskind is so-called Stochastic Block Model (SMB) [39]. We divide the nodes into Mclasses, denoted g1 . . . gM , where each node is a member of exactly one class. Themodel is defined by M×M matrix C where for two nodes which belong to groupsgi, gj the probability of a link between them is Cgigj . In other words, in SMB,we have “communities” M which are usually more tightly linked internally (Cii

elements are higher), and there is a smaller probability of links between them.In the adjacency matrix representation of the network in suitable ordering, thecommunities form blocks - see Fig. In a classical SBM, linking withing each blockis random, and the model has some of the same properties as random network.

52

6.2 Dynamics on networks

Often we are interested not in the dynamic of the network itself, but some dy-namical model ”living” on the network. Some of the most studied models onnetworks are inspired by

• disease spreading: SIS, SIR and similar models

• opinion formation: voter models

• synchronization: Kuramoto oscillators

Usually, the model consists of some state variable attached to each node (e.g.two or three state variable in SIS and SIR model, a phase of the Kuramotooscillator) and a dynamic describing dynamics of the variables, where usually theevolution of state si of a node i depends just on states sj of it’s neighbours.

The range of models provides a plethora of possibilities of dynamics, and alsoa range of methods used to study them.

In this work, we will focus just on Ising model - some of it’s advantages are wecan often understand it’s behaviour at least qualitatively using approximationsnear equilibrium, we can use the large body of knowledge about the model builtin statistical physics, and the behaviour of the model is easier to interpret interms of global state variables such as energy. This provides us a stronger toolkitthan models which are locally strongly non-equilibrium and have to be treatede.g. as a Markov process.

Also, Ising model has the virtue of being one of the simplest examples of class,with few parameters.

6.3 Ising model on networks

The Hamiltonian of Ising model on a network is essentially the same as for lattice- for each node i there is spin variable σi taking values −1, 1. Spins in two linkednodes i, j interact with energy Jσiσj, where Ji,j is an adjacency matrix multipliesby interaction strength energy J . There is also an external field h interactingwith each spin with energy hσi. The energy of the system is described by theHamiltonian

H =∑i,j

Jσiσj +∑j

hσj (6.11)

where Ji,j represents the couplings. If the coupling terms are positive for alli, j, the model is ferromagnetic and the ground state is either all spins having value−1 or 1. If all Ji,j < 0 the model is antiferromagnetic. From the more general classof models where the terms are mixed, “spin glasses” are models where couplingparameters are random variables drawn from a distribution symmetric aroundzero.

Boltzmann distribution of the spin configurations is

p(σ) =1

Ze−β

∑i,j Jσiσj−β

∑j hσj (6.12)

53

where β > 0 is the inverse temperature and Z partition function and σ = σ1...N

is the vector of sigmas. We will denote < · > an expected value of function f(σ)with respec to the Boltzmann measure

< f(σ) >=∑

σl∈{−1,1}Nf(σ)p(σ) (6.13)

In thi s notation, the average magnetization per spin is

MN(β, h) =1

N

∑i∈[1,N ]

< σi > (6.14)

and other thermodynamic quantities similarly). In the thermodynamic limit wewill have M(β, h) = limN→∞MN .

6.3.1 Mean-field description

A simple network on which we can solve the Ising model is the complete graph.In mean field approximation, we will express an individual spin as it’s mean valueand a fluctuation terms

si =< si > +δsi (6.15)

The interaction term of the Hamiltonian than becomes

sisj = (< si > +δsi)(< sj > +δsj)

=< si >< sj > +δsi < sj > +δsj < si > +δsiδsj(6.16)

In the mean field approximation, we will assume the fluctuation term δsiδsjis small, and we can decompose the interaction term to interactions just with themean field.

sisj ≈< si >< sj > +δsi < sj > +δsj < si >

=< si >< sj > +(si− < si >) < sj > +(sj− < si >) < si >

= si < sj > +sj < si > − < si >< sj >

(6.17)

Because of symmetry on the complete graph, all mean values si are the same,we have

< si >= m

and the interactions can be replaced by an interaction with the mean field

sisj = sim+ sjm−m2 = m[si + sj −m] (6.18)

and the Hamiltonian H simplifies to

HMF = m∑j

J(2σj −m) + h∑j

σj

=JN(N − 1)m2

2+ (h−mJ(N − 1))

∑j

σj

(6.19)

54

so the spins interact only with the effective field h = h−mJ(N −1), and the sys-tem factorizes to non-interacting spins. Partition function is simply the productof single-spin interactions

ZMF = e−βJN(N−1)m2

(2 cosh β[h− 2mJ(N − 1)]N (6.20)

We know that for single spins < σi = tanh βh. At the same time < σi >= mand we get the self-consistency condition for m:

m = tanh βh+ β[2mJ(N − 1)] (6.21)

This can be solved for h = 0 and we get the usual analysis of solutions: abovesome TC = 2J(N − 1) there is only a trivial solution m = 0. Bellow Tc, thereare two possible solutions. TC is the critical temperature. Often the interactionconstant is re-scaled J = J

N−1so as to get finite critical temperature in the

N →∞ limit, TC = 2J .Free energy F of the system is

F = −T lnZ

= JN(N − 1)m2 −NT ln 2−NT ln (coshβ[h− 2mJ(N − 1)])

=1

2NTCm

2 −NT ln 2−NT ln (coshβ[h−mTC ])

(6.22)

In re-scaled variable f = FN

we get free energy per spin

f =1

2TCm

2 − T ln 2− T ln (coshh−mTC

T) (6.23)

6.3.2 Tree-like approximation

A key idea to analyze the Ising model on random graphs is to use the fact thatexpectations of local quantities coincide with the corresponding values for theIsing model on suitable random trees [ref]

[ expand ]

6.3.3 Nucleation on random graphs and scale-free net-works

Behaviour of the 1st order phase transition in Ising spins on a random graph,switching from one orientation to the other in presence of external field h is de-scribed by the free energy profile 6.23. Making the assumptions of magnetizationm being small and external field h being small and the system being close , wecan take the lower terms of the Taylor expansion of f around 0, and get theLandau free energy profile

f ≈ mh+T − TC

2m2 +

TC

12m4 (6.24)

In the ferromagnetic phase with zero field, this has two symmetrical stablesolutions, corresponding to spins being mostly oriented as +1 or −1, separatedby an energy barrier which scales linearly with N .

55

Similar argument can be made also for scale-free networks. For a rigorousderivation, see [28]. Less formally, we can make following approximation: let’sreplace the original graph with a fully connected weighted network, with networkweights

kikj< k > N

The sum of couplings for each node is the same as in the original graph and theapproximation could be understood as averaging over all realizations of networkwith a given degree sequence ki. Another interpretation is here we study so called”annealed network” - a network where link connectivity is rapidly changing (fasterthan than the spins), so each spin feels an effective mean field of all other nodes,weighted by their degrees. We have a system with Hamiltonian

HAN = J∑i,j

kikjσj

< k > N+ hh

∑j

σj (6.25)

When we use effective magnetization

m =∑j

kjσj

< k > N

we proceed in the same way is in Equation 6.19. What is notable in the weightingfactor is the strong influence of the highly connected nodes - ’hubs’.

Critical temperature in this model is

TC =2J

ln [< k2 > /(< k2 > −2 < k >)]≈ J

< k2 >

< k >(6.26)

and the behaviour depends on the the exponent γ of the degree distribution.In particular if γ ≤ 3 thermal fluctuations can not destroy the ferromagneticorder for any finite temperature, in the thermodynamic limit??

6.4 Simulation study of nucleation in Ising model

on a Barabasi-Albert network

We can use this analytic solution from the previous section as a test case for theother approach we are using, simulations using forward flux sampling. In thenumerical experiments, we first construct an ensemble of graphs with the givenproperties (usually we use use n = 100). Then we initially set the spins to −1,and set an external field h in the orientation which makes this branch of thesolution only a local minimum (meta-stable). We evolve the system with theGlauber dynamic, using the standard Metropolis-Hastings Monte Carlo updaterule, e.g. in one step we select a spin randomly, compute ∆E as the energy changeon flipping the spin and then flipping it with probability

min{e−∆E/kBT , 1}

. Fore more details about the algorithmic implementation details is in Ap-pendix A.

56

20 40 60 800

0.2

0.4

0.6

0.8

1

−12

−11

−10

−9

−8

−7

−6

Probability of reaching next order parameterTransition rate [1/MCSS]

P(up

)

Tran

sitio

n ra

te [l

og_1

0]

Figure 6.6: Graph of the nucleation progress (top, red line indicates criticalorder parameter) and sample of a critical core (bottom, red nodes in the core)of Barabasi-Albert network with N = 100 nodes, average degree 4, T = 1.0,h = 0.5.

57

As the transition probability of overcoming the energy barrier can be ex-tremely small, we use Forward Flux Sampling as a technique to sample rareevents. (See Appendix A for more details.) Using FFS, we are able to measureseveral critical parameters: transition rates, size of the critical cluster, and energyof the critical cluster.

For illustration of the nucleation process, at Figure 6.6 we present the progressand critical core for small network with N = 100 nodes and average degree 4.The order parameter is the number of up spins.

We replicated the nucleation experiment of Chen et al. [19], i.e. computingtransition rates and critical nuclei sizes of Barabasi-Albert networks with N =1000 nodes and average degree 6. The order parameter is the number of up spins,first interface at λ0 = 130, last interface at λm = 880 with m = 200 interfacesevery 3 spins, at every interface they perform 1000 experiments (i.e. collecting upto 1000 samples per interface). Temperature is set to T = 2.59 with field h = 0.7.

They report obtaining ΦA,0 = 1.24× 10−4MCSS−1 and P (λm|λ0) = 4.48× 10−46,resulting in RHom = 5.55× 10−50MCSS−1 = 10−49.26MCSS−1, and critical nucleussize λFFS

c = 474.We ran the experiments in the same settings, only setting λm = 7301. At every

interface, we collect at least 1000 samples. We express the rate multiplicativeerror computed from the sample as the standard deviation in the exponent. Werun n = 100 independent trials at every parameter setting with independentlysampled networks.

We obtained ΦA,0 = 1.87× 10−3 ± 2.75× 10−4MCSS−1 and P (λm|λ0) =10−44.34±1.79 = 4.57× 10−45, resulting inRHom = 10−47.07±1.76MCSS−1 = 8.48× 10−48MCSS−1,and critical nucleus size λFFS

c = 471.05±5.39. See Figures 6.10 and 6.12 for visualcomparison within a more general parameter setting.

6.4.1 Core degree distribution throughout a transition

To examine the order at which different vertices are added to the core over thetransition path, we measure the degree distributions of the core at various orderparameters for a sample of n = 100 Barabasi-Albert networks of N = 1000nodes and average degree 4. For every network, 6 independent FFS are run withT = 1.0, h = 0.5.

The progression is illustrated in Figure 6.7. Until a critical order parameter isreached, the core consists of moslty small degree vertices. Above this threshold,higner-degree vertices are added more rapidly while low-degree vertices are addedat a more stable rate.

1While Chen et al. used λm = 880, the rate change from interface at 600 spins until 900spins is under 0.1%.

58

1 2 5 10 2 5 1000

0.2

0.4

0.6

0.8

1

Order 50 Order 100 Order 151 Order 201 Order 252 Order 302 Order 353 Order 403 Order 418 (crit. at 413.4)Order 454 Order 504 Order 555 Order 605 Order 656 Order 707 Order 757 Order 808 Order 858 Order 909 Order 959 Order 980

Node degree bin

Frac

tion

of n

odes

up

0.0

503.

6

195.

815

6.6

59.2

28.8

22.0

14.0

8.0

4.2

3.4

1.4

1.5

1.1

0.5 0.0

Figure 6.7: The histograms of the degrees of up-spin nodes at various order pa-rameters during a nucleation transition. The degrees are binned in exponentiallysized buckets, the numbers above the graph indicate the number of nodes in adegree bucket. The recorded odred closest to the critical order is highlighted.

59

6.5 Ising model on clustered scale-free networks

Having verified the agreement of our simulation both with theory and previousnumerical experiment, we move to the study of clustered networks.

To study the influence of clustering coefficient, we studied a set of scale-freenetworks where we kept all other parameters the same and varied the clusteringcoefficient. We are using the Holme-Kim model [37] which is a generalization ofthe Barabasi-Albert model, additionally parameterized by 0 ≤ p△ ≤ 1. Holme-Kim model with p△ = 0 reduces to the Barabasi-Albert model. While clusteringcoefficient can’t be prescribed directly in this model, higher p△ gives rise to higherclustering coefficent. In the graphs used in the following experiments (namely forHolme-Kim graphs with average degree 6), the empiric relation of p△ and theclustering coefficient is the following:

p△ clustering coefficient0.0 0.031± 0.0870.1 0.080± 0.130.3 0.18± 0.190.5 0.29± 0.220.7 0.40± 0.240.9 0.53± 0.240.98 0.58± 0.23

Again, we use forward flux sampling to allow us study of very rare events– transition rates out of the metastable state were as low as 10−140MCSS−1.In all cases first construct an ensemble of graphs with the given properties.For generating networks, we use the Holme-Kim model average degree 6 andp△ ∈ {0 . . . 0.99}. Then we study the transition from a meta-stable state wherealmost all spins have value −1 while the energetically favourable orientation inthe external field h would be +1. For practical reasons, we used number of spinsup as the order parameter. For details of the simulation implementation seeAppendix A, for details about the software implementation and tools used seeAppendix B.

As an illustrative example, we can look at a small network, N = 100 nodes,average degree 4 and p△ ∈ {0.5, 0.9}. See Figures 6.8 and 6.9 for results – wealso show one sampled critical cluster. Two main differences from the sparsecase in extending Figure 6.6 are already apparent – transition rate increases withgrowing clustering coefficient, and critical cluster size decreases.

60

20 40 60 800

0.2

0.4

0.6

0.8

1

−9

−8

−7

−6

−5

Probability of reaching next order parameterTransition rate [1/MCSS]

P(up

)

Tran

sitio

n ra

te [l

og_1

0]

30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

−7.5

−7

−6.5

−6

−5.5

−5

Probability of reaching next order parameterTransition rate [1/MCSS]

P(up

)

Tran

sitio

n ra

te [l

og_1

0]

Figure 6.8: Graph of the nucleation progress of a Holme-Kim network with N =100 nodes, average degree 4,p△ = 0.5 (top) and p△ = 0.9 (bottom), T = 1.0,h = 0.5. Compare to Figure 6.6.

61

Figure 6.9: A sample of graphs and critical cores for p△ = 0.5 (left) and p△ = 0.9(right) in the setting of Figure 6.8. Compare to Figure 6.6.

Transition rate

The first quantity we compute is the transition rate for Holme-Kim networksaverage degree 6. First, we consider the dependency on p△ ∈ {0 . . . 0.99} withN = 1000 nodes and various values of T and h. See Figure 6.10 for the results.As an ablation test, we perform one line of experiments with only 100 samples perinterface. Then we consider the dependency on N ∈ {50 . . . 10000} with T = 1.0,h = 0.5 and various values of p△. See Figure 6.11 for the results and Figure 6.14for a rescaled plot with p△ = 0.9. The error bars include both model parametervariance (i.e. differences in sampled network) and FFS rate computation error.We ran 10 ≤ n ≤ 100 independent trials at every parameter setting.

We can observe a dramatic increase in the transition rate with increasingclustering coefficient.

Critical cluster size

The next property we studied was the critical cluster size. The simulation wasdone with the same conditions as for the transition rate of the previous section.The dependence on p△ are in Figure 6.10, the dependence on N are in Figure 6.13,with a rescaled plot for p△ = 0.9 in Figure 6.14.

Here we can observe two our main results. First, increased clustering dramati-cally deceases the critical cluster size. Second, networks with high local clusteringcoefficient seem to exhibit different scaling properties. While for the tree-like net-works the mean field prediction holds and the critical size is linear function ofthe network size, the scaling we observe in dense networks is sub-linear.

62

0 0.2 0.4 0.6 0.8 1

−140

−120

−100

−80

−60

−40

−20

0

T=2.59, h=0.7 (100 samples/interface)T=2.59, h=0.7 (1000 samples/interface)T=2.59, h=0.5 (1000 samples/interface)T=2.59, h=0.3 (1000 samples/interface)T=2.59, h=0.1 (1000 samples/interface)

p_triangle

Tran

sitio

n ra

te [l

og10

, 1/M

CSS]

0 0.2 0.4 0.6 0.8 1

−140

−120

−100

−80

−60

−40

−20

0

T=2.59, F=0.7 (1000 samples/interface)T=2.3, h=0.7 (1000 samples/interface)T=2.0, h=0.7 (1000 samples/interface)T=1.7, h=0.7 (1000 samples/interface)

p_triangle

Tran

sitio

n ra

te [l

og10

, 1/M

CSS]

Figure 6.10: Transition rate dependence on p△ and h (top), respectively p△ andT (bottom). Holme-Kim model with N = 1000 nodes and average degree 6,temperature and external field T = 1.0, h = 0.5. The red cross indicates the ratereported by Chen et al.

63

0 200 400 600 800 1000

−120

−100

−80

−60

−40

−20

0

p=0.0p=0.5p=0.9

N

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og10

, 1/M

CSS]

0 2k 4k 6k 8k 10k

−1800

−1600

−1400

−1200

−1000

−800

−600

−400

−200

0

p=0.0p=0.5p=0.9

N

Tran

sitio

n ra

te [l

og10

, 1/M

CSS]

Figure 6.11: Transition rate dependence on the network size N and p△ forN = 50 . . . 1000 (top) and N = 200 . . . 10000 (bottom). Holme-Kim model withaverage degree 6, temperature and external field T = 1.0, h = 0.5.

64

0 0.2 0.4 0.6 0.8 1

250

300

350

400

450

500

550

T=2.59, h=0.7 (100 samples/interface)T=2.59, h=0.7 (1000 samples/interface)T=2.59, h=0.5 (1000 samples/interface)T=2.59, h=0.3 (1000 samples/interface)T=2.59, h=0.1 (1000 samples/interface)

p_triangle

Criti

cal c

lust

er s

ize

0 0.2 0.4 0.6 0.8 1

150

200

250

300

350

400

450

500

T=2.59, F=0.7 (1000 samples/interface)T=2.3, h=0.7 (1000 samples/interface)T=2.0, h=0.7 (1000 samples/interface)T=1.7, h=0.7 (1000 samples/interface)

p_triangle

Criti

cal c

lust

er s

ize

Figure 6.12: Critical cluster size dependence on p△ and h (top), respectively p△and T (bottom). Holme-Kim model with N = 1000 nodes and average degree6, temperature and external field T = 1.0, h = 0.5. The red cross indicates thecritical size reported by Chen et al.

65

0 200 400 600 800 10000

50

100

150

200

250

300

350

400

450 p=0.0p=0.5p=0.9

N

Criti

cal s

ize

0 2k 4k 6k 8k 10k0

1000

2000

3000

4000

5000p=0.0p=0.5p=0.9

N

Criti

cal s

ize

Figure 6.13: Critical cluster size dependence on the network size N and p△ forN = 50 . . . 1000 (top) and N = 200 . . . 10000 (bottom). Holme-Kim model withaverage degree 6, temperature and external field T = 1.0, h = 0.5.

66

0 2k 4k 6k 8k 10k

−12

−10

−8

−6

−4

−2

0

N

Tran

sitio

n ra

te [l

og10

, 1/M

CSS]

0 2k 4k 6k 8k 10k0

200

400

600

800

1000

N

Criti

cal s

ize

Figure 6.14: Transition rate (top) and critical cluster size (bottom) dependenceon the network size N for p△ = 0.9. Rescaled from Figures 6.13 and 6.11.

67

Conclusion

In previous chapters we studied nucleation in progressively more abstract contextsand system, starting from classical nucleation theory and ending with hysteresisin complex networks. We have illustrated the nucleation path going via moreenergetically favourable nanophase in alkali halide crystal, influence of surfaceenergy heterogeneity on heterogeneous nucleation, and influence of clustering onnucleation in complex network. All of these directions can be extended - in caseof complex networks, it would be intersting to consider more different models,and also develop a better theoretical understanding.

One claim from the Introduction remaining unaddressed is: the broad theorycan be applied also to social systems.

We can partially support the claim by making a formal analogy betweendecision making of bounded rational agents and spins in Ising model on a net-work. When choosing between two options, let’s call them A and B, with utilitiesU(A), U(B) an information-theoretic bounded agent i will choose A with a prob-ability

pi(A) =eβUi(A)∑i=A,B eβUi

(6.27)

where inverse temperature β can be interpreted as a parameter controllingrationality of the agent, increasing temperature meaning more randomness [56].In machine learning, the function transforming utilities to probabilities is knownas softmax(). Now, if we assume the utility of the agent can be decomposed intoa part which tracks agreement and disagreement with neighbours

∑j Cδd(i),d(j)

where d(i), d(j) are decisions of agents i, j, and a part which makes one of theoptions better, by a simple linear transformation and representing the choice asa spin variable taking values −1, 1, we get

Ui =∑j

Jσiσj + hσi (6.28)

Now, if we let the agents update the option they are taking in a randommanner, choosing one agent at a time and making him choose afresh, we get veryclose to the Ising model.

68

A. Numerical simulationsimplementation

We developed a software tool gIsing to run the numerical simulations of the Isingmodel on general graphs.

A.1 Forward-flux sampling algorithm

We implement standard forward-flux sampling [3, 1] with several improvementsto speed up the computation and automate parameter estimation. Namely, thesoftware has the option to automatically set the boundary interfaces and alsoautomatically select intermediate interface spacing.

Order parameter

We use the number of up-spins as the main order parameter rather than the sizeof the largest cluster. In general networks, the assumption that a minimal-energycore of certain order will be connected does not hold (contrary to e.g. grids):Assume 5-node ”star” network with one central node and 4 nodes attached onlyto the central node. Then the minimal energy core of order 2 are 2 non-centralvertices; the most likely transition trajectory first flips 2 non-central vertices,then the center vertex, and then the rest of the graph.

This parameter would be unusual in nucleation on grids literature, but waspreviously used when studying networks in Chen et al. [19]. It also has thepractical advantages that it is very fast to compute, e.g. by maintaining a counterof up-spins, and it changes by at most 1 at every spin update (unlike connectedcomponent sizes that can change by more, possibly skipping over interfaces).

First and last interface search

FFS uses a series of interfaces between the initial and final states to calculaterate constants and generate transition paths - the first and last interfaces can bespecified manually or automatically set to boundaries of the mesostable regionsaround all-up and all-down states. We choose the first and last interface auto-matically. To do that, we sample several independent trajectories (5 by default,each of 1000N MCSS by default) from the all-up (resp. all-down) state, discardtheir first 1/3 (to reach the mesostable state) and then create an order distribu-tion. The order of interface A (resp. B) is then chosen as the 0.9999-quantile(resp. 0.0001-quantile) of this distribution. The up-crossing rate at interface Awill then be in the order of 1

1000N, and the energy gradient at interface B should

be directed strongly upwards while being well after the critical order threshold.

Adaptive intermediate interfaces

Intermediate interfaces can be chosen to be uniformly spaced between A and B(by specifying their number), or can be inserted automatically. After collectingenough samples at an interface of order k, we search for optimal next interface

69

placement. The step size s is optimized to reach P (Ik+s|Ik) = 0.5. The rationaleis that this ensures:

• Ensures the number of interfaces is O(− log ΨA,B).

• At each interface, the total number trajectories is at most 2S where S isthe number of samples per interface1.

We approximate this optimal step size by exponentially increasing or decreas-ing previous step size, each time running the populating algorithm with smallertarget sample size on the temporary interface (10 by default). Note that minimalstep size is 1, we also set the maximal step size to B−A

20to have better progress

granularity, improve critical order accuracy and to prevent few edge cases.

Populating the first interface

The first interface is populated by running several trajectories from the all-downstate, creating a new sample whenever the order parameter crosses the interfacein upwards direction. The rate at interface A is then computed from the timesbetween subsequent crossings of A upwards. Note that the time from the start isnot included; we also assume that the state is well-mixed after passing throughthe meso-stable region and reaching A. This is ran until there are S samples. Ontimeout (e.g. after 1× 105 sweeps without crossing A by defalt), the trajectoryis reset and the timeout value is added to the times between crossings (if thetrajectory has crossed A at least once). However, timeouts are very infrequenthere.

Populating subsequent interfaces

Subsequent interfaces are populated from the previous interface samples: at eachiteration, and independent sample is chosen from the previous population and atrajectory is run until it reaches the next interface, reaches interface A, or reachesa timeout. The timeout used is 1× 105 sweeps by default and timeout trajectoriesare not used in rate calculations as they happen mostly on the potential plateaunear critical size (but still infrequently).

Spin updates

The algorithm uses the standard Metropolis-Hastings Monte Carlo update rule,e.g. one step is selecting a spin randomly, computing ∆E as the free energy changeon flipping the spin and then flipping it with probability min{e−∆E/kBT , 1}. Thecomplexity of one update is then O(d),where d is the average degree.

Largest cluster search

The largest cluster search is implemented as connected component search in thegraph induced by the nodes with the target spin values, optionally dropping afraction of the edges when sampling. The search is performed as a depth-first

1Note that an interface pair with e.g. P (Ik+s|Ik) = 0.001 would require 1001S trajectories,while 10 intermediate interfaces would together require only 20 trajectories.

70

search from every node, keeping a ”visited” Boolean flag for every node. Thecomplexity of a single search is then O(N +M) where N is the number of nodesand M the number of graph edges.

Sampling a subset of the edges gives an unbiased estimate of the expectedlargest cluster size (under edge selection) but the high variance breaks interfacecrossing detection: With few samples, the states selected for may be the onesthat admit higher variance in the order parameter. Also, the inconsistency maymake a state skip over several interfaces (when tightly spaced), complicating andpotentially compromising the algorithm.

Our solution is to fix the subsets of edges that will be discarded before largestcomponent search. While this would have the potential to create artifacts witha small number of samples (e.g. ignoring an edge in almost all samples), largeclustering sample number should mitigate this effect.

The other option we consider is to use the exact largest cluster size (no edgedropping) as the order parameter.

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B. Implementation notes

We developed a software tool gIsing to run the numerical simulations of the Isingmodel on general graphs.

The software is written in Python 3.6 with an extension module written in C.The main Python libraries used are NetworkX (general network library), Numpy(numerical and vector data manipulation) and Plotly (plotting). The C moduleimplements a series of random spin updates and order parameter measurement,both for a given number of steps and until certain order parameters are reached.Measuring cluster size is performed only once every given number of steps due toits complexity.

B.1 Execution speed

We benchmarked the speed of the elementary operations of our algorithm: spinupdate and largest cluster computation. We focus on these operations ratherthan the entire FFS since other algorithm components use negligible part of theCPU time, and the number of total iterations heavily depends on the graph andhigh-level computation parameters. We express the speed in nodes per secondignoring the effect of edges since it is not significant for small average degrees(e.g. less up to 20).

The benchmarks were ran on AMD Ryzen 5 2600 3.4 GHz processor, singlethreaded, using Python 3.6.7, GCC 7.4.0 and Ubuntu 18.04. The graphs consid-ered were Barabasi-Albert graphs on 103 – 105 vertices and average degrees 2 –20, and 2D toroidal grid graphs 20× 20 – 100× 100.

The speed of node updates is 5.7× 106 ± 7.4× 105 node updates per second.The speed of clustering is 1.6× 108 ± 1.3× 108 nodes per second and is moresensitive to average node degrees.

B.2 GPU implementation and drawbacks

We have implemented the low-level parts of the algorithm in the TensorFlowframework to run on a GPU, aiming for a significant speedup compared to the Cextension. Although the implementation is very non-trivial and includes severaloptimizations, we have not succeeded in utilizing the computational potential ofa GPU. You can find our TF2.0-based implementation at https://github.com/gavento/graph-ising/tree/old-tf. It seems that a more low-level approach(e.g. writing specialized CUDA kernels) and specialized algorithms would berequired. Below we discuss the drawbacks of GPU computation of parallel spinupdates and connected component search.

Our GPU implementation was similarly fast or slower than a single-threadedC code ran on a CPU, making it an inefficient use of a GPU and indicating thata large speedup would be needed. Also note that it is not generally possible tocombine subroutines on the GPU and CPU on the same data as the transfer andsynchronization costs are usually prohibitive.

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B.2.1 Parallel spin updates

The vectorized nature of GPU invites to do parallel spin updates in the Metropolis-Hastings algorithms. Indeed, the core update operation can be made very fast.The slowest part is collecting adjacent node data in general graphs (the GPU ar-chitectures are not optimized for sparse matrix multiplication or random node ac-cess). Since the spin updates should be uncorrelated, we need to select a smaller,preferably independent node set (i.e. set with no edges among its nodes).

Sampling a random spin subset to be updated seems like a straightforwardsolution but the fraction of vertices updated may need to be very small to keepthe number of simultaneously updated nodes sufficiently low. In graphs with apower-law degree distribution, the high-degree nodes would have a higher chanceto be updated simultaneously, distorting the process.

Decomposing the nodes into several independent sets needs at least χ subsetswhere χ is the coloring number of the underlying graph. Even with such a de-composition, it is unclear what effect would the implied (partially) fixed updateorder have on the process. One mitigation would be to alternate between severaldecompositions but the trade-off is unclear and a large number of decompositionsmay have a negative effect on the algorithm.

B.2.2 Connected component search

Computing cluster size reduces to connected component size in the subgraphG′ induced by the selected spin values and edges selected for sampling. Ouralgorithm assigns every node a unique value cv (e.g. its ID). In every step, wesimultaneously update cv ← maxu∈N(v)∪{v}cu. When c∗ does not change in a step,the connected components of G′ are each filled with a distinct value, and we onlyneed to look for the most-frequent one.

The number of steps is between the radius and diameter of G′, which isgenerally O(log |G|) for scale-free networks but is Θ(|G|1/d) for d-dimensionalgrids, making this method much slower. Similarly to spin updates, connectedcomponent search on a GPU (possibly unavoidably) suffers from slow access toadjacent graph nodes.

73

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List of Figures

1.1 Landscape metaphor: physical landscape and abstract potentiallandscape. Photo by Jay Huang. . . . . . . . . . . . . . . . . . . . 4

1.2 Competition of surface and volume terms, creating the nucleationbarrier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Movement of clusters in the space of sizes. a. one step correspondsto addition of one particle. c. critical cluster on top of the hill . . 7

2.1 Free energy ∆G(N) as a function of cluster size N in a nucleationregime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Shape dependence of strain energy. (For c/a→∞, E = 3/4). . . 142.3 Schematic representation of the energies of clusters of sizes N ,

respective of N + 1 and the intermediate state of higher energy. . 182.4 Definition of the incubation time tinc. . . . . . . . . . . . . . . . . 202.5 Variation of free energy in the critical region for sizes close to Nc 22

3.1 Suzuki phase unit cell in an NaCl crystal in the 100 plane. Cl ionsin violet and Pb ions in yellow. Vacancies indicated by letter V. . 24

3.2 Selected configurations of clusters in KCl system . . . . . . . . . . 263.3 Energies of Pb-vacancy clusters in the {111} Na plane of NaCl

crystal. Relative positions of the clusters represent energies offormation from their closest constituent parts. . . . . . . . . . . . 28

3.4 Example of a supercell used for layer energy calculations. Pb atomsare yellow and Na atoms are blue. . . . . . . . . . . . . . . . . . . 29

3.5 Examples of periodic arrangements in the {111} plane. Na ions inblue and Pb ions in yellow. . . . . . . . . . . . . . . . . . . . . . 30

3.6 PbCl2 crystal in the 001 plane . . . . . . . . . . . . . . . . . . . . 313.7 NaCl crystal in the {111} plane, Cl ions in violet . . . . . . . . . 32

5.1 Schematics of the simulation cell: the lower heterogeneous surfacehas favorable conditions for nucleation. The upper surface hassurface energy preventing nucleation. The boundary conditionsare periodic in remaining directions. . . . . . . . . . . . . . . . . 41

5.2 Three examined cases of surface heterogeneity - shade of gray rep-resents the difference from a homogeneous surface, which would beuniform gray. From the left: 1. regular stripes 2. random patternof species 3. correlated random variable . . . . . . . . . . . . . . . 41

5.3 Decrease of nucleation energy barrier ∆G with increasing “con-trast” of surface pattern in case of surface patterned with regularstripes. Heterogeneity amplitudes from the top: 0.2, 0.4, 0.8 inunits of h. Size of the cluster N is simply the number member spins. 42

5.4 Decrease of nucleation energy barrier ∆G with increasing “con-trast” of surface pattern in case of random surface. Heterogeneityamplitudes from the top: 0.2, 0.4, 0.8 in units of h. Size of thecluster N is simply the number member spins. . . . . . . . . . . . 43

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5.5 Decrease of nucleation energy barrier ∆G with increasing “con-trast” of surface pattern in case of random surface. Heterogeneityamplitudes from the top: 0.2, 0.4, 0.8 in units of h. Size of thecluster N is simply the number member spins. . . . . . . . . . . . 43

6.1 Degree distributions for four real world networks, from [9]: in-ternet connectivity at the router level, protein-protein interactionnetwork, email network and a citation network. The green linesshows Poisson distribution with the same average degree as in thedata, illustrating the fact that many real-world networks have a de-gree distribution which can not be accounted for by random graphmode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.2 Local clustering coefficient. Values of Ci are indicated next to nodes. 476.3 Cayley tree. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.4 Wattz-Strogatz model. . . . . . . . . . . . . . . . . . . . . . . . . 506.5 Evolution of a Barabasi-Albert network. In each step, the newly

added node is indicated by a white circle, the old nodes by greycircles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.6 Graph of the nucleation progress (top, red line indicates critical or-der parameter) and sample of a critical core (bottom, red nodes inthe core) of Barabasi-Albert network with N = 100 nodes, averagedegree 4, T = 1.0, h = 0.5. . . . . . . . . . . . . . . . . . . . . . . 57

6.7 The histograms of the degrees of up-spin nodes at various orderparameters during a nucleation transition. The degrees are binnedin exponentially sized buckets, the numbers above the graph indi-cate the number of nodes in a degree bucket. The recorded odredclosest to the critical order is highlighted. . . . . . . . . . . . . . 59

6.8 Graph of the nucleation progress of a Holme-Kim network withN = 100 nodes, average degree 4,p△ = 0.5 (top) and p△ = 0.9(bottom), T = 1.0, h = 0.5. Compare to Figure 6.6. . . . . . . . . 61

6.9 A sample of graphs and critical cores for p△ = 0.5 (left) and p△ =0.9 (right) in the setting of Figure 6.8. Compare to Figure 6.6. . . 62

6.10 Transition rate dependence on p△ and h (top), respectively p△ andT (bottom). Holme-Kim model with N = 1000 nodes and averagedegree 6, temperature and external field T = 1.0, h = 0.5. Thered cross indicates the rate reported by Chen et al. . . . . . . . . 63

6.11 Transition rate dependence on the network size N and p△ for N =50 . . . 1000 (top) and N = 200 . . . 10000 (bottom). Holme-Kimmodel with average degree 6, temperature and external field T =1.0, h = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.12 Critical cluster size dependence on p△ and h (top), respectivelyp△ and T (bottom). Holme-Kim model with N = 1000 nodes andaverage degree 6, temperature and external field T = 1.0, h = 0.5.The red cross indicates the critical size reported by Chen et al. . . 65

6.13 Critical cluster size dependence on the network size N and p△ forN = 50 . . . 1000 (top) and N = 200 . . . 10000 (bottom). Holme-Kim model with average degree 6, temperature and external fieldT = 1.0, h = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

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6.14 Transition rate (top) and critical cluster size (bottom) dependenceon the network size N for p△ = 0.9. Rescaled from Figures 6.13and 6.11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

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List of publications

1. J Kulveit, P Demo, K Polak, AM Sveshnikov, and Z Kozısek. Formation ofstructured nanophases in halide crystals. The Journal of chemical physics,134(14):144504–144504, 2011.[49]

2. Jan Kulveit. Ising model simulations as a testbed of nucleation theory.Acta Polytechnica. 55(1):29–33, 2015.[50] Jan Kulveit and Pavel Demo.

3. Kozısek, Z., Demo, P., Sveshnikov, A., Kulveit, J. (2015). HomogeneousNucleation of Ni Liquid Near Critical Supercooling. Advanced Science,Engineering and Medicine, 7(4), 339-342.

In review

1. J Kulveit, P Demo. Heterogeneous nucleation on a surface with heteroge-neous surface energy. arXiv preprint arXiv:1605.02670

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