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Coupled flux nucleation model

ap-plied to the metallic glass AMZ4

Kopplad fl¨

odesmodell applicerad p˚

a det metalliska glaset AMZ4

Project description

Master thesis, 30 credits, Computational material science Spring 2021

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Abstract

Additive manufacturing (AM), also known as 3D-printing, has made it possible to produce components made of bulk metallic glass (BMG) which have remark-able properties compared to parts made of conventional alloys. A metallic glass is a metastable noncrystalline alloy that form if a melt is quenched with a suffi-cient cooling rate. Research on systems with low critical cooling rates have made the maximum dimensions of these alloys to grow to what is called BMG’s. The high local cooling rate obtained during AM makes it in principle possible to bypass the dimension restrictions that otherwise have been present when creating these alloys but the procedure is complex. It is believed that oxygen impurities in the powder feedstock material used during AM of Zr-based alloys makes it favourable for nucleation of stable crystalline phases at lower activation energies which hin-ders fully glass features to develop. The purpose of this thesis is to investigate how the limiting solute concentration in the bulk of the AM produced alloy AMZ4 (Zr59.3Cu28.8Al10.4Nb1.5(at%)) impact the nucleation. Using a numerical model

based on classical nucleation theory (CNT) that couples the interfacial and long range fluxes makes it possible to study how impurities impact the nucleation event. However, missing oxygen dependent data makes this a study on how limiting solute impact the nucleation in AMZ4. The numerical model is validated against earlier work and the results obtained from the simulations on AMZ4 shows a strong con-nection between the nucleation event and the limiting solute concentration. Further investigations on phase separation energies and the production of concentration dependent time-temperature-transformation (TTT) diagrams are needed to fully describe the connection to oxygen concentration. Nevertheless, the implemented model captures important features that the classical model cannot describe which needs to be taken into account when describing the nucleation in AMZ4.

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Abstrakt

Friformsframst¨allning (eng. additive manufacturing (AM)), ocks˚a k¨ant som 3D-printing, har gjort det m¨ojligt att producera komponenter gjorda av bulkmetallglas (eng. bulk metallic glass (BMG)) vilka har anm¨arkningsv¨arda egenskaper j¨amf¨ort med delar gjord av konventionella legeringar. Ett metalliskt glas ¨ar en metasta-bil icke kristallin legering som skapas om en sm¨alta sl¨acks med en tillr¨acklig kyl-hastighet. Forsking p˚a system med l˚aga kritiska kylhastigheter har gjort att de maximala dimensionerna av dessa legeringar har ¨okat till vad som kallas BMG’s. Den h¨oga lokala kylhastigheten som erh˚alls under AM g¨or att dimensionsrestrik-tionerna principiellt kan kringg˚as vilka annars ¨ar n¨arvarande vid skapandet dessa legeringar men proceduren ¨ar komplex. Det ¨ar trott att orenheter av syre i pulver-r˚avarumaterialet som anv¨ands vid AM av Zr-baserade legeringar g¨or det f¨ordelaktigt f¨or k¨arnbildning av stabila kristallina faser vid l¨agre aktiveringsenergier vilket hin-drar fulla glas egenskaper fr˚an att utvecklas. Syftet med denna uppsats ¨ar att unders¨oka hur den begr¨ansande l¨osningen p˚averkar k¨arnbildningsf¨orloppet i den AM producerade legeringen AMZ4 (Zr59.3Cu28.8Al10.4Nb1.5(at%)). En numerisk

modell baserad p˚a klassisk k¨arnbildningsteori (eng. classical nucleation theory (CNT)) som kopplar gr¨ansskikt- och l˚angdistans-fl¨odet g¨or det m¨ojligt att studera hur orenheter p˚averkar k¨arnbildningsf¨orloppet. Syreberoende data g¨or dock detta till en studie om hur den begr¨ansande l¨osningen p˚averkar k¨arnbildningen i AMZ4. Den numeriska modellen valideras mot tidigare arbeten och resultaten fr˚an simu-leringarna av AMZ4 visar ett starkt samband mellan k¨arnbildningsf¨orloppet och den begr¨ansade l¨osningskoncentrationen. Vidare studier r¨orande fas-separeringsenergier och framst¨allningen av koncentrationsberoende tid-temperature-transformation (eng. time-temperature-transformation (TTT)) diagram beh¨ovs f¨or att till fullo beskriva kopplingen till syrekoncentrationen. Den implementerade modellen f˚angar dock vik-tiga egenskaper som den klassiska modellen inte kan beskriva vilka m˚aste tas h¨ansyn till n¨ar k¨arnbildning i AMZ4 ska beskrivas.

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Acknowledgements

This thesis has been an experience into the unknown and it has demanded both patience and ingenuity on a new level. It has also given me the opportunity to dive much deeper into a topic than ever before which has been an oddly satisfying expe-rience with late nights and a lot of debugging and rephrasing. These past two years have went by faster then I could ever imagine, I want to thank all the lecturers in-volved in teaching me everything from numerical computations to index gymnastics. A specially big thanks goes to Martin Fisk, associate professor in material science at Malm¨o University and Anders Ericsson, PhD student at the Division of Solid Me-chanics at Lund University as mentors during this semester. Their help have been invaluable and given me motivation to further seek knowledge and understanding in this subject.

May 25, 2021 Mattias Tidefelt

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Contents

1 Introduction 1

2 Background 2

2.1 Additive manufacturing . . . 2

2.2 Metallic glasses . . . 3

2.3 Synthesising of bulk metallic glasses . . . 5

3 Classical nucleation theory 6 3.1 Coupled flux nucleation . . . 11

3.2 Steady state nucleation . . . 16

3.3 Growth of stable nucleus . . . 17

3.4 Numerical implementation . . . 18

4 Result and discussion 20 4.1 Material parameters: model validation . . . 20

4.2 Coupled flux model validation . . . 21

4.3 Material parameters: AMZ4 . . . 25

4.4 Solute concentration impact on nucleation in AMZ4 . . . 28

4.4.1 TTT diagrams for AMZ4 . . . 32

4.4.2 Quenched in distributions . . . 34

5 Conclusions 36

A Derivation of 𝑘+(𝑛) and 𝑘−(𝑛 + 1) A1

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1

Introduction

Every material requires its own manufacturing method. The methods have been re-fined from the first attempt to create a new material to what then becomes something that is used at an industrial scale to supply commercial demands. A manufacturing method that has started to get more and more recognition is additive manufacturing (AM), also known as 3D printing [1]. As the name suggests, the manufacturing is preformed by continuously adding material in comparison to conventional methods where the material is removed from a slab when milling or added as full body volume when casting. Because of the high local cooling rates that can be obtained during AM, so called bulk metallic glasses (BMG’s) [2] can be manufactured by freezing the amorphous structure of a liquid metal into a metastable solid state. Metallic glasses are a new class of materials known for their ceramic strength features such as high yield strength and low fracture toughness while otherwise keeping or enhancing the metallic properties.

The cooling rate required to form a metallic glass is strongly dependent on the composition of the alloy [2]. The so called ”𝑛𝑜𝑠𝑒” in the time-temperature-transformation (TTT) diagram, that represent an initiation of nucleation, shifts depending on the composition and by this the cooling rate needed to form a glass. The composition of an alloy is seldom pure as it often contains trace elements or other unwanted impurities which can impact the crystallisation behaviour. As the feedstock material for manufacturing metallic glasses with AM is a powder it natu-rally contains oxygen impurities and this is believed to create unwanted nucleation in Zr-based alloys which hinders fully glass features to develop.

A lot of studies have been made on the AM processing of metallic glasses with se-lective laser melting (SLM) [3–7]. A commonly used method to numerically describe nucleation is the classical nucleation theory (CNT) [8]. This method have been used to study the nucleation in the AM produced metallic glass AMZ4 (Zr59.3Cu28.8Al10.4

Nb1.5(at%)) [9] in earlier work done by Ericsson et al. [3]. However, this model does

not capture diffusion limited nucleation. The impact of oxygen concentration have been studied for SiO2 precipitation in a pure Si system as a diffusion limited

nucle-ation event by Kelton et al. [10]. The used model captures important features in solid state phase transformations and it should be applicable to other systems as well.

The purpose of this thesis is to numerically investigate how the limiting solute concentration in the bulk impact the nucleation in AMZ4. As impurities are dilute, using a model based on CNT that describe nucleation with a coupled long range and interfacial flux derived by Kelton et al. [11] makes it possible to investigate a diffusion limited nucleation event in AMZ4. With this model the connection between the nucleation and the oxygen concentration in the powder feedstock material can be investigated.

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2

Background

To connect with the specific manufacturing method of AMZ4 a brief description of additive manufacturing is presented here. To understand how and why nucleation in metallic glasses occur, a short description of what metallic glasses are and how they can be produced in bulk are also presented here.

2.1

Additive manufacturing

The process name AM include various types of layer-by-layer manufacturing meth-ods [1]. Common feedstock materials used during the processing are different liquids, pastes or powders that are fused together by solidifying one layer at a time and by this growing a full body. The parts that are manufactured all requires a model gen-erated by computer aided drawing (CAD) files which are converted into a format that the processing machines can read. When a part is manufactured some surface refining is usually needed as the surfaces of the as-manufactured parts tend to be quite rough. In recent years the use of AM have been growing in the area of so called rapid prototyping (RP) as the possibility to create prototypes on site without requiring special tools or moulds is superior to conventional prototyping methods in many ways. The most common method of RP is stereo lithography (SLA) where the feedstock material is liquid polymers that are stabilised through photo polymerisa-tion using an UV laser. A common method that uses paste feedstock is the fused deposition modeling (FDM) which might be the most commonly and commercially used AM method. The method has been developed to handle different material types such as polymers, ceramics and metals but the accuracy of the parts made are often quite low.

The development of methods that use powder based feedstock materials have been used to start manufacturing high quality parts in metal [1]. Some commonly used methods are selective laser sintering (SLS) and SLM where the difference mainly is the temperature that the feedstock material is heated to during the processing. The powder feedstock material is heated close to the melting temperature, 𝑇𝑚, in a

chamber where a laser melts a geometry in the powder layer. These processes are more complex than the other methods but remarkable properties of the manufac-tured parts have made this into an area of strong research. The high temperature gradient that is obtained from the AM process makes it possible to manufacture alloys that cannot be produced with conventional casting and forging methods.

Common for all AM methods are that they do not require the machine tools and castings that regular manufacturing require but manufacturing a part is time consuming so it is economically favourable to use for small batches. Further, the left over feedstock material can readily be reused and no excess material is removed during the process which makes it more environmentally friendly. Finally, the layer-by-layer processing makes it possible to create geometries that otherwise would be impossible to create as no constraints from tools or casting geometries are present but the size limitation because of the machines used make AM only viable for

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com-ponents with smaller dimensions. For further discussions the reader is refereed to [1].

2.2

Metallic glasses

Metals are known for their crystalline structures [12]. The formation of these struc-tures are initiated by nucleation events in the liquid melt as it solidifies during which the formed nuclei grow and grains are formed. For multi component systems (i.e alloys), the composition of the grains can vary which gives different crystalline structures and properties. Between these grains, interfaces called grain boundaries are created which also gives the alloy typical properties. The grain boundaries con-sists of more loosely packed atoms with no specific crystalline ordering and act as favourable diffusion paths and dislocation sinks.

Depending on what rate a liquid is cooled from the liquidus temperature, 𝑇𝑙,

different types of crystallisation is achieved [2]. In figure 1 a TTT-diagram is shown where the initiation of crystallisation in the liquid is represented by the parabolic curve. As seen in the figure there is a possibility to form a supercooled liquid and avoid crystallisation if the cooling is fast enough. As the liquid is cooled the viscosity decrease and the rate at which this occur is dependant on the cooling rate. When the viscosity change to that of a solid a glass metal is formed and the temperature at which this occur is called the glass transition temperature, 𝑇𝑔, which by this is a

kinetic parameter compared to 𝑇𝑙.

Figure 1: A typical TTT-diagram. The ”𝑛𝑜𝑠𝑒” is the tip of the crystallisation zone and sets the required cooling rate needed to form a metallic glass. Inspired by [12]. A measurement of an alloys glass forming ability (GFA) is the so called reduced glass transition temperature given by [2]

𝑇𝑟 𝑔 = 𝑇𝑔 𝑇𝑙

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The higher value of 𝑇𝑟 𝑔 the higher the viscosity is and it has been suggested that

values of 𝑇𝑟 𝑔 ≥ 2/3 completely suppress the formation of homogeneous crystalline

nucleation. This suggests that a glass should be able to form independent on the cooling rate used during manufacturing. However, this is not the case as the rate must be high enough to miss the nose of the TTT-diagram at temperature, 𝑇𝑛,

but with a low 𝑇𝑙 and a high 𝑇𝑔 the required cooling rate to form a glass becomes

smaller. For each glass forming alloy, a critical cooling rate 𝑅𝑐 exists which can be

approximated as

𝑅𝑐 = Δ𝑇

𝑡𝑛

, (2)

where Δ𝑇 and 𝑡𝑛 corresponds to the under cooling to temperature 𝑇 from 𝑇𝑙 and the

time at the nose of the compositions corresponding TTT-diagram.

The driving force in phase transformation is the chemical potential, 𝛿𝜇, which corresponds to the Gibbs free energy change per atom/molecule between the phases [12]. For supercooled liquids and glasses 𝛿𝜇 depends on the magnitude of Δ𝑇 . Above 𝑇𝑚, 𝛿𝜇 is positive and below 𝑇𝑚, 𝛿𝜇 is negative. The change in enthalpy, Δ𝐻,

entropy, Δ𝑆, and the volume, 𝑉 , are discontinuous when a crystalline solid phase form from a liquid but for a glass these changes are continuous as the supercooled liquid transform into a glassy state at 𝑇𝑔 [13]. As the the glassy phase is metastable

it is in a higher energy state than what the crystalline phase is. The driving force between a crystalline and supercooled phase can be approximated as the difference between the corresponding two free energy curves.

The critical cooling rate vary vastly depending on the composition of the alloy and it have been shown that in general the ”nose” of the TTT-diagram shifts more towards the right the more components that are present in an alloy [2]. This have made the research on multi component systems a topic of interest. Early successful compositions required cooling rates of around 106 K/s but newly formed composi-tions have shown critical cooling rates as low as 0.067 K/s. The critical cooling rate determines the maximum cross section of the alloy that can be formed. Initially the samples that could be produced ranged from cross section of about 20-50 μm whereas now cross sections of up to a few centimeters have been produced in Pd-, Zr-, Pt- and Cu-based systems [14]. What dimension and composition that should be called a BMG or not have been debated, but as of now a BMG is vaguely a multi component alloy that can form a homogeneous noncrystalline phase with a cross section larger than a millimeter.

The properties of metallic glasses are typically high yield strength and low ductility in ranges in that of ceramics while still having metallic properties such as high electrical and thermal conductivity. In figure 2 the new material class have been highlighted in an Ashby chart showing the yield strength versus fracture toughness relative to other classes of materials. Metallic glasses should not be used in high temperature applications as the amorphous state of the metal is metastable and stable crystalline phases can form if an activation energy, usually temperature or stress, is applied. As the glass is amorphous, the lack of grain boundaries gives an absence of the diffusion paths that are typically found in polycrystalline alloys and

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thus the corrosion resistance of fully developed glass structures have shown to exceed those of traditionally cast or composite alloys. The soft magnetic properties have also shown to exceed those of traditionally cast alloys as the glass do not exhibit the magnetic domains that polycrystalline alloys experience which gives a close to non existence of the hysteresis loops otherwise found in soft magnetic alloys. For further discussions on the material properties the reader is refereed to [2].

Figure 2: Approximate Ashby chart with the addition of the new material class metallic glasses. The properties of the metallic glasses in terms of yield strength and fracture toughness have those in between engineering metals and ceramics. Inspired by [15].

2.3

Synthesising of bulk metallic glasses

Metallic glasses can be produced in various shapes and sizes depending on the initial state of the alloy and how it is processed [2]. Different areas of applications can be meet by utilising the shape of the glass that is produced. Vapor state process methods are the earliest reported successful attempts to achieve small quantities of amorphous alloys while solid state processes have been used to synthesise large quantities of amorphous powder. Liquid state processes are the only methods that are used to produce BMG’s and this area is rapidly growing in both demand and ongoing research as the large dimensions of BMG’s give the possibility to scale the production to commercial levels. The discovery of BMG’s comes from the evolution of the rapid solidification processes (RSP’s) where a commonly used method is melt spinning. The setup of this processing technique have the main purpose to rapidly conduct heat away from the liquid melt at a rate sufficient for glass formation. To achieve this rapid solidification the geometries that can be formed are thin films and ribbons.

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A lot of research have been made in finding compositions which lower 𝑅𝑐 and

with this the dimensions have grown to what we call BMG’s [2] . Typical alloys that are used are Ti-, Zr- and Fe-based alloys. The alloys used to produce BMG’s have usually a quite low 𝑅𝑐 of around 10

1-103K/s and nowadays different casting

methods and water quenching can also be used to produce BMG’s. To achieve good GFA the purity of the feedstock material is of high importance as impurities can form stable phases with lower activation energy that act as heterogeneous nucleation sites. It has been shown that impurities of oxygen content in the feedstock material affect the required Δ𝑇 and 𝑅𝑐 needed to achieve homogeneous amorphous Zr-based

BMG’s. The oxygen content have also shown to affect the thermal stability of the BMG that is formed.

During the last years the possibility to form BMG’s with AM have become a topic of research [3–7] as the nature of the process gives high cooling rates and the geometric limitations of the produced components can in principle be avoided. A popular AM method used when forming BMG’s is SLM. A lot of research is ongo-ing in this area to handle issues with porosity and unwanted nucleation because of the temperature history caused by cyclic reheating of the fused/as-solidified amor-phous material. Nucleation of crystalline zones in an amoramor-phous alloy is not always unwanted though as the possibility to create crystalline BMG-composites give the possibility to form more ductile crystalline-glassy composites. Thus, the research in understanding the nucleation behaviour in BMG’s is important to be able to predict the manufacturing of BMG’s with AM.

3

Classical nucleation theory

Phase transformations where a stable new phase is formed from a metastable phase is driven by stochastic fluctuations. These fluctuations are typically measured as a radius 𝑟 or a number of atoms/molecules 𝑛. According to LaMer [16], the growth of a new phase is considered to contain three steps in time, an initial distribution of species which through fluctuations (1) initiate a microscopic nucleation event (2) where the formed nuclei finally grow to macroscopic sizes (3). In figure 3 the solute concentration of this process is illustrated in time.

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Figure 3: Time evolution of solute concentration during a phase transformation according to LaMer’s mechanism. The nucleation process refers only to the time period where stable nuclei forms, (2). Before this event fluctuations in the solute occur that with time give the right conditions (1). The nucleation event is followed by the growth of these stable nuclei (3). Inspired by [16].

Classical nucleation theory (CNT) [8] has its basis in a sharp interface formu-lation where the work of formation needed to form a nucleus of size 𝑛 is expressed in terms of the change in Gibbs free energy. For polymorphic nucleation under con-stant pressure the work of formation needed to form a spherical cluster of size 𝑛 can be expressed as

𝑊(𝑛) = 𝑛𝛿𝜇 + (36𝜋)1/3𝑣¯2/3𝑛2/3𝜎, (3) where ¯𝑣and 𝜎 corresponds to the mean atomic/molecular volume and the interfacial free energy per unit area. This expression is a simplification and for small clusters the effect from the ordering gradient at the interface is more prominent. For large cluster sizes this approximation works well though. Noting that 𝑛¯𝑣 = 4/3𝜋𝑟3, the work of formation can be expressed in terms of radius as

𝑊(𝑟) = 4𝜋 3¯𝑣

𝑟3𝛿 𝜇+ 4𝜋𝑟2𝜎 . (4)

The work needed to form a cluster is a competition between the loss of energy be-cause of volumetric formation and the gain of energy bebe-cause of the Gibbs-Thomson effect that the interface create. This gives an energy maximum at a critical size 𝑛∗ which is illustrated in figure 4(a). At this energy the cluster has formed an unstable equilibrium phase which means that

𝜕𝑊(𝑛) 𝜕 𝑛

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bellow this limit the atoms/molecules tend to dissipate from the cluster and above this limit continuous growth of this cluster is mostly favourable which create a critical region. To be certain that a nuclei is stable, the cluster size should be evaluated at least 𝑘𝐵𝑇 in terms of energy away from 𝑊 (𝑛

), which is illustrated in figure 4(b).

(a) (b)

Figure 4: (a) The competition of energies during nucleation. At the critical cluster size an unstable equilibrium phase is formed compared to the initial phase. (b) Around the critical cluster size a critical region exist where the growth or dissipation of clusters is uncertain. The evaluation of stable nuclei should be done outside this region. (a) is inspired by [12] and (b) is inspired by [8].

The critical number of atoms/molecules and the corresponding critical work of for-mation can readily be obtained by using Eq. (5) which gives

𝑛∗ = 32𝜋 3¯𝑣 𝜎3 |Δ𝑔|3 (6) and 𝑊(𝑛∗) = 16𝜋 3 𝜎3 Δ𝑔2 , (7)

where the change Gibbs free energy per unit volume have been introduced as Δ𝑔 = 𝛿 𝜇/¯𝑣.

When a homogeneous medium is equilibrated quickly, the probability of a fluc-tuation is 𝑃(𝑛) ∝ exp  Δ𝑆 𝑘𝐵  , (8)

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of forming a cluster of size 𝑛 proportional to the Boltzmann weight as 𝑃(𝑛) ∝ exp  −𝑊(𝑛) 𝑘𝐵𝑇  . (9)

From the above Boltzmann weighted probability the equilibrium distribution of clusters 𝑁 containing 𝑛 atoms/molecules can be expressed as

𝑁𝑒 𝑞(𝑛) = 𝑁0exp  −𝑊(𝑛) 𝑘𝐵𝑇  , (10)

where 𝑁0 corresponds to the number of solute atoms/molecules per volume in the

initial phase. This distribution assumes that there is no flux between the initial phase and the new phase and do not describe the nucleation process well. It does however, give us a mathematical basis for the time-dependant nucleation. If re-versible nucleation is to be described, the cluster population 𝑁 (𝑛) depends on the forward and backward interfacial attachment rates 𝑘+ and 𝑘−. This is illustrated in figure 5 together with the nucleation rate, 𝐼 (𝑛), which corresponds to the transfer between two states.

Figure 5: Histogram of the reverisble cluster population as a function of cluster size. The population is connected with the forward and backward rate constants and the difference between two of these states give the nucleation rate. Inspired by [11]

The first kinetic model of nucleation was created by Wolmer and Weber where they assumed that clusters of 𝑛 atoms/molecules, 𝐸𝑛, grow or shrink slowly by the

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reactions [8]. This gives the forward and backward rate constants from 𝐸𝑛−1+ 𝐸1 𝑘+(𝑛−1)  𝑘−(𝑛) 𝐸𝑛 𝐸𝑛+ 𝐸1 𝑘+(𝑛)  𝑘−(𝑛+1) 𝐸𝑛+1. (11)

From these reactions the transient distribution with the rate constants can be ex-pressed as 𝜕 𝑁(𝑛, 𝑡) 𝜕 𝑡 =𝑁 (𝑛 −1, 𝑡) 𝑘+(𝑛 − 1) − 𝑁 (𝑛, 𝑡) (𝑘+(𝑛) + 𝑘−(𝑛)) + 𝑁 (𝑛 + 1, 𝑡) 𝑘−(𝑛 + 1), (12) where 𝑡 is the time. This expression have the form of a master equation and implies that the solutions to the differential equations have to be solved on a grid in both space and time. Likewise the transient nucleation rate, as illustrated in figure 5, can be expressed as the difference between the forward and backward flux from two following states as

𝐼(𝑛, 𝑡) = 𝑁 (𝑛, 𝑡) 𝑘+(𝑛) − 𝑁 (𝑛 + 1) 𝑘−(𝑛 + 1). (13) Turnbull and Fisher assumed that the transition between two states followed the curve of a typical energy barrier where they called the maximum energy the activated complex [17] as illustrated in figure 6.

Figure 6: Activation energy between two states. The maximum energy called the activated complex is used to derive the Boltzmann weighted forward and backward rate constants. Inspired by [8, 17].

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From this and with the Boltzmann weight they constructed expressions for the forward and backward rates as

𝑘+(𝑛) = 𝑂 (𝑛)𝛾 exp  −𝛿𝑊(𝑛) 2𝑘𝐵𝑇  𝑘−(𝑛 + 1) = 𝑂 (𝑛)𝛾 exp  𝛿𝑊(𝑛) 2𝑘𝐵𝑇  , (14)

where 𝑂 (𝑛) and 𝛾 represent the number of possible attachment sites and an atomic-/molecular jump frequency, the derivation of these expressions is found in appendix A. Assuming that the jump frequency of the atoms/molecules is that of the diffusion in the initial (parent) phase, the frequency can be expressed as

𝛾 = 6𝐷

𝜆2

, (15)

where 𝐷 and 𝜆 are the diffusion coefficient of the parent phase and the atomic/molecular jump distance. For spherical clusters the possible attachment sites is approximately 4𝑛2/3 and the forward and backward attachment rates can thus be written as

𝑘+(𝑛) = 4𝑛2/3 6𝐷 𝜆2 exp  −𝛿𝑊(𝑛) 2𝑘𝐵𝑇  𝑘−(𝑛 + 1) = 4𝑛2/36𝐷 𝜆2 exp  𝛿𝑊(𝑛) 2𝑘𝐵𝑇  , (16)

where 𝛿𝑊 (𝑛) the is work of formation needed to form a cluster of size (𝑛 + 1) given by

𝛿𝑊(𝑛) = 𝑊 (𝑛 + 1) − 𝑊 (𝑛) = Δ𝜇 + (36𝜋)1/3¯𝑣2/3 

(𝑛 + 1)2/3− 𝑛2/3𝜎 . (17) The total number of nucleus that has formed during a set anneal time is given by [8]

𝜒 = ∫ 𝑡

0

𝐼(𝑡)𝑑𝑡. (18)

For long annealing times it can be expressed as

𝜒(𝑡) = 𝐼𝑠𝑡(𝑡 − 𝜃), 𝑡  𝜃, (19)

where 𝐼𝑠𝑡

and 𝜃 corresponds to the steady state nucleation rate that will be discussed in section 3.2 and the induction time which is the time lag before any nucleus can be observed.

3.1

Coupled flux nucleation

For phase transformations in condensed matter the initial and the new phase does seldom have the same concentration and this is something that the classical model cannot properly describe. The stochastic fluctuation in the neighbourhood of the

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nuclei must be included and linked to the stochastic flux at the interface in order to account for the effects of long range diffusion. In figure 7 an illustration of the approach used to approximate the inclusion of long range diffusion first derived by Russell [18] and further developed by Kelton et al. [11] is shown. An extra shell is added to account for nearest neighbour diffusion and thus an extra dimension with the forward and backward rate constants 𝛼 and 𝛽 for the shell with the occupants 𝜌 are added to the system.

Figure 7: Schematic representation of the coupled flux model. Compared to the classical model an extra shell with occupants 𝜌 is added to account for long range diffusion which gives rise to the extra rate constants 𝛼 and 𝛽. Inspired by [11]

Using the same expression for the work of formation given by Eq. (4), the equilibrium distribution from Eq. (10) is expressed as

𝑁𝑒 𝑞(𝑛, 𝜌) = 𝑁0exp  −𝑊(𝑛) 𝑘𝐵𝑇  𝑃( 𝜌), (20)

where 𝑃( 𝜌) is the normalised probability for having 𝜌 solute atoms/molecules in the nearest neighbour shell. 𝑃( 𝜌) can be computed from the entropy change between an atom/molecule located in the parent phase or in the shell. The entropy is related to the number of configurations as

𝑆= 𝑘𝐵ln Ω, (21)

where Ω is the number of configurations. Both atoms in the shell and the parent phase will contribute to the entropy change which gives

Ω𝑠 = 𝜌𝑚 𝑛! 𝜌!( 𝜌𝑚 𝑛 − 𝜌)! (22) and Ω𝑝 = 𝑁𝑠! 𝑁0!(𝑁𝑠− 𝑁0)! , (23)

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where 𝜌𝑚

𝑛 and 𝑁𝑠 corresponds to the maximum number of atom/molecule sites in

the shell and the number of sites in parent phase. This gives the atomic fraction as 𝑁0/𝑁𝑠 = 𝑎𝑡% and for a spherical shell 𝜌

𝑚

𝑛 is approximately given as 4𝑛

2/3. In

synergy to Eq. (8) the probability of a fluctuation in the shell is then

𝑃( 𝜌) ∝ exp © ­ « Δ𝑆𝑠 ℎ𝑒𝑙 𝑙 0→𝜌 + Δ𝑆 𝑝 𝑎𝑟 𝑒𝑛𝑡 𝑝 ℎ𝑎 𝑠𝑒 𝑁0→𝑁0−𝜌 𝑘𝐵 ª ® ¬ . (24)

Using Stirling’s approximation, the above equation can be used to arrive at the expression 𝑃( 𝜌) = 𝛼(𝑛) 𝜌𝑚 𝑛! 𝜌!( 𝜌𝑚 𝑛 − 𝜌)!  𝑁0 𝑁𝑠− 𝑁0 𝜌 , (25)

where 𝛼(𝑛) corresponds to a normalisation constant. The normalisation constant has the conditionÍ𝜌𝑚

𝑛 𝜌=0𝑃( 𝜌) = 1 and is given by 𝛼(𝑛) = 1 Í𝜌𝑚 𝜌=0 𝜌𝑚𝑛! 𝜌!( 𝜌𝑚𝑛−𝜌)!  𝑁0 𝑁𝑠−𝑁0 𝜌. (26)

For the coupled flux model, assuming that the number of atoms/molecules in the clusters and their nearest neighbour shells change by the gain or loss of one atom/molecule at a time, the rate constants must be described within a two-domensional (𝑛,𝜌) space [19]. A schematic illustration of the (𝑛,𝜌) space and how the rate constants are connected is illustrated in figure 8.

Figure 8: The (𝑛,𝜌) space for the first 3 𝑛 and corresponding 𝜌𝑚

𝑛 which follows

approximately 4𝑛2/3. The connection between the different states are also illustrated here. Inspired by [19]

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From figure 8 the cluster distribution 𝑁 (𝑛, 𝜌) as a function of 𝑘+, 𝑘−, 𝛼 and 𝛽 connected to possible transition states can be represented as

. (27)

The master equation (12) is changed with the coupled flux model to give the time-dependent size distribution with the added shell attachment rates as

𝜕 𝑁(𝑛, 𝜌) 𝜕 𝑡 =𝛼(𝑛, 𝜌 −1) · 𝑁 (𝑛, 𝜌 − 1, 𝑡) − (𝛼(𝑛, 𝜌) + 𝛽(𝑛, 𝜌)) · 𝑁 (𝑛, 𝜌, 𝑡) + 𝛽(𝑛, 𝜌 + 1) · 𝑁 (𝑛, 𝜌 + 1, 𝑡) + 𝑘+(𝑛 − 1, 𝜌 + 1) · 𝑁 (𝑛 − 1, 𝜌 + 1, 𝑡) + 𝑘−(𝑛 + 1, 𝜌 − 1) · 𝑁 (𝑛 + 1, 𝜌 − 1, 𝑡) − 𝑘+(𝑛, 𝜌) + 𝑘−(𝑛, 𝜌) · 𝑁 (𝑛, 𝜌, 𝑡) . (28)

Assuming detailed balance from the constrained equilibrium hypothesis (i.e no net flux through clusters at equilibrium) gives the relation [19]

𝑘+(𝑛, 𝜌) 𝑁 (𝑛, 𝜌) = 𝑘−(𝑛 + 1, 𝜌 − 1)𝑁 (𝑛, 𝜌 − 1) (29) from which the forward and backward interfacial attachment rate constants now can be constructed as 𝑘+(𝑛, 𝜌) = 𝜌 6𝐷0 𝜆2 exp  − 𝛿𝑊𝑛 2𝑘𝐵𝑇  [𝐺 (𝑛, 𝜌)] 𝑘−(𝑛 + 1, 𝜌 − 1) = 𝜌 6𝐷0 𝜆2 exp  𝛿𝑊𝑛 2𝑘𝐵𝑇   1 𝐺(𝑛, 𝜌)  , (30)

where 𝐷0 is the interfacial diffusion coefficient. These are the same rates as those used in the classical model with the addition of a correction factor which takes the

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change of entropy in the shell into account. The correction factor is given by 𝐺(𝑛, 𝜌) =  𝛼𝑛 𝛼𝑛+1 𝜌𝑚 𝑛! 𝜌𝑚 𝑛+1! 1 𝜌 ( 𝜌𝑚 𝑛+1− 𝜌 + 1)! ( 𝜌𝑚 𝑛 − 𝜌)! ·  𝑁0 𝑁𝑠− 𝑁0  −1/2 (31) and the derivation of these constants are found in appendix B. To account for the added shell used in the model, the rates that describe the change of atoms/molecules in the shell of the clusters are needed. Setting up a detailed balance equation once again as

𝛼(𝑛, 𝜌 − 1)𝑁 (𝑛, 𝜌 − 1) = 𝛽(𝑛, 𝜌)𝑁 (𝑁, 𝜌) (32) and assuming that the rates are symmetric, the rate constants for attachment and detachment to the shell can be expressed as

𝛼(𝑛, 𝜌 − 1) = 𝜉 𝜌 𝐷 𝜆2  𝜌𝑚 𝑛 − 𝜌 + 1 𝜌 1/2 𝑁0 𝑁𝑠− 𝑁0 1/2 𝛽(𝑛, 𝜌) = 𝜉 𝜌 𝐷 𝜆2  𝜌𝑚 𝑛 − 𝜌 + 1 𝜌 −1/2 𝑁0 𝑁𝑠− 𝑁0 −1/2 , (33)

where 𝜉 is an added constant taking into account that the atoms/molecules do not immediately disperse into the parent phase when leaving the shell. For the derivation of 𝛼(𝑛, 𝜌 − 1) and 𝛽(𝑛, 𝜌) the reader is advised to follow the methodology used to derive the interfacial attachment rates shown in appendix B. For a dilute binary solution 𝜉 is approximated to 𝜉 = (4𝜋)2/3 2 (3¯𝑣) 1/3𝜆2(𝑁 𝑠𝐶∞) 1/2, (34)

where 𝐶∞ is the solute concentration far away from the cluster i.e 𝐶∞ = 𝑁0.

To evaluate the quantities obtained with the coupled flux model, effective results are obtained by summing quantities over 𝜌 for each 𝑛 [11]. The transient nucleation rate from Eq. (13) is thus changed to

< 𝐼(𝑛, 𝑡) >=

𝑚=𝜌𝑛𝑚

Õ

𝑚=0

[𝑘+(𝑛, 𝑚) 𝑁 (𝑛, 𝑚, 𝑡) − 𝑘−(𝑛 + 1, 𝑚 − 1)𝑁 (𝑛 + 1, 𝑚 − 1, 𝑡)]. (35)

To evaluate the new interfacial attachment rates, effective forward and backwards rates are expressed as

h𝑘±(𝑛)i = Í𝜌𝑚 𝑛 𝜌=0𝑃( 𝜌) 𝑘 ±(𝑛, 𝜌) Í𝜌𝑛𝑚 𝜌=0𝑃( 𝜌) . (36)

The average number of solute atoms/molecules in the neighbourhood of clusters of size 𝑛 is defined as h𝜌 (𝑛)i = Í𝜌𝑛𝑚 𝜌=0 𝜌 𝑁(𝑛, 𝜌) Í𝜌𝑚𝑛 𝜌=0 𝑁(𝑛, 𝜌) , (37)

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which gives the average concentration in the shell as h𝐶 (𝑛)i = h𝜌 (𝑛)i

(4𝜋)1/3(3𝑛¯𝑣2/3𝜆). (38)

This expression predict higher concentrations near subcritical clusters while lower concentrations in the neighbourhood of large clusters as expected from diffusion limited growth.

3.2

Steady state nucleation

The transient distribution will not converge to the equilibrium distribution but rather find a lower steady state distribution [8]. The steady state distribution gives 𝑁(𝑛, 𝜌) → 0 when 𝑛 → ∞ rather than the equilibrium distribution which gives 𝑁(𝑛, 𝜌) → ∞ when 𝑛 → ∞ which makes no physical sense. With this in considera-tion the steady state distribuconsidera-tion can be described as

𝑁𝑠𝑡(𝑛, 𝜌) → 𝑁𝑒 𝑞(𝑛, 𝜌) 𝑎𝑠 𝑛 → 0 𝑁𝑠𝑡(𝑛, 𝜌) → 0 𝑎𝑠 𝑛 → ∞

(39) As mentioned earlier, assuming an equilibrium distribution means that no net flux is exchanged between states. From this, analytical expressions for both the induction time and the steady-state nucleation rate at the critical cluster size have been de-rived. An analytical approximation of the time-independent steaty-state nucleation rate derived by Kelton et al. used for the classical nucleation model is given by

𝐼𝑠𝑡 = 𝑁𝑒 𝑞(𝑛∗) 𝑘+(𝑛∗) 𝑍, (40) where 𝑍 is the the Zeldovich factor which describe the probability for a cluster to continue grow past the critical size instead of shrink. For spherical clusters it is expressed as 𝑍 =  |𝛿𝜇| 6𝜋𝑘𝐵𝑇 𝑛∗ 1/2 . (41)

A popular analytical expression for the induction time in the classical model have been derived by Kaschiev and is given by

𝜃𝑘(𝑛 ∗

) = 2

3𝜋𝑘+(𝑛) 𝑍2. (42)

Kelton et al. have derived analytical approximations for the coupled flux model based on the classical model [19]. The expressions from the classical theory are scaled with the lowest shell and forward interfacial attachment rates which give the time-independent steady-state nucleation rate as

𝐼𝑠𝑡 𝑙 𝑓 ≈ 𝛼(𝑛∗,0) 𝑘+(𝑛∗,1) · 𝐼𝑠𝑡 (43)

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and the induction time as

𝜃𝑙 𝑓(𝑛∗) ≈

𝑘+(𝑛∗,1) 𝛼(𝑛∗,0)

· 𝜃 (𝑛∗). (44)

This is done because the rates are dominant at low 𝜌 for low concentrations. This means that for a dilute case, 𝑘+(𝑛, 1) ≈ 𝑘+(𝑛∗) and thus using Eq. (40) and (42) in the above corresponding equations yield

𝐼𝑠𝑡 𝑙 𝑓 = 𝛼(𝑛 ∗ ,0) 𝑍 𝑁𝑒 𝑞(𝑛 ∗) (45) and 𝜃𝑙 𝑓(𝑛∗) = 2 3𝜋𝑍2 1 𝛼(𝑛∗,0) . (46)

These expressions where first introduced by Russell but have been further developed by Kelton et al. because of the previous lacking agreement with the computed transient values.

3.3

Growth of stable nucleus

To evaluate the growth of stable nuclei and by this be able to construct TTT-diagrams, expressions are needed to evaluate the clusters that are formed and what volume these clusters occupy. A model used by Ericsson et al. [3] uses a Lagrange-like approach for growth of nuclei as

𝑟(𝑡 + 𝛿𝑡) = 𝑟 (𝑡) + 𝑢(𝑟)𝛿𝑡, (47)

where 𝑢(𝑟) and 𝛿𝑡 corresponds to the growth rate and the incremental time step. The growth rate is given by

𝑢(𝑟) = 𝜕𝑟 𝜕 𝑡 = 16𝐷 𝜆2  3¯𝑣 4𝜋 1/3 sinh  ¯ 𝑣 2𝑘𝐵𝑇  Δ𝐺𝑣− 2𝜎 𝑟   , (48)

where Δ𝐺𝑣 corresponds to the molar volume change in Gibbs free energy, Δ𝐺𝑣 =

Δ𝐺/𝑉𝑚. Using the Johnson-Mehl-Avrami-Kolmogroc (JMAK) equation the

crys-tallised volume fraction can be expressed as

𝑥(𝑡𝑚) = 1 − exp − 4𝜋 3 𝑚 Õ 𝑖=1 𝜒(𝑡𝑖)𝑟 (𝑡𝑖) 3 ! , (49)

where 𝜒(𝑡𝑖) and 𝑟 (𝑡𝑖) corresponds to the number of nuclei produced and the radius

at time 𝑡𝑖. To account for the probability of clusters dissolving as discussed around

figure 4, the cluster size used for evaluation of particle growth from stable nuclei is taken such as 𝑛𝑒𝑣 𝑎𝑙 = 𝑛 ∗+ 1 2 r 𝑘𝐵𝑇 𝜋 𝜎 . (50)

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If the solution is stepped in time this means that Eq. (18) can be approximated as 𝜒(𝑡𝑖) = 𝐼 (𝑛𝑒𝑣 𝑎𝑙, 𝑡𝑖)𝛿𝑡. (51) As the highest growth velocities occur at the steady-state nucleation rate, expres-sions for the growth using the steady-state rates and maximum growth rates have been derived by taking 𝑟 → ∞ in Eq. (48) and reformulating Eq. (49). This gives an expression for the time corresponding to the crystallised volume fraction as

𝑡(𝑥) =  − 3 𝜋 𝐼𝑠𝑡𝑢3 𝑚 𝑎𝑥 ln(1 − 𝑥) 1/4 , (52)

where the maximum growth rate is given by

𝑢𝑚 𝑎𝑥 = 16𝐷 𝜆2  3¯𝑣 4𝜋 1/3 𝑠𝑖𝑛 ℎ  ¯𝑣Δ𝐺𝑣 2𝑘𝐵𝑇  . (53)

Another model that can be used to describe the growth is based on mass balance and used by Fisk et al. [20]. By taking the number solute concentration far from the nuclei, 𝐶∞, to be a function of stable nuclei formed, the time-dependent parent

phase concentration can be expressed as

𝐶(𝑡) = 𝐶0− 𝑓 𝐶𝑝, (54)

where 𝐶𝑝 and 𝑓 corresponds to the solute concentration in the precipitate and the

volume fraction of formed stable nuclei. For spherical particles 𝑓 is calculated as

𝑓(𝑡𝑚) = 4𝜋 3 𝑚 Õ 𝑖=1 𝜒(𝑡𝑖)𝑟 (𝑡𝑖) 3 . (55)

By taking the capillary effect into account the growth rate can be expressed as 𝜕𝑟 𝜕 𝑡 = 𝐷 𝑟 𝐶(𝑡) − 𝐶𝑒 𝑞 𝐶𝑝− 𝐶𝑟  1 + 𝑟 ∗ 𝑟  , (56)

where 𝐶𝑒 𝑞 and 𝐶𝑟 corresponds to the equilibrium concentration and the

concentra-tion at the curved interface [12]. Further 𝐶𝑟 can be approximated as

𝐶𝑟 = 𝐶𝑒 𝑞exp 2𝜎¯ 𝑣 𝑘𝐵𝑇 𝑟  . (57)

3.4

Numerical implementation

The set of coupled differential equations given by Eq. (35) can be solved using a matrix formulation as [19]

𝑑𝑵 𝑑 𝑡

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where C and N corresponds to the rate coefficient matrix and the cluster distribu-tion. This can be done because only the distribution 𝑁 (𝑛, 𝜌, 𝑡) is time-dependent while the rate constants are not. The most straight forward approach is to solve the system using an explicit finite difference formulation as

𝑁(𝑛, 𝜌, 𝑡 + 𝛿𝑡) = 𝑁 (𝑛, 𝜌, 𝑡) +

𝜕 𝑁(𝑛, 𝜌, 𝑡) 𝜕 𝑡

· 𝛿𝑡, (59)

but the rapid changes in the population makes the problem stiff and thus other ap-proaches must be taken if sufficiently small time steps are wanted. By expressing the derivatives in terms of the new distributions the transient solution can be expressed implicit as

𝑁(𝑛, 𝜌, 𝑡 + 𝛿𝑡) =

𝑁(𝑛, 𝜌, 𝑡)

1 − C𝛿𝑡 . (60)

This expression remains stable even if the time step is large and thus a big amount of computation time is saved. By accessing C as B = 1 − C𝛿𝑡, a better conditioned matrix is obtained which further increase stability.

To limit the grid in 𝑛-space a lower limit ˜𝑢 is set to unity and an upper limit ˜𝑣 is set to at least 𝑛𝑒𝑣 𝑎𝑙 + 20 to be able to evaluate cluster sizes without interference

of simulation boundaries. With this the time-dependent cluster size distribution 𝑁(𝑛, 𝜌, 𝑡) in matrix form is expressed as

𝑁(𝑛, 𝜌, 𝑡) = [𝑁 (1, 0, 𝑡), 𝑁 (1, 1, 𝑡), 𝑁 (1, 2, 𝑡), 𝑁 (1, 3, 𝑡), 𝑁 (1, 𝜌𝑚 ˜ 𝑢, 𝑡), 𝑁(2, 0, 𝑡), 𝑁 (2, 1, 𝑡), 𝑁 (2, 2, 𝑡), 𝑁 (2, 3, 𝑡), · · · , 𝑁 (𝑛, 𝜌𝑚 ˜ 𝑣, 𝑡)] 𝑇 . (61)

The restricted (𝑛, 𝜌) space in figure 8 and the corresponding transitions between states from Eq. (27) give rise to the boundary conditions and together with the above expression for 𝑁 (𝑛, 𝜌, 𝑡), the rate coefficient matrix takes the form

𝐶𝑖, 𝑗 =                        𝐶1,1 𝐶1,2 0 0 0 0 0 0 0 · · · 𝐶1,2 𝐶2,2 𝐶2,3 0 0 𝐶2,6 0 0 0 · · · 0 𝐶3,2 𝐶3,3 𝐶3,4 0 0 𝐶3,7 0 0 · · · 0 0 𝐶4,3 𝐶4,4 𝐶4,5 0 0 𝐶4,8 0 · · · 0 0 0 𝐶5,4 𝐶5,5 0 0 0 𝐶5,9 · · · 0 𝐶6,2 0 0 0 𝐶6,6 𝐶6,7 0 0 · · · 0 0 𝐶7,3 0 0 𝐶7,6 𝐶7,7 𝐶7,8 0 · · · 0 0 0 𝐶8,4 0 0 𝐶8,7 𝐶8,8 𝐶8,9 · · · 0 0 0 0 𝐶9,5 0 0 𝐶9,8 𝐶9,9 · · · 0 0 0 0 0 0 0 0 𝐶10,9 · · · . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .                        . (62)

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distri-bution is taken as a monomer distridistri-bution which gives 𝑁(𝑛, 𝜌, 0) =          𝑁𝑒 𝑞(𝑛, 0) if 𝑛 = 1 and 𝜌 = 0 0 if 𝑛 = 1 and 𝜌 ≠ 0 0 if 𝑛 ≠ 1 . (63)

The computational cost is not considered to be more then moderate for small 𝑛∗ but quickly grows to memory demands over that what a personal computer can produce because of the rapidly growing coarse C matrix which size strongly depend on 𝜌𝑚

𝑛.

4

Result and discussion

The results from a validation of the implemented numerical coupled flux model is presented here followed by the results obtained from the study on how the limiting solute concentration impact the nucleation in AMZ4. The simulated results are obtained using Matlab scripts [21].

4.1

Material parameters: model validation

To validate the numerical nucleation model, parameters used in previous work by Kelton et al. [10] are used. These parameters are used as insufficient information on the parameters used in the article describing the behaviour of the model [19] makes the reproduction of the specific results impossible. However, the author is the same in both studies and the same model have been used which means that the parameters used in the simulations should work well to validate the implemented numerical model. The parameters are simplified and the purpose of previous work have been to investigate how the coupled flux model captures the impact of solute concentration during different anneals.

The bulk diffusivity used is given by

𝐷 =1.3 · 10−5exp −2.53eV 𝑘𝐵𝑇



. (64)

In the previous study, the difference in the interfacial and bulk diffusion coefficient have not been accounted for and they have been chosen to be the same, i.e 𝐷0= 𝐷. Using an ideal solution model, the Gibbs free energy difference per atom/molecule is approximated as the difference between oxygen in SiO2precipitates and dissolved

oxygen in a Si crystal and is expressed as Δ𝜇 = −𝑘𝐵𝑇ln  𝐶 𝐶𝑒 𝑞  , (65) where 𝐶𝑒 𝑞 is given by 𝐶𝑒 𝑞 =2.21 · 1027exp −1.03eV 𝑘𝐵𝑇  . (66)

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The interfacial energy is taken from a linear fit to measured data at temperatures between 650°C and 800°C and can be expressed as

𝜎 =2 · 10−4+ 26.61 · 10−2𝑇 [J/m2]. (67) Other parameters that are used during the simulations are given below in table 1.

Table 1: Parameters used during the validation of the numerical coupled flux model.

Parameter Symbol Value Unit

Average atomic volume ¯𝑣 3.45· 10−29 [m3] Atomic jump distance 𝜆 2.15· 10−10 [m] Atomic concentration - 2.45·10−5 [at%] Initial concentration 𝑁0 7.10· 1023 [1/m3]

Temperature 𝑇 600 [°C ]

4.2

Coupled flux model validation

To emphasise the impact of taking the long range diffusion into account in the coupled flux model, comparisons with the classical model are presented together with comparisons to the study done by Kelton et al. [19].

One of the main features of the coupled flux model is that the rate constants are concentration dependent which makes the nucleation process more inert with lower concentrations compared to the classical model. The forward and backward interfacial attachment rate constants from the classical model given by Eq. (16) and the effective rate constants from the coupled flux model given by Eq. (36) are shown in figure 9(a) where the concentration dependence is clearly seen. The behaviour of the computed rates are in agreement with [19] but the magnitudes differ.

Another of the main features is highlighted by using Eq. (38) from which figure 9(b) is obtained. With the added shell the coupled flux model captures a concentra-tion profile that the classical model cannot describe. A higher concentraconcentra-tion than the average is observed at subcritical clusters while the concentration is lower than the average around stable nuclei. This behaviour is also observed in [19] but the magnitude of the concentration saturation differ.

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(a) (b)

Figure 9: (a) Forward and backward interfacial attachment rates from the classical model (dashed red and blue lines respectively) compared with the effective forward and backward interfacial attachment rates from the coupled flux model (solid red and blue lines respectively). (b) Average concentration profile of the nearest neighbour shell computed from the transient cluster distribution.

In figure 10 the effective transient solution < 𝑁 (𝑛) > from the coupled flux simulation is shown together with the transient solution from the classical model simulation. As seen, the distributions almost match but the time required to reach steady state vary by several orders of magnitude. This clearly demonstrate the importance of the concentration dependence which is missing in the classical model.

(a) (b)

Figure 10: The transient solution captured at different times during the simulation of the cluster distribution computed using (a) the classical model and (b) the effective distribution using the coupled flux model. For both models the solution converges towards a steady-state cluster distribution. The equilibrium distribution, the critical cluster size and the cluster size used for evaluation of stable nuclei are also shown.

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In figure 11 the effective cluster population, rate of change of cluster population and the nucleation rate are shown for some different values of 𝑛. The behaviour of the quantities resemble that from the evaluation done by Kelton et al. [19] but an overshoot in the nucleation rate for clusters smaller than 𝑛∗ is present here which does not occur in [19]. The missing agreement in the behaviour could be linked to the specific parameters used in the article and the overshoot is typically something observed in the classical model for clusters smaller than the critical size [8].

Figure 11: From top: Effective cluster population, rate of change of cluster pop-ulation and nucleation rate. Blue, red, yellow, magenta and green corresponds to 𝑛=6, 9, 12, 17, 23, 𝑛∗ =8. In the first figure the cluster sizes increases with time to their steady state values. From the second figure a decreasing peak is observed in the rate of change for each cluster size. The last figure shows a convergence towards a common steady-state nucleation rate for all cluster sizes.

The transient nucleation rate computed from Eq. (13) and (35) evaluated at 𝑛𝑒𝑣 𝑎𝑙 versus the analytical expressions for the steady-state nucleation rate given by Eq. (40) and (45) are shown in figure 12. For the case studied here the coupled flux model predict a steady-state nucleation rate of several orders of magnitude less than that computed with the classical model which should be the case in diffusion limited nucleation.

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(a) (b)

Figure 12: Transient nucleation rate evaluated at 𝑛𝑒𝑣 𝑎𝑙 and the analytical computed

value of the steady-state nucleation rate using (a) the classical model and (b) the coupled flux model.

The coupled flux model should also give higher induction times. The number of nuclei formed computed with Eq. (51) from the two models are shown in figure 13 where the induction time have been computed from the slope of the population curve at the transient time which corresponds to the time where the nucleation rate reaches steady-state. The induction time corresponds to the time at which the linear slope is equal to zero. The coupled flux model give induction times orders of magnitude larger compared to the classical model which is as expected in accordance to the transient solutions shown in figure 10.

(a) (b)

Figure 13: Numerical computed population curve evaluated at 𝑛𝑒𝑣 𝑎𝑙 where the

in-duction time have been calculated from a linear fit to the population curve. (a) The classical model and (b) the coupled flux model.

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To emphasise the difference in the models the values obtained from both the classical model and the coupled flux model using the transient evaluations and the analytical expressions from Eq. (40), (42), (45) and (46) are presented below in table 2.

Table 2: Transient and analytic steady-state rates and induction times for SiO2

precipitation in Si using the classical model and the coupled flux model.

Model 𝐼𝑠𝑡 𝐴𝑛𝑎𝑙 𝑦𝑡𝑖 𝑐 [m 3/𝑠] 𝐼𝑠𝑡 𝑇 𝑟 𝑎𝑛𝑠𝑖 𝑒𝑛𝑡 [m 3/𝑠] 𝜃𝑛∗ 𝐴𝑛𝑎𝑙 𝑦𝑡𝑖 𝑐 [s] 𝜃𝑛𝑒 𝑣 𝑎𝑙 𝑇 𝑟 𝑎𝑛𝑠𝑖 𝑒𝑛𝑡 [s] CNT 7.18· 1014 6.72· 1014 7.06· 10−2 3.33· 10−2 Coupled flux 1.16· 109 1.14· 109 4.24· 104 6.73· 104

It is clearly seen that the coupled flux model captures the behaviour of diffusion limited nucleation in a way that the classical model cannot describe by lowering the steady-state nucleation rate and raising the induction time. The model also captures the expected concentration profile around subcritical clusters in diffusion limited nucleation event. From this the use of the implemented coupled flux model is validated to use further for investigation of the nucleation behaviour in AMZ4.

4.3

Material parameters: AMZ4

The parameters that are used are the same as the ones used in previous work done on SLM processing of AMZ4 by Ericsson et al. [3]. The interfacial energy have been chosen such as to fit a TTT-diagram of AMZ4 using the classical model with no connection to concentration dependence. Further, the phase separation energies used are those between the supercooled liquid and crystalline AMZ4 and do not connect specifically to oxygen rich precipitates. Using the coupled flux model with these parameters will result in a convergence towards the result obtained with the classical model with higher solute concentration but the connection to oxygen cannot be captured. The temperature interval is taken such as 𝑇𝑔 < 𝑇 < 𝑇𝑚 to use the

parameters in the vicinity of their validity.

The effective bulk diffusion comes from an Einstein-Stokes relation which de-scribes diffusion of particles in a liquid and can be expressed as

𝐷 = 𝑘𝐵𝑇

3𝜋𝜆𝜂, (68)

where 𝜂 is the viscosity. Once again the interfacial diffusion coefficient is taken to be the same as the bulk diffusion, i.e 𝐷0 = 𝐷, because of lack of data. This will however affect the results as it is an important parameter and with oxygen dependent experimental data this would be an interesting parameter to further investigate. The relation in Eq. (68) have shown to not represent diffusion in glass and supercooled liquids well because of the drastic change in viscosity. Kelton et al. have developed a temperature dependent viscosity expression for capturing diffusion in glasses that

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requires high cooling rates such as Ti- and Zr-based systems [22]. The Blodgett-Egami-Nussunow-Kelton (BENK) expression for the viscosity is expressed as

𝜂 = 𝜂0exp  𝐸(𝑇 ) 𝑇  , (69)

where 𝜂0 and 𝐸 (𝑇 ) represents the viscosity at infinite temperature and the free

energy barrier (expressed in Kelvin) given by

𝐸(𝑇 ) = 𝐸+ 𝑇𝐴(𝑞𝑇𝑟)𝑧Φ(𝑇𝐴− 𝑇 ), (70) where 𝑞 and 𝑧, 𝐸∞ represents universal fitting parameters and a fitting parameter

for the energy barrier approximated as 𝐸∞ = 6.466𝑇𝐴. Further, 𝑇𝐴, 𝑇𝑟 and Φ( 𝑋)

corresponds to a universal scaling temperature approximated as 𝑇𝐴 = 2.02𝑇𝑔, a

reduced temperature and a heavy side step function.

The chemical potential per atom is derived from measurements of the heat capacity of liquid and crystal AMZ4 and is expressed as

𝛿 𝜇= −Δ𝐺

𝑚(𝑇 )

𝑁𝐴

, (71)

where Δ𝐺𝑚(𝑇 ) and 𝑁𝐴 corresponds to the change in Gibbs free energy per mole and

Avogadro’s constant. Δ𝐺𝑚(𝑇 ) is expressed the usual way as

Δ𝐺𝑚(𝑇 ) = Δ𝐻𝑚(𝑇 ) − 𝑇 Δ𝑆𝑚(𝑇 ), (72)

where Δ𝐻𝑚(𝑇 ) and Δ𝑆𝑚(𝑇 ) corresponds to the enthalpy and entropy of fusion per

mole which are obtained by fitting data with the Kubaschewski equations derived by Heinrich et al. These expressions take the form of

Δ𝐻𝑚(𝑇 ) = Δ𝐻𝑓 − 𝑎− 𝑐 2  𝑇2 𝑚 − 𝑇 2 + 𝑏  𝑇−1 , − 𝑇 −1 + 𝑑 3  𝑇3 𝑚 − 𝑇 3 Δ𝑆𝑚(𝑇 ) = Δ𝑆𝑓 − (𝑎 − 𝑐) (𝑇𝑚− 𝑇 ) + 𝑏 2  𝑇−2 , − 𝑇 −2+ 𝑑 2  𝑇2 𝑚 − 𝑇 2, (73)

where 𝑎, 𝑏, 𝑐 and 𝑑 corresponds to fitting parameters.

The expression used for the interfacial energy is denoted as the Mondal-Kumar-Gupta-Murty (MKGM) model [23] which gives an increase in the interfacial energy at temperatures below 𝑇𝑚. This is unusual but earlier studies have shown the same

behaviour in Cu-Zr based systems, for further discussion on this the reader is refereed to [3]. The model assumes the interfacial region as a monolayer and the nucleus as a hard sphere, a reduced analytical expression is given by

𝜎(𝑇 ) =  𝑃 𝜌𝑝𝜆 3Δ𝐺𝑣(𝑇 ) 32𝜋  Δ𝐺𝑚 𝑑(𝑇 ) − Δ𝐺 𝐿 𝑑(𝑇 ) 1/2 , (74)

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where 𝑃, 𝜌𝑝, Δ𝐺 𝑚

𝑑(𝑇 ) and Δ𝐺 𝐿

𝑑(𝑇 ) corresponds to the packing factor, the planar

atomic density, the activation energy for diffusion in the monolayer and the activa-tion energy in the liquid. Further, assuming an Arrhenius temperature dependence Δ𝐺𝑚 𝑑(𝑇 ) and Δ𝐺 𝑙 𝑑(𝑇 ) are estimated as Δ𝐺𝑚 𝑑(𝑇 ) = 𝑘𝐵𝑇ln 3𝜋𝑣𝜆3𝜂𝑇 𝑔 𝑘𝐵𝑇𝑔 ! Δ𝐺𝐿 𝑑(𝑇 ) = 𝑘𝐵𝑇ln  3𝜋𝑣𝜆3𝜂(𝑇 ) 𝑘𝐵𝑇  , (75)

where 𝑣 and 𝜂𝑇 𝑔 corresponds to the atomic vibration frequency and the viscosity at

𝑇𝑔.

Because of the lack composition dependent Gibbs free energy, the value of 𝐶𝑒 𝑞

used in the growth model from Eq. (56) is chosen bluntly from the knowledge that it should be small in comparison to the initial concentration. Studies on AMZ4 have shown that during AM processing with SLM, crystalline zones with the composition Zr4Cu2O forms [7] which have been used as the precipitate concentration. This

study also suggests that these precipitates grow to a cubical shape.

The parameters used in Eq. (56), (69), (70), (73) and (74) are presented in table 3 together with the other required parameters used during the simulations.

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Table 3: Parameters and fitting parameters used during the simulations of the nucleation in AMZ4.

General parameters Symbol Value Unit

Molar volume 𝑉𝑚 11.536· 10−6 [m3/mol]

Average atomic volume 𝑣¯ 1.892· 10−29 [m3]

Atomic jump distance 𝜆 3.306· 10−10 [m]

Atomic concentration - 10−1,10−2,10−3,10−4 [at%]

Transient temperature 𝑇 400,450,500,550 [°C ]

Steady-state temperature 𝑇 400:50:850 [°C ]

Diffusivity parameters

Glass transition temperature 𝑇𝑔 667 [ K ]

High temperature viscosity 𝜂0 4.7167· 10−5 [ Pas ] Chemical potential fitting parameters

Entalpy of fusion Δ𝐻𝑓 8.578· 103 [ J/mol ]

Entropy of fusion Δ𝑆𝑓 7.131 [ J/mol ]

Melting temperature 𝑇𝑚 1203 [ K ]

Fit constant a 5.224· 10−4 [ J/(molK2)]

Fit constant b 1.031· 107 [ J/mol ]

Fit constant c 6.230· 10−3 [ J/(molK2)]

Fit constant d -6.047· 10−7 [ J/(molK3)]

Interfacial energy parameters

P/𝜌𝑝 𝛼 1.28 [-]

Viscosity at 𝑇𝑔 𝜂𝑇 𝑔 1.5· 10

16 [Pas]

Atomic vibration frequency 𝑣 1013 [1/s]

Growth from Mass conservation

Equilibrium concentration 𝐶𝑒 𝑞 10−6 [%at]

Oxygen cluster concentration 𝐶𝑝 14.3 [%at]

4.4

Solute concentration impact on nucleation in AMZ4

The parameters used for AMZ4 give substantially larger critical cluster sizes com-pared to the simulations of the nucleation in Si. As temperatures increase this result in a collapse of Eq. (31) as the factorials goes towards infinity with the large values of the corresponding 𝜌𝑚

𝑛. This has set the limit of the temperature investigation

to 550°C for the transient simulations. Fortunately the expressions used for the steady-state nucleation rate and the induction time given by Eq. (45) and (46) are not dependent on the interfacial attachment rates and these have been used to investigate the solute concentration at higher temperatures.

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not only in 𝑛 but also in 𝜌, this effect is more dominant the higher the concentration of solute present as seen in figure 14. It is also seen how this phenomenon makes it favourable for faster growth as the overall wider distribution shows a higher density of clusters with saturated shells for the high concentration distribution whereas the low concentration makes clusters to grow mostly with few shell occupants.

(a)

(b)

Figure 14: Transient distribution 𝑁 (𝑛, 𝜌) captured at different times for two different concentrations, a) 𝐶∞ = 0.001%at and b) 𝐶∞ = 0.1%at. The temperature during

both simulations is set to 500°C.

To further visualise the transient behaviour for the different concentrations simulated in figure 14, contour plots are shown in figure 15 where it is seen how the distribution grows in 𝑛 as the shells get filled. In figure 15(a) it is seen how the peak in the cluster population for the low concentration has close to no shell occupants while the higher concentration in 15(b) shows that the peak contains more shell occupants the larger the cluster is.

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(a) (b)

Figure 15: Contour plots of the transient distributions shown in figure 14, (a) cor-responds to 𝐶∞ = 0.001%at and (b) 𝐶∞ = 0.1%at. To highlight the event a lower

limit is set to h𝑁𝑒 𝑞(𝑛

)i · 10−8.

Taking a cross section of the transient distribution at 𝑛∗ for the different simu-lation temperatures and concentrations that have been investigated shows how the peak of the distribution shifts towards clusters with more occupants in the shell with higher concentrations. Lower temperatures yield a higher density together with a smaller width of the distribution, for low concentrations this width effect is negligible though.

Figure 16: Cross sections of the steady-state transient distributions evaluated at 𝑛∗, magenta, yellow, red and blue corresponds to the temperatures 550°C, 500°C, 450°C and 400°C.

Using the transient computed values for the steady-state nucleation rate ob-tained from Eq. (35) and the computed values from the analytical expression given by Eq. (45), the concentration dependent nucleation rates as functions of temper-ature are presented in figure 17(a). To account for the critical region the transient solution is evaluated at 𝑛𝑒𝑣 𝑎𝑙 as the steady-state nucleation rate should be the same

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for all cluster sizes. It is clearly seen that the concentration of solute impact the steady-state nucleation rate by orders of magnitude but follows a similar trend in terms of temperature dependence.

The concentration dependent induction times as functions of temperature from both the analytical expression given by Eq. (46) and the values computed from the linear fit to the population curve at the transient time obtained with Eq. (51) are presented in figure 17(b). The induction time depend on the evaluated cluster size and thus the transient evaluation is done at the critical cluster size. As for the nucleation rate, a clear dependence of the solute concentration is observed as the induction times grow with lower concentration and a similar trend is observed in terms of temperature dependence.

(a) (b)

Figure 17: (a) Steady-state nucleation rates from analytical expression (blue) and transient evaluation evaluated at 𝑛𝑒𝑣 𝑎𝑙 (red). (b) Induction time from analytical

expression (blue) and transient evaluation evaluated at 𝑛∗ (red).

As seen in figure 17 the computed values from the analytical expressions mostly show good agreement with the transient values. A slight offset is observed though, this is more prominent in the result from the steady-state nucleation rate but the inverse offset is also observed for the induction times. The good agreement motivates the analytical expressions given by Eq. (45) and (46) to be used to examine the whole temperature interval. These results are shown in figure 18 where at high temperatures, a rapid decrease in the steady-state nucleation rate but a convergence of the induction times are observed. The different concentrations give an offset and for the steady-state nucleation rate this have a stronger impact at lower temperatures while the reverse is observed for the induction time.

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(a) (b)

Figure 18: Analytical evaluation at 𝑛∗ of concentration dependent (a) steady-state nucleation rate and (b) induction time. The concentration gives an offset in the computed values but the trends are the same.

4.4.1 TTT diagrams for AMZ4

The crystallisation can be evaluated by taking different amount of transient be-haviour into consideration. All of the following simulations have been evaluated at the time at which 5% of the volume have crystallised.

Using the expression for the time corresponding to a set crystallised volume fraction from Eq. (52) and the maximum growth rate given by Eq. (53) together with the steady-state nucleation rate computed from Eq. (45) concentration depen-dent steady-state TTT-diagrams are obtained which are shown in figure 19(a). A clear dependence of the limiting solute concentration is seen and the required critical cooling rate needed to form an amorphous alloy is increased as the limiting solute is increased. This is what is predicted by the coupled flux model as the concentration of solute atoms should limit the rate at which clusters grow and by this the time it takes to form stable nuclei. The resulting crystallisation curves resembles more and more those obtained in earlier work as the concentration of limiting solute is increased [3, 6].

Semi-steady-state crystallisation curves can be obtained by evaluating the crys-tallisation using Eq. (49) with the analytical expression for the steady-state nucle-ation rate given by Eq. (45) but with the transient expression for the growth from Eq. (48). This enables transient growth behaviour to be evaluated for the temper-atures that the transient model cannot simulate. The fully transient description for the allowed temperature interval is obtained with the nucleation rate given by Eq. (35) together with the growth rate from Eq. (48) evaluated with the expression for the crystallisation from Eq. (49). The results from these two models are seen in figure 19(b) where it can be seen that the behaviour of the semi-steady-state TTT-diagrams seems to lie in between the behaviour predicted from the steady-state and

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transient crystallisation curves. The temperature at the ”𝑛𝑜𝑠𝑒” of the diagrams have also decreased compared to figure 19(a).

(a) (b)

Figure 19: Concentration dependent TTT diagrams created using (a) analytical ex-pressions for the steady-state nucleation rate and the maximum growth rate and (b), analytical expressions for the steady-state nucleation rate and transient evaluation of the growth rate (solid lines) and fully transient evaluation (dashed black lines).

With the growth model expressed by Eq. (56) a crystallisation from diffu-sion controlled mass balance can be simulated. Using the steady-state nucleation rate computed with Eq. (45) and evaluating the crystallisation with Eq. (49), semi-steady-state TTT-diagrams derived from transient mass conservation can be constructed. Together with the fully transient crystallisation curves computed using the nucleation rate from Eq. (35) and the growth model given by Eq. (56) evaluated with the crystallisation from Eq. (49) the results in figure 20 are obtained.

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Figure 20: Concentration dependent TTT diagrams created using analytical expres-sions for the steady-state nucleation rate and a transient formulation based on mass conservation for the growth rate (solid lines) and fully transient descriptions (dashed black lines).

As seen in the figure the predicted crystallisation curves differ vastly from those predicted in figure 19, no crystallisation is observed at all for concentrations lower than 𝐶∞ = 0.01. This is believed to be caused by the parameters used during the

simulations as the interfacial energy in the model have been fitted to TTT-diagrams using Eq. (48) with the classical model which have no connection to limiting solute concentration. Further, this growth model is the only one that directly takes any concentration dependence into account which should affect the result in a more severe way. What can be seen however is that the temperature at the ”𝑛𝑜𝑠𝑒” of the diagrams have further decreased compared to figure 19 and a much stronger connection to the limiting solute concentration is observed.

4.4.2 Quenched in distributions

By comparing the transient nucleation rate computed using Eq. (35) at a reference temperature with an initial distribution as a steady-state distribution form both a higher and lower temperature anneal against an initial monomer distribution the behaviour in figure 21 is obtained. From an AM perspective this is of interest as it can be seen as a reheating of a previously annealed zone. A sharp increase in the nucleation rate is observed with the initial distribution from a lower temperature while the initial distribution from a higher temperature gives a nucleation rate very close to the initial monomer distribution. The reason for this is that a lower temper-ature gives a higher driving free energy that result in a higher population of clusters compared to a higher temperature which gives a lower population. As these distri-butions are quenched in as initial distridistri-butions a rapid increase in nucleation will be observed from the larger quenched in population, with time this extra contribution

Figure

Figure 1: A typical TTT-diagram. The ”
Figure 2: Approximate Ashby chart with the addition of the new material class metallic glasses
Figure 3: Time evolution of solute concentration during a phase transformation according to LaMer’s mechanism
Figure 5: Histogram of the reverisble cluster population as a function of cluster size
+7

References

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