Modelling of precipitation in an Fe-C-Cr alloy

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STOCKHOLM SVERIGE 2019,

Modelling of precipitation in an Fe-C-Cr alloy

JOHAN FRISK

KTH

SKOLAN FÖR INDUSTRIELL TEKNIK OCH MANAGEMENT

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was to provide validation and further development of the model developed by Bonvalet et al (Acta Materialia 100, 2015, p. 169–177). The model was implemented in a Python code, which utilizes the TC-Python SDK from Thermo-Calc Software. Comparisons between the present work implementation and the commercial software module TC-PRISMA were carried out for precipitation simulations in a Fe–0.16 wt% C–4.0 wt% Cr alloy. Moreover, a systematic study of the influence of the interfacial energy and the number of nucleation sites on precipitation kinetics was performed. The results indicate that the present work model and TC-PRISMA calculate growth and coarsening of precipitates in a similar manner, but the models differ in how nucleation-related parameters are treated. Most significantly, the two models calculate the driving force in different ways. This causes the precipitation kinetics to be shifted in time between the two models.

Sammanfattning

Modellering av utskiljningsprocesser inom material är av både akademiskt och industriellt intresse. Verktyg och programvaror som modellerar utskiljning kan användas inom ett ICME- ramverk (från engelska: integrated computational materials engineering) för att designa material med önskvärda egenskaper. Målet med detta examensarbete är att bepröva och fortsatt utveckla en urskiljningsmodell som presenterats av Bonvalet et al (Acta Materialia 100, 2015, s. 169–177). Modellen har implementerats i Python-kod, och koden har baserats på utvecklings-verktyget TC-Python SDK skapat av Thermo-Calc Software AB. Modell- implementeringen i Python har jämförts med det kommersiella programvarupaketet TC- PRISMA genom simuleringar av urskiljning av karbider i en Fe-legering med 0,16 vikt% C och 4,0 vikt% Cr. Vidare har inverkan av ytenergi och antal kärnbildningspunkter på modellens resultat studerats. Sammantaget tyder resultaten på att modellen som implementerats och TC- PRISMA beräknar tillväxt och förgrovning av karbiderna på snarlika sätt, men de två modellerna skiljer sig åt i hur de behandlar kärnbildningsparametrar. Den största skillnaden mellan modellerna är att den drivande kraften för kärnbildning beräknas på två olika sätt. Detta leder till att utskiljningsförfarandet förskjuts i tid mellan de två modellerna.

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Therefore, one interesting area of research, from a material’s design perspective, is the modelling of carbide precipitation in steels. Such models may be used in the study and optimization of the precipitation transformation on a material’s structure [2]. Through such optimization, stronger and better materials may be developed, and such materials could be used in lighter and more sustainable products.

A model has been proposed by Bonvalet et al. (2015) [3] for precipitation in multicomponent alloys. It simulates precipitation, dealing concomitantly with homogeneous nucleation, growth, and coarsening in low supersaturated non-dilute multicomponent alloys. The aim of this work is to further the development of the model by implementing it in a more generic version than what has been done previously. Moreover, this work also aims to provide further validation of the proposed model by applying it to an Fe-C-Cr alloy which has previously been investigated, both experimentally and by performing TC- Prisma simulations [4]. Finally, the differences between TC-PRISMA software and the model implemented in this work are investigated in terms of the parameters driving the phase transformation.

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2 2 Background

Fe-C-Cr alloys possess desirable material properties, with high strength and good toughness [5]. Often, they are processed in order to create a martensitic microstructure which give a good combination of these two properties. However, due to the low ductility of a martensitic microstructure, they are often tempered after quenching and prior to use. During the tempering process, secondary phases may precipitate from the matrix phase [6]. The precipitation process in tempered martensitic steels is affected by the as-quenched microstructure. The martensitic structure has a high defect density, and these defects act as nucleation sites for the secondary phases [5]. This affects the mechanical properties of the tempered steel. As such, controlling the precipitation process may be used for controlling the mechanical properties of tempered martensitic steels. In previous works, nanoprecipitation has been studied as a way to simultaneously increase both the strength and hardness of martensitic steels [7].

Due to advances in field of integrated computational materials engineering (ICME) [2], it is possible to optimize the microstructure of tempered martensite using computer-aided design.

This approach allows for better understanding of the fundamental intricacies of precipitation processes without costly experimental work. However, it requires well-developed models in order to be usable for such design purposes. To facilitate the further development and design of advanced martensitic steels, practical models which account for precipitation kinetics are necessary [5].

The alloy investigated in this work is an Fe–0.16 wt% C–4.0 wt% Cr alloy. Since the alloy contains carbon, an interstitial element in the material, and chromium, a substitutional one, modelling precipitation kinetics in such an alloy requires handling elements which diffuse at markedly different speeds. This is interesting from a research perspective, since the model proposed by Bonvalet et al. has previously only been applied to alloys which do not feature fast diffusion elements, such as carbon [3]. Moreover, the Fe-C-Cr combination is the foundation of some modern steels, such as hot-work tool steels [5]. This demonstrates the usefulness of precipitation models with predictive capabilities, since their potential area of use in alloy- and process optimization may be important for industry applications.

The phase diagram for the alloy investigated in this work is displayed in Figure 1, for 0.16 wt%

C, where the point marked A indicates the weight fraction of chromium and the ageing temperature used for the tempering treatment and thus the simulations in this work. Under these conditions, the phase diagram indicates that in its equilibrium state the alloy contains only M7C3 and BCC, where M7C3 is a carbide precipitated from a BCC matrix. Interestingly, previous experimental work has found the alloy to contain two types of precipitates when tempered at 700 °C for 1000 h [4]. The two documented precipitates are M7C3 and M23C6. M7C3

is a chromium-rich carbide which has a trigonal crystal structure [8]. M23C6 is also a chromium- rich carbide, but it has a face centred cubic structure [8]. Therefore, it is of interest to study the processes which gives rise to this non-equilibrium behaviour.

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4 3 Precipitation

Precipitation in solid materials is a phase transformation involving the formation and growth of a new phase from a supersaturated parent phase [11] [12]. This process may be divided into three steps: the nucleation of the new phase, the growth of the new nuclei, and the coarsening of these precipitates. Strictly speaking, nucleation, growth, and coarsening are not discrete steps as they do overlap in time. Indeed, nuclei are not all created at the same time [12].

To achieve precipitation in the simplest case, an initially homogeneous solid solution is held in a two-phase region of the material’s phase diagram. In practice, precipitation is commonly induced by quickly quenching a material from a single-phase region of the phase diagram, where a homogeneous solid solution is obtained, and then annealing it at a temperature where the material, at equilibrium, would form a multi-phase microstructure [12]. A schematic display of such a process is displayed in Figure 2 representing the phase diagram of a binary A-B alloy.

In this figure, the material is homogenized at position 1 in a single-phase region of the phase diagram. Following this, the material is quenched rapidly to position 2 in order to hinder intermediary transformations. Finally, it is held at a temperature in the two-phase region of the phase diagram, position 3. At the annealing temperature, the parent phase is then supersaturated and there will be a driving force for precipitation in order to decrease the Gibbs’

free energy of the system [8].

Figure 2 - Schematic phase diagram illustrating the quenching and annealing of a binary alloy. Figure adapted from [8].

In this part of the present report, the governing equations for nucleation, growth, and coarsening of a precipitate phase 𝛽 from a matrix phase 𝛼 for both binary and multicomponent alloys are presented.

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limited solubility in the matrix phase [11]. The most common theoretical model describing the nucleation process is the classical nucleation theory (CNT) [13] [14] . In CNT, a spontaneously formed embryotic nucleus must overcome a nucleation barrier (a maximum in the Gibbs’ free energy change due to the formation of the nucleus, see Figure 3) to be thermodynamically stable and grow. An embryotic nucleus which is smaller than the critical size, corresponding to the maximum in Gibbs’ free energy change, see Figure 3, may overcome the nucleation barrier due to thermal fluctuations [6]. Moreover, it is assumed that the system reaches a steady-state with respect to the number of nuclei created per unit time and per unit volume [6]. The number of nuclei created per unit time and per unit volume, called the nucleation rate, can be calculated through:

with 𝑛₀ the number of nucleation sites, 𝑓^𝛽 the volume fraction of the precipitate phase, 𝑍 the Zeldovich factor, 𝛽^∗ the atomic attachment rate, ∆𝐺^∗ the nucleation barrier, 𝑘_𝑏 the Boltzmann constant, 𝑇 the temperature, 𝑡 the time, and 𝜏 the incubation time. This equation applies only to homogenous nucleation, with only one type of precipitate and where nuclei are defined only by their size [11] [6]. In the following are described the different parameters driving the nucleation rate considering spherical precipitates.

3.1.1 Nucleation barrier and driving force

The nucleation barrier is determined by the change in Gibbs free energy when forming the precipitate phase. For spherical nuclei, which are assumed to be unaffected by elastic contribution from the matrix, this change is calculated through the balance of two terms. The driving force for nucleation decreases the Gibbs’ free energy while the creation of an interface between the matrix and the precipitate increases the Gibbs’s free energy [8] [11] [15]:

with 𝑟 the radius of the nucleus, ∆𝐺_𝑚 the driving force for nucleation (∆𝐺_𝑚≤ 0, when nucleation is favourable) and 𝜎 the interfacial energy. ∆𝐺 has a maximum value with respect to 𝑟 when

𝑑∆𝐺

𝑑𝑟 = 0. This maximum value, ∆𝐺^∗, is the nucleation barrier. The dependency of ∆𝐺 on 𝑟 is displayed in Figure 3. The nucleation barrier is expressed as:

𝐽𝑛= 𝑛0(1 − 𝑓^𝛽)𝑍𝛽^∗exp⁡(−∆𝐺^∗

𝑅𝑇 )(1 − exp (−𝑡

𝜏⁡))⁡ Eq. 1

∆𝐺 =4𝜋𝑟³

3 ∆𝐺_𝑚+ 4𝜋𝜎𝑟² ^{Eq. 2}

∆𝐺^∗=16𝜋𝜎³

3∆𝐺𝑚2 Eq. 3

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Figure 3 - Schematic representation of the change in Gibbs free energy of a spherical nucleus with respect to its radius, diagram from [8]

In order to calculate the nucleation barrier, ∆𝐺^∗, the driving force for nucleation, ∆𝐺_𝑚, must be calculated, see Eq. 2 [11] [6]. In Figure 4, a schematic diagram of Gibbs-free-energy curves of a matrix phase α and precipitate phase β in a binary A-B alloy is displayed. Under the assumption of local equilibrium, the chemical potential of each element 𝑖 at the interface in two or more coexisting phases must be equal, i.e. 𝜇_𝑖^𝛼= ⁡ 𝜇_𝑖^𝛽. The chemical potential is the change in Gibbs’

free energy stemming from an incremental increase in the number of atoms of element 𝑖 in a phase, at constant temperature, pressure, and moles of other components (𝜇_𝑖^𝛼 =^𝜕𝐺^𝛼

𝜕𝑁_𝑖|

𝑇,𝑃,𝑁) and is represented in Figure⁴ by the intersection of the tangents with the axes. The overall free energy of the system can be reduced by moving atoms of element 𝑖 from the 𝛼⁡phase to the 𝛽 phase. The driving force ∆𝐺_𝑚⁡for the nucleation of the precipitate phase is then calculated through the parallel tangent construction, as displayed in Figure 4 b), and is equal to the difference between the two tangents [6] [11].

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The radius at which ∆𝐺 reaches its maximum value (i.e. when ^𝑑∆𝐺

𝑑𝑟 = 0) is the critical radius, 𝑟^∗, and is expressed as:

Precipitates with radii larger than the critical radius will grow, as this decreases the free energy.

Contrariwise, precipitates with radii smaller than the critical radius will tend to be dissolved [11].

3.1.3 Zeldovich factor

Nucleation is an inherently stochastic process, and a precipitate close to the critical radius may be perturbed into growing or dissolving by thermal fluctuations [12]. In order to account for this mechanism, the Zeldovich factor is used. The Zeldovich factor is written as follows [3]:

𝑟^∗= 2𝜎

−∆𝐺_𝑚 ^{Eq. 4}

𝑍 = 𝛺√𝜎

2𝜋𝑟^∗²√𝑘_𝐵𝑇 ^{Eq. 5}

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here 𝛺 is the atomic volume. The Zeldovich factors describes the probability of a precipitate at the critical radius growing into a stable precipitate [6].

3.1.4 Attachment rate

The growth of the precipitate is dependent on the mass transfer of solute atoms from the matrix to the precipitates through volume diffusion, and on the mobility of the matrix-precipitate interface [6]. Either mechanism may be the limiting one, which is known as the volume diffusion case or the interface reaction case respectively. In this work, only the volume diffusion case is considered. However, due to a lack of expressions available in the literature of the attachment rate in the volume diffusion case [16], this work utilizes a formulation which is derived by assuming the interface reaction case. The attachment rate, 𝛽^∗, is calculated as follows [3]:

where 𝑎 is the lattice parameter for the precipitate, and 𝐷_𝑖𝑖^𝛼⁡⁡is the diagonal diffusion coefficient in the matrix phase for element 𝑖 and 𝐶_𝑖^𝛼 is the concentration of the same element 𝑖 in the matrix phase. Whichever of 𝐶_𝑖^𝛼𝐷_𝑖𝑖^𝛼 ⁡has the lowest value will act as the limiting factor for attachment of solute atoms and thus dictate the attachment rate [6].

An alternative definition for the attachment rate has been proposed by Svoboda et al. [17]:

where⁡𝑋_𝑖^𝛽⁡and 𝑋_𝑖^𝛼 are the mole fractions of element 𝑖 at the interface of the precipitate and of the matrix, respectively, and 𝐷_𝑖 is the corresponding diffusion coefficient in the matrix phase [17]. Nevertheless, it has been shown by Bonvalet and Ågren [16] that this expression of the attachment rate is not consistently-derived. Indeed, it was just an identification of a pre-factor by comparing the growth law in a binary alloy and in a multicomponent alloy [16]. In this work, Eq. 6 is used for calculations of the attachment rate.

3.1.5 Interfacial energy

The interfacial energy of a newly formed precipitate is difficult to measure experimentally [6].

Usually, it is approximated to be constant during the precipitation process. A commonly employed model to estimate the interfacial energy is the extended Becker’s model [18]. The coherent interfacial energy can be calculated as follows:

𝛽^∗= 4𝜋𝑟^∗²

𝑎⁴ min(𝐷_𝑖𝑖^𝛼, 𝐶_𝑖^𝛼) ^{Eq. 6}

𝛽^∗= 4𝜋𝑟^∗²

𝑎⁴ [∑(𝑋_𝑖^𝛽− 𝑋_𝑖^𝛼)² 𝑋_𝑖^𝛼𝐷_𝑖

𝑘

𝑖=1

]

−1

Eq. 7

𝜎_𝑐 = 𝑛_𝑠𝑧_𝑠

𝑁_𝐴𝑧_𝑙∆𝐸_𝑠 Eq. 8

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where ∆𝐸_𝑠 is the energy of solution, 𝑧_𝑙 is the coordination number of an atom in the bulk crystal lattice, 𝑁_𝐴 is Avogadro’s number, 𝑧_𝑠⁡is the number of cross bonds per atom in the interface, and 𝑛_𝑠 is the number of atoms per unit area.

3.1.6 Incubation time

The incubation time for a cluster refers to the time prior to the system entering the steady-state nucleation regime described by Eq. 1 [6]. The incubation time can be calculated as follows:

where 𝑍 and 𝛽^∗ are the Zeldovich factor and the attachment rate, respectively.

3.2 Growth and coarsening

When a precipitate has nucleated, it may grow to reduce its surface-to-volume ratio, due to the Gibbs-Thomson effect. This is known as the growth and coarsening regimes of precipitation [8]. As previously mentioned, precipitates with radii larger than the critical radius will grow larger with time. Moreover, the critical radius itself is time dependent [8]. Since the supersaturation and, thus, the driving force for the transformation decrease with time, the critical radius will increase with time, see Eq. 4. In this work, the concentration profiles are assumed homogeneous in the precipitates.

In the case of solid-state transformations, growth of precipitates is often a volume-diffusion- controlled process and depends on the transfer of atoms through the bulk material [12]. In the case of a binary A-B alloy, the flux may be described by Fick’s first law:

where 𝐉 is the flux vector, 𝐃 is the diffusion matrix of the solute atoms in the matrix, and 𝛁⃗⃗ 𝐂 is the concentration gradient of the solute atoms. In this work, bold notation indicates a matrix or a vector. In Figure 5 a schematic concentration profile in 1D around the interface is displayed, where the concentration of solute atoms is higher in the precipitate phase and the concentration of solute atoms in the matrix, far from the precipitate, is the mean-field concentration of the alloy.

𝜏 = 2

𝜋𝑍𝛽^∗ ^{Eq. 9}

𝐉 = −𝑫𝛁⃗⃗ 𝐂 Eq. 10

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Figure 5 - Schematic concentration profile around the matrix-precipitate interface where 𝛼⁡is the matrix phase, 𝛽 is the growing precipitate phase, 𝑧^∗ is the interface position, 𝑐̅ is the mean-field composition of the alloy. Diagram from [8]

At the interface of a binary A-B alloy, the rate of partitioning of the solute atoms must equal the flux in order to respect mass balance:

In Eq. 11, 𝐶^𝛽(𝑟) and 𝐶_⁡^α(𝑟) are the interfacial concentration in solute atoms in the precipitate and in the matrix, respectively, 𝐷 is the diffusion coefficient of the solute atoms in the matrix, and ⁡𝑧 is the distance to the matrix-precipitance interface [6].

In the case of multicomponent alloys, this equation must be generalized. Then the mass balance equation writes [12]:

where 𝐶_𝑘^∞ is the composition of element 𝑘 in the matrix far from the interface, 𝐶_𝑘^𝛼 is the composition of element 𝑘 in the matrix, ⁡𝐶_𝑗^𝜶(𝑟)⁡and 𝐶_𝑗^𝛽(𝑟) are the compositions at the interface for the matrix and the precipitate, respectively [19], and 𝐷_𝑗𝑘 is the diffusion coefficients matrix.

While Eq. 11 and Eq. 12 may appear similar to each other, the multicomponent case requires a diffusion matrix rather than a diffusion coefficient, as the influence of some element on the diffusion of others element must be considered [12].

In the case of binary systems, the assumption of local equilibrium at the interface between matrix and precipitate allows for the direct calculation of the growth rate and is sufficient for modelling growth and coarsening [8]. However, for multicomponent systems this is no longer the case. When applying Eq. 12 on a multicomponent system with 𝑁 − 1 independent elements, there will be a system of 2𝑁 − 1 unknowns, but only 𝑁 − 1 equations. This is due to the

⁡𝑑𝑟

𝑑𝑡(𝐶^𝛽(𝑟) − 𝐶_⁡^α(𝑟)) = −𝐷𝑑𝐶

𝑑𝑧 ^{Eq. 11}

(𝐶_𝑗^𝛽(𝑟) − ⁡𝐶_𝑗^𝜶(𝑟))𝑑𝑟

𝑑𝑡 = ⁡ ∑𝐷_𝑗𝑘

𝑟 (𝐶_𝑘^∞− 𝐶_𝑘^𝛼)

𝑁

𝑘=2

Eq. 12

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interfacial compositions being unknown. When assuming local equilibrium at the interface [12], this gives rise to the following equality of chemical potentials between the two phases [6]:

The mechanical equilibrium in the case of spherical precipitates [12] writes:

As such, Eq. 12, in combination with Eq. 13 and Eq. 14, thus result in a system of 2𝑁 − 1 equations in 2𝑁 − 1 unknowns. However, solving such a system of equations is computationally expensive, as the equations are non-linear.

In the work of Philippe and Voorhees [20], an analytical expression for the growth rate in the multicomponent case was derived. This was done under the assumption of small supersaturation for which a Taylor expansion of the chemical potential around their equilibrium values was possible. Under the assumptions of small supersaturation, the difference in composition between the matrix and the precipitate at the interface may be approximated by the difference in equilibrium compositions [20]. Together with the assumption of local equilibrium at the interface, an analytical expression for the growth rate of a precipitate in a multicomponent alloy can be derived. This growth rate may be calculated as follows:

where ∆𝑪̅ is the equilibrium tie-line between the matrix and the precipitate (∆𝐂̅ = 𝐂̅^𝛂− 𝐂̅^𝛃, where the bar notation indicates equilibrium compositions in the matrix and the precipitate, respectively), ∆𝑪^∞ is the supersaturation vector (∆𝐂^∞= 𝐂_𝟎^𝛂− 𝐂̅^𝛂, i.e. the difference between mean-field composition of the alloy, 𝐂_𝟎^𝛂, and the equilibrium matrix composition, 𝐂̅^𝛂), 𝑉_𝑚^𝛽 is the molar volume of the precipitate, 𝐆^⁡α⁡is the hessian of the molar Gibbs’ free energy of the matrix (𝐆^⁡α⁡=^𝜕^𝟐^𝐺^𝛼

𝜕𝐶^𝛼2 , the curvature of the matrix curve displayed in Figure 4, evaluated at the equilibrium composition), and 𝐌 is the mobility matrix such that 𝐌𝐆^𝛂= 𝐃, [20]. It is worth mentioning that (∆𝐂̅)^𝑇𝐆^⁡α⁡∆𝐂^∞ is the driving force for the transformation as has been shown by Philippe et al. [19].

Moreover, using Eq. 15, Philippe and Voorhees [20] showed that the interfacial compositions for the matrix and the precipitates may be calculated as follows:

𝜇_𝑖^𝛼(𝐶₂^𝛼, 𝐶₃^𝛼, … , 𝐶_𝑁^𝛼, 𝑃^𝛼) = ⁡ 𝜇_𝑖^𝛽(𝐶₂^𝛽, 𝐶₃^𝛽, … , 𝐶_𝑁^𝛽, 𝑃^𝛽), 𝑖 = 1, 2, 3, …⁡, 𝑁⁡ ^{Eq. 13}

𝑃^𝛽 = 𝑃^𝛼+2𝜎

𝑟 ^{Eq. 14}

𝑑𝑟

𝑑𝑡= 1

𝑟(∆𝑪̅)^𝑇𝐌^−𝟏∆𝑪̅((∆𝐂̅)^𝑇𝐆^⁡α⁡∆𝐂^∞−2𝜎𝑉_𝑚^𝛽

𝑟 ) ^{Eq. 15}

𝐂^𝛂(𝒓) = 𝐂^∞− 𝐃^−𝟏∆𝐂̅

(∆𝐂̅)^𝑻𝐌^−𝟏∆𝐂̅((∆𝐂̅)^𝑇𝐆^𝛂−2𝜎𝑉_𝑚^𝛽

𝑟 ) ^{Eq. 16}

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In Eq. 17, 𝐆^β is the hessian of the molar Gibbs’ free energy of the matrix (𝐆^⁡β⁡=^𝜕^𝟐^𝐺^β

𝜕𝐶^β2 , the curvature of the precipitate Gibbs energy curve displayed in ^{Figure 4}, evaluated at the equilibrium composition). Since larger particles possess a lower free energy due to the Gibbs- Thomson effect, coarsening (also known as Ostwald ripening) will be prevalent at the later stages of a precipitation process [12]. The Gibbs-Thomson effect stems from difference in pressure between the inside of the precipitate and the matrix, in the case of a curved surface [8]. As Gibbs’ free energy is a function of pressure, this pressure difference will shift the Gibbs’- free-energy curves. For phases which are not stochiometric, this will also result in a shift in composition. In Figure 6, the Gibbs-’Thomson effect is visualized as the difference in Gibbs’ free energy between a precipitate phase with a flat interface and precipitate phase with a curved interface. In the case of spherical precipitates, this difference is 𝜎^𝑑𝑂

𝑑𝑛= ⁡𝜎^2𝑉^𝑚

𝛽

𝑟 ⁡ [8]. This construction is carried out under the assumption of local equilibrium, see section 3.1.1.

Figure 6 - Schematic diagram of difference in Gibbs' free energy between a flat and a curved interface for precipitate phase 𝛽. These are represented by 𝐺^𝛽 and 𝐺_𝑟^𝛽 respectively. ^𝑑𝑂

𝑑𝑛 is the change of the interface area following the addition of one new atom to the interface [8].

The larger precipitate particles dispersed in the matrix will grow over time while smaller particles shrink and dissolve, meaning the size evolution is competitive [17]. This phenomenon is displayed schematically in Figure 7, where the number of precipitates decreases with time while the mean size of the precipitates increases with time.

𝐂^𝜷(𝒓) = 𝐂̅^𝛃+ 𝐆^𝛃^−𝟏𝐆^𝛂⁡(𝐂^∞− 𝐃^−𝟏∆𝐂̅𝑟𝑑𝑟

𝑑𝑡) ^{Eq. 17}

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Figure 7 – a) Schematic evolution of number density of precipitates with time, and b) Schematic diagram of growth and coarsening of spherical precipitates

In Figure 7, coarsening is schematically displayed in a material which is annealed. In a), the evolution of the number density of precipitates with time is displayed. During the initial stages of the precipitation process, nucleation takes place and the number density of precipitates increases with time. Following this, nucleation stops and the number density is constant, since no new precipitates are formed and no precipitates have had sufficient time to dissolve. Finally, the smallest precipitates start to dissolve, and the number density decreases. In b), the behaviour of the precipitates is displayed schematically. The number of precipitates decrease with time, but the mean size increases as coarsening occurs. This behaviour is further exemplified in Figure 8, where a creep-resistant X20CrMoV12.1 steel containing M23C6carbides is annealed for different times and the carbides are shown to coarsen.

In the following, growth refers to the regime when nucleation has stopped but no precipitates have yet been dissolved. In a similar manner, coarsening refers to the regime following the dissolution of the first precipitates. These two regimes are actually not distinct and do overlap – both in practice and in the code used in this work. However, this nomenclature is used for the sake of clarity and in the explanation of results.

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Figure 8 - SEM images of coarsening of M23C6 carbides in creep-resistant X20CrMoV12.1 steel annealed for different times and at different temperatures [21].

In summary, precipitation is a process governed by both thermodynamics and kinetics.

Initially, embryotic nuclei of a new thermodynamic phase are formed from a supersaturated parent phase in a nucleation process described by Eq. 1. Following this, the size of the nucleated precipitates evolves in a manner which is described by Eq. 15. This evolution is initially mainly sustained by a reduction of supersaturation in the parent phase. However, at later stages of precipitation the size evolution is competitive, with larger precipitate particles growing at the expense of smaller particles, which shrink and dissolve. In the next section of this work, approaches for modelling precipitation processes are presented.

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15 4 Modelling of precipitation

Modelling of precipitation requires coupling nucleation with kinetic descriptions of growth and coarsening. Several approaches have been proposed for both research purposes and industry use [6], among which three of the most prevalent are kinetic Monte Carlo Simulations [22] [23]

, phase-field simulations [24], and mean-field approaches [3] [25]. The use of kinetic Monte Carlo simulations has been shown to be suitable descriptors of precipitation processes, but they are computationally expensive [22] [23]. Furthermore, phase-field models have been applied to precipitation and shown to be useful for modelling microstructure evolution [24] [26].

However, phase-field models are also computationally expensive. Due to the ample availability of both thermodynamic and kinetic databases, mean-field models that explicitly employ nucleation, growth, and coarsening theories may be an attractive and computationally efficient alternative to kinetic Monte Carlo and phase-field models [3]. Kampmann-Wagner approaches are frequently used for modelling precipitation [6] [27]. This work utilizes a Kampmann- Wagner approach.

4.1 The Kampmann-Wagner approach applied to precipitation

Kampmann and Wagner introduced a particle size distribution of the precipitates (PSD) to model precipitation in multicomponent systems [27]. In their approach, the PSD and the time for the process are discretized. For each discrete population of precipitates, a number density (number of precipitates per unit volume), a volume fraction, and a size (commonly the radius of spherical precipitates) is calculated at each discrete time step [3] [27].

In each class in the PSD the number density of precipitates is set by the nucleation rate, as defined in Eq. 1, multiplied by the time step. All precipitates within one population evolve in the same manner during the precipitation process, according to the growth rate [27], see Eq.

15. The number density of a class 𝑖 of precipitates can be calculated as follows:

where 𝑁_𝑣_𝑖 is the number density of the class, 𝐽_𝑛 is the nucleation rate, and ∆𝑡 is the discretized time step.

All precipitates within one population or class are assumed to be of the same size, given by the Zeldovich radius. The use of the Zeldovich radius rather than the CNT critical radius makes certain that all nuclei above the chosen radius will grow, as nuclei at or close to the critical radius may still dissolve due to thermal fluctuations [6]. The Zeldovich radius can be calculated from the critical radius by:

Two types of approaches are used to solve the evolution of the PSD as a function of time:

- Lagrange-like models, where the number density of precipitates in each population is kept constant while their size evolve [3].

- Euler-like models, where the size of each population is kept constant while varying their number density.

𝑁_𝑣_𝑖 = 𝐽_𝑛∆𝑡 Eq. 18

𝑅_𝑧 = 𝑅^∗+1 2√𝑘_𝑏𝑇

𝜋𝜎 ^{Eq. 19}

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The two methods have been shown to be equivalent [28]. In Figure 9, the schematic evolution with time of a PSD in a Lagrange-like model is displayed. At the first time step, displayed a), only one class of precipitates has been nucleated. Following this, in b), more classes of precipitates have been nucleated. Finally, in c), the classes which have a mean radius which is larger than the critical radius grow, while classes with a mean radius which is smaller will shrink and, ultimately, dissolve.

Figure 9 - Schematic diagram of the evolution with time of the PSD in a Lagrange-like model. a) displays the nucleation of the first class of precipitates, b) displays the PSD following nucleation of more classes, c) displays the growth and shrinkage

In the Kampmann-Wagner approach, the nucleation rate is calculated at the start of each time step. Following nucleation, the growth or shrinkage of each class of precipitates is calculated in accordance with the growth rate. The new volume fraction of each class 𝑖 is calculated from the new size 𝑅_𝑖 as:

The total volume fraction of precipitates is then calculated through the summation of the volume fraction of all classes, 𝑓_𝑡𝑜𝑡= ∑ 𝑓_𝑖 _𝑖. In order to calculate the matrix concentration, the molar fraction is assumed to be equal to the volume fraction in this work. The new supersaturation in the matrix of element 𝑖 is then calculated in the case of a two-phased material in the following manner:

𝑓_𝑖 =4𝜋

3 𝑅_𝑖³𝑁_𝑣𝑖 ^{Eq. 20}

𝐶_𝑖^𝛼= ^𝐶⁰

𝛼−𝑓_𝑡𝑜𝑡^𝛽 𝐶_𝑒𝑞^𝛽

1−𝑓_𝑡𝑜𝑡^𝛽 Eq. 21

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17

Using the new supersaturation in the matrix, the driving force for the next time step is calculated. This process is carried out at each time step.

4.2 Modelling in TC-PRISMA [29]

TC-PRISMA is a commercial software for modelling precipitation [29]. It is based on a Kampmann-Wagner approach, as described in section 4.1, to simulate nucleation, growth, and coarsening of precipitates in multicomponent and multiphase alloys [25] [29]. In TC-PRISMA, there are two models for calculating the growth rate of precipitates: the simplified model and the advanced model [20]. The advanced model and the simplified model are described in the following sections

4.2.1 The advanced model

The advanced model has been derived by Chen, Jeppsson, and Ågren [25], and calculates the growth-rate by computing the operating tie-line by solving flux-balance equations, see Eq. 23 [20]. Furthermore, some of the model parameters are different than those introduced in chapter 3 [29]. It uses the derivation of the attachment rate provided by Svoboda et al. [17], see Eq. 7, and a different time-dependence factor in the expression for the nucleation rate, which is calculated as follows [29]:

The growth rate in the advanced model is calculated by numerically solving 2𝑁 − 1 equations.

Of these, 𝑁 equations stem from the equality of chemical potentials (see Eq. 13), and 𝑁 equations stem from following system of equations:

where 𝑀_𝑖 is the atomic mobility of component 𝑖, 𝜇_𝑖^𝛼⁡is the chemical potential in the matrix at the mean-field concentration 𝜇_𝑖^𝛽 is the chemical potential at the interface in the precipitate, and 𝜀_𝑖 is the effective diffusion distance factor [25]. While Eq. 12 and Eq. 23 are similar, Eq. 23 is constructed in such a way to handle chemical potentials and atomic mobilities rather than mole fractions and diffusion coefficients. This is done in order to handle information available in thermodynamic databases.

4.2.2 The simplified model

The simplified model is based on the advanced model, but instead of computing the operating tie-line it uses the tie-line for the bulk concentration [29]. In the simplified model, the growth rate is calculated as follows:

where 𝐾 is a constant and, in the case of spherical precipitates, calculated through:

𝐽𝑠= 𝑛0(1 − 𝑓^𝛽)𝑍𝛽^∗exp (−∆𝐺^∗

𝑘𝑏𝑇 ) exp (−𝑡

𝜏⁡)⁡ Eq. 22

𝑑𝑟

𝑑𝑡(𝐶_𝑖^𝛽− ⁡𝐶_𝑖^𝜶) = ^⁡𝐶

𝛼𝑖𝑀𝑖(𝜇_𝑖^𝛽−𝜇_𝑖^𝜶)

𝜀_𝑖𝑟 ,⁡⁡𝑖 = 1, 2, 3, …⁡, 𝑁 _{Eq. 23}

𝑑𝑟 𝑑𝑡=𝐾

𝑟(∆𝐺𝑚−2𝜎𝑉_𝑚^𝛽⁡

𝑟 )⁡ Eq. 24

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In Eq. 25, the interfacial concentrations of element 𝑖, 𝐶_𝑖^𝛽(𝑟) and 𝐶_𝑖^𝛼(𝑟), are taken from the tie- line of the mean-field matrix concentration [29].

4.3 Model by Bonvalet et al. [3],

Bonvalet et al. [3] proposed a model for nucleation, growth and coarsening in non-dilute multicomponent alloys based on the theoretical work of Philippe et al. [20]. The precipitation tool has been shown to be a good descriptor of precipitation in NiCrAl superalloys [3]. This model is a Lagrange-like model which utilizes the assumption of small supersaturation and the growth equation derived by Philippe et al. (see Eq. 15). It takes off-diagonal elements of the diffusion matrix into account and it does not require fully coupled CALPHAD calculations, since Philippe et al. derived an analytical expression for the growth rate [3]. In comparison, the advanced TC-PRISMA module requires new CALPHAD calculations to be carried out at each time step to compute the operating tie-line [25]. Rather, the calculation of the Hessian of the Gibbs free energy curves, 𝐆^⁡𝛂⁡ and 𝐆^⁡𝛃⁡, and the diffusion matrix may be carried out prior to the simulation and do not need to be calculated at each time step, which allows for this model to be in principle more computationally efficient [3]. However, one has to keep in mind that this way of computing the growth rate is based on the assumption of low-supersaturated solid solutions which is not the case in the advanced TC-PRISMA model.

As the proposed model does not explicitly calculate the operating tie-line at each time step, it is possible it lacks some of the predictive capabilities of the advanced TC-PRISMA model.

However, as the proposed model allows for the variation of the interfacial composition, which the simplified TC-PRISMA model does not, it may possess better predictive capabilities than the simplified model. Moreover, due to TC-PRISMA being commercial proprietary software, it does not offer the possibility of altering the source code. This may be an issue in research applications, where investigating different variations of the model (e.g. including heterogeneous nucleation) may be of interest. On the other hand, the proposed model is not proprietary, and any code implementation of the model may be altered in accordance with the needs of the researcher. As such, the model by Bonvalet et al. may be a useful complement for research purposes.

4.3.1 Further validation of model by Bonvalet et al.

While there is potential to the proposed model for use as a research tool, it has not been tested extensively. To further test and develop the model proposed by Bonvalet et al. a number of issues must be considered:

- The model has previously not been employed on alloys which contain fast diffusers, such as carbon

- The model has previously not been employed on alloys which feature more than one phase precipitating concomitantly

- The current code for application of the model is not coupled to thermodynamic and kinetic databases for CALPHAD calculations, and all such calculations must be carried out manually outside the current code framework

- The current code may only handle a fixed number of precipitate classes

In section 5, this work presents results and improvements with respect to these issues.

𝐾 = ∑(𝐶_𝑖^𝛽(𝑟) − 𝐶_𝑖^𝛼(𝑟))²𝜀𝑖⁡ 𝐶_𝑖^𝛼(𝑟)𝑀_𝑖

𝑁

𝑖=2

Eq. 25

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5 Modelling of precipitation in Fe–0.16 wt%C–4.0 wt%Cr

Previous works have found TC-PRISMA to be able of capturing the main physical features of precipitation in Fe-C-Cr alloys [4] [30] (see chapter 1). This part of the report compares the results from simulations carried out in TC-PRISMA with simulations carried out using the model proposed by Bonvalet et al. and implemented in Python™ in this work. This implementation was carried out in order to remedy the issues discussed in section 4.3.1. The simulations are ran for an Fe–0.16 wt%C–4.0 wt%Cr alloy which is tempered at 700 °C using the thermodynamic database TCFE9 [10] and the kinetic database MOBFE2 [31]. The choice of the alloy was done due to its previous use in experimental and modelling work by Hou et al. [4].

5.1 Calculation of model parameters

To apply the model by Bonvalet et al. to the chosen alloy, thermodynamic calculations were carried out using the Thermo-Calc software [9]. Specifically, the equilibrium concentrations and the Gibbs’ free energy curves of the precipitates, M7C3 and M23C6, and of the BCC matrix were computed using Thermo-Calc equilibrium calculations and the TCFE9 database. It is worth noting that in the work by Hou et al. [4], the TCFE7 database was used. However, there is a significant difference between TCFE7 and TCFE9 in regard to precipitation calculations using TC-PRISMA. Therefore, the latest database, TCFE9, was chosen for this work as it ought to be an improvement of the previous version. The difference between the two databases is displayed in Appendix A. In order to calculate the equilibrium concentrations for M23C6, which is metastable at the temperature of this work, M7C3 was excluded from the Thermo-Calc calculation. This is done by excluding the Gibbs’ free energy curves of all other phases than BCC and M23C6. As such, an artificial equilibrium is calculated, which corresponds to a system which may only contain BCC and M23C6. Tables 1-3 summarizes equilibrium Cr and C concentrations in the matrix and precipitates. Table 4 displays the concentrations of the BCC matrix when M7C3 was excluded from the Thermo-Calc calculation.

Table 1 - BCC matrix equilibrium composition in Fe–0.16 wt%C–4.0 wt%Cr at 700 °C , calculated using TCFE9

Element Mole fraction

C

^0.00008

Cr

^0.02941

Table 2 - Equilibrium composition in the M7C3 carbide in Fe–0.16 wt%C–4.0 wt%Cr at 700 °C , calculated using TCFE9

C

^0.3

Cr

^0.57117

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Table 3 - Equilibrium composition in the M23C6 carbide in the Fe–0.16 wt%C–4.0 wt%Cr alloy at 700 °C , calculated using TCFE9

C

^0.20690

Cr

^0.45233

Table 4 - BCC matrix composition in Fe–0.16 wt%C–4.0 wt%Cr at 700 °C , when excluding the M7C3 carbide, calculated using TCFE9

C

^0.00012

Cr

^0.02769

The Hessians of the Gibbs’ free energy curves, 𝐆^⁡𝛂⁡ and 𝐆^⁡𝛃⁡, i.e. curvature of the Gibbs’ free energy curves, needed to calculate the growth rate, see Eq. 14, of the three phases were initially calculated numerically. These calculations used Gibbs’ free energy curves which were extracted from Thermo-Calc databases. The curves were extracted for a composition range close to the equilibrium compositions of the three phases. However, the Gibbs’ free energy curves of the precipitates displayed large fluctuations with respect to composition. This is presented for M7C3

in Figure 11 and for M23C6 in Figure 12. Consequently, the numerical differentiation of the Gibbs’ free energy curves proved unreliable. To remedy this, higher-order-polynomial surfaces were fitted to the Gibbs’ free energy curves, calculations of the curvature of the fitted surfaces at the equilibrium compositions of the precipitate phases and the matrix phase were then carried. However, it is worth noting that the polynomial surfaces may be a poor fit to the Gibbs’

free energy curves. Especially the fit for the M23C6 carbide proved poor.

Figure 10 - The Gibbs' energy curve of the BCC matrix (in black), as a function of concentration, and the approximated polynomial surface in Fe–0.16 wt%C–4.0 wt%Cr at 700 °C , calculated using TCFE9

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Figure 11 - The Gibbs' energy curve of the M7C3 carbide (in black), as a function of concentration, and the approximated polynomial surface (in color) in Fe–0.16 wt%C–4.0 wt%Cr alloy at 700 °C , calculated using TCFE9

Figure 12 - The Gibbs' energy curve of the M23C6 carbide (in black), as a function of concentration, and the approximated polynomial surface (in color) in Fe–0.16 wt%C–4.0 wt%Cr alloy at 700 °C, calculated using TCFE9

The Hessians (evaluated at the equilibrium composition of the matrix) for BCC, M7C3, and M23C6 are presented in Table 5, Table 6, and Table 7, respectively.

Table 5 - The Hessian of Gibbs’ free energy of the matrix phase (BCC) evaluated at the equilibrium composition in Fe–0.16 wt%C–4.0 wt%Cr alloy at 700 °C , calculated using TCFE9. [J/mole]

𝑮_𝑪,𝑪^𝑩𝑪𝑪= 𝟏. 𝟎𝟗𝟔𝟒 × 𝟏𝟎^𝟖 𝑮_{𝑪𝒓,𝑪}^𝑩𝑪𝑪= −𝟏. 𝟗𝟗𝟑 × 𝟏𝟎^𝟓 𝑮_{𝑪,𝑪𝒓}^𝑩𝑪𝑪= −𝟏. 𝟗𝟗𝟑 × 𝟏𝟎^𝟓 𝑮_{𝑪𝒓,𝑪𝒓}^𝑩𝑪𝑪 = 𝟐. 𝟖𝟎𝟐𝟗 × 𝟏𝟎^𝟓

Table 6 - The Hessian of Gibbs’ free energy of M7C3 evaluated at the equilibrium composition in Fe–0.16 wt%C–4.0 wt%Cr at 700 °C, calculated using TCFE9. [J/mole]

𝑮_𝑪,𝑪^𝑴^𝟕^𝑪^𝟑= 𝟏. 𝟏𝟏𝟒𝟓 × 𝟏𝟎^𝟓 𝑮_{𝑪𝒓,𝑪}^𝑴^𝟕^𝑪^𝟑= −𝟗. 𝟐𝟖𝟐 × 𝟏𝟎^𝟑 𝑮_{𝑪,𝑪𝒓}^𝑴^𝟕^𝑪^𝟑= −𝟗. 𝟐𝟖𝟐 × 𝟏𝟎^𝟑 𝑮_{𝑪𝒓,𝑪𝒓}^𝑴^𝟕^𝑪^𝟑 = 𝟓. 𝟎𝟖𝟒 × 𝟏𝟎^𝟒

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Table 7 - The Hessian of Gibbs’ free energy of M23C6 evaluated at the equilibrium composition for the Fe–0.16 wt%C–4.0 wt%Cr alloy at 700 °C , calculated using TCFE9. [J/mole]

𝑮_𝑪,𝑪^𝑴^𝟐𝟑^𝑪^𝟔 = −𝟗. 𝟗𝟏𝟖𝟒 × 𝟏𝟎^𝟓 𝑮_{𝑪𝒓,𝑪}^𝑴^𝟐𝟑^𝑪^𝟔= −𝟔. 𝟐𝟖𝟓𝟒 × 𝟏𝟎^𝟐 𝑮_{𝑪,𝑪𝒓}^𝑴^𝟐𝟑^𝑪^𝟔 = −𝟔. 𝟐𝟖𝟓𝟒 × 𝟏𝟎^𝟐 𝑮_{𝑪𝒓,𝑪𝒓}^𝑴^𝟐𝟑^𝑪^𝟔= 𝟔. 𝟏𝟓𝟒𝟗 × 𝟏𝟎^𝟒

The reduced diffusion matrix in the matrix phase was calculated for the equilibrium composition of the matrix, as this is an assumption made in the theoretical work by Philippe and Voorhes [20]. The calculations were carried out using the Thermo-Calc diffusion module (DICTRA) [9]. In the present work, the diffusion matrix is assumed to be constant during the transformation. The values calculated are presented in Table 8.

Table 8 - The calculated reduced diffusion matrix at the equilibrium composition of the matrix phase at 700 °C. [𝑚²𝑠⁻¹]

D

ij

C Cr

C

^{4.57 × 10}⁻¹¹ −4.46 × 10⁻¹⁸

Cr

−8.377 × 10⁻¹⁴ 6.18 × 10⁻¹⁸

While there are established ways of doing the calculations listed in this section, it is highly time consuming and it has to be done manually. Thus, it is impractical to use the model proposed by Bonvalet et al. in its current code implementation when switching between different alloys, compositions and temperatures. Therefore, there is a need for implementing the model in a more generalized form. In this work, this is done by implementing the model using TC-Python, the Python™ language-based SDK available from Thermo-Calc that allows for coupling Thermo-Calc calculations and databases. This implementation is presented below, in section 5.2.

5.2 Implementation of the model by Bonvalet et al. in Python™

In this work, the model proposed by Bonvalet et al. [3] has been implemented in Python™ in order to facilitate coupling with Thermo-Calc through the TC-Python SDK. This implementation of the model is largely based on the previous implementation in FORTRAN by Bonvalet et al. However, a number of improvements to the code have been made:

• Since the code is written in Python™, the use of libraries such as Numpy, Scipy, and Matplotlib is possible.

• The code has been generalized in order to handle more than one type of precipitate phase, which facilitate further validation of the model. Moreover, it also makes the proposed model more easily applicable.

• The implementation in Python™ allows for direct coupling with Thermo-Calc calculations and databases. This provides a means of automating the calculations of simulation parameters discussed in section 5.1, which further facilitates the use of the model as a research tool. It also allows for running TC-PRISMA simulations through the TC-Python SDK, which facilitates comparisons with TC-PRISMA.

• The limitations on the number of classes of precipitates in one simulation has been removed.

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In order to test the issues presented in 4.3.1 and the validity of the code in comparison to its’

previous implementation in FORTRAN, results from simulations of precipitation in Fe–0.16 wt%C–4.0 wt%Cr are presented in the following section and compared against other approaches.

5.3 Evaluation of new implementation of model proposed by Bonvalet et al.

An evaluation of the model implementation carried out in this work is presented in this section.

First, the current implementation in Python™ is compared with the previous implementation in FORTRAN with respect to the precipitation of M7C3 carbide in a BCC matrix at 700 ºC.

Thereafter, the new implementation is compared with TC-PRISMA in the case of precipitation of only one carbide (M7C3 or M23C6) and in the case of precipitation of two carbides concomitantly (M7C3 and M23C6) at the same temperature. Finally, a systematic study of the influence of the number of nucleation sites and of the interfacial energy on the simulation is presented.

5.3.1 Comparison between Python™ and FORTRAN implementations

The first simulation of M7C3 precipitation kinetics at 973⁡K for 5.3 × 10⁵ seconds has been performed using the parameters presented in Table 9. The number of nucleation sites and the molar volume are the default values in TC-PRISMA for this alloy and the interfacial energy is the one used by Hou et al. [4]. For the sake of comparison, both implementations used the same calculation parameters.

Table 9 – Simulation parameters for comparison between implementations of model in Python^TM and FORTRAN.

Simulation parameter Value

Number of nucleation sites 2.207 × 10²²

Interfacial energy [J/m²] 0.415

Molar volume [m³] 7.272 × 10⁻⁶

Temperature[K] 973

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Figure 13 - Evolution with time of the M7C3 precipitates mean size obtained with the Python implementation (blue curve) and with the previous FORTRAN implementation (orange curve).

In Figure 13, the evolution of the mean radius with time in both implementation is displayed.

At the initial stages of the nucleation, the mean radii increase. Following this, the mean radii are almost constant, and, finally, the mean radii increase again. The evolution of the mean radius is almost identical for the two implementations. The small discrepancy stems from the two implementations saving numbers in different formats. These results indicate that the implementation in Python^TM has been carried out correctly. A similar agreement between the implementations are found in the case of volume fraction and number density as functions of time.

5.3.2 Comparison between present work and TC-PRISMA, for M7C3

A simulation of homogeneous M7C3 precipitation kinetics at 973⁡K for 5.3 × 10⁵ seconds has been performed both with the current work approach and implementation and with TC- PRISMA. The parameters used for running the simulations are presented in Table 9. Both models use the same parameters. The number of nucleation sites and the molar volume are the default values in TC-PRISMA for this alloy and the interfacial energy is the one used by Hou et al. [4]. The evolution of the mean radius, the number density and the volume fraction of M7C3

precipitates as a function of time are presented in Figure 14, Figure 16, and Figure 17 respectively.

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Following the plateau, the mean radii start to increase again. While the two curves display a similar shape, the proposed model displays a larger mean radius for the entire simulation.

Moreover, this work’s model predicts both an earlier start and end of the plateau than TC- PRISMA does.

The three distinct regimes may correspond to nucleation, growth, and coarsening. Since when nucleation stops, and growth starts, no new precipitates are added, this induces a different behaviour in the mean size evolution. Moreover, the later increase in mean radius for the two implementations may correspond to coarsening. This is investigated in Figure¹⁵ which displays the evolution with time of the cubes of the mean radii.

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Figure 15 - Mean radius cubed of M7C3 precipitates obtained with this work’s model (blue curve) and with TC- PRISMA (orange curve).

In Figure 15, the cubes of the mean radii as a function of time are displayed for both models.

Initially, the cubes of the mean radii increase sharply for both models. Following this, both models display a regime of slower increase with time. At the end of the simulation, both models predict an almost linear proportionality between the cubes of the mean radii and time. It is known that during coarsening, the increase in mean radius should be linearly proportional to the third root of time [32]. As such, this plot indicates that in Figure 14, the last regime may correspond to the coarsening stage of precipitation. It is worth noting that the slope resulting from the proposed model is higher than the slope resulting from TC-PRISMA, which indicates that the growth rate in the proposed model is higher than the one in TC-PRISMA.

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Figure 16 - Evolution with time of number density of M7C3 precipitates obtained by the present work’s model (blue curve) and TC-PRISMA (orange curve)

In Figure 16 the evolution with time of the number density of precipitates is displayed for both tool results. Initially, both curves are increasing. Following this, both curves reach a plateau.

Finally, following the plateau, the number density decreases for both models. Except at the very early stages of the simulations, TC-PRISMA predicts a higher number density than the proposed model. It is worth noting that the plateau is predicted to start earlier by the proposed model than by TC-PRISMA, as is the start of the decrease following the plateau.

The three distinct regimes displayed by both curves correspond to nucleation, growth, and coarsening. During the initial stages, with increasing number density, nucleation of new precipitates causes the increase in the number density. Following the stop of nucleation, no new precipitates are added to the material. This corresponds to the plateau with a stable amount of precipitates. During coarsening, the number of precipitates per unit volume decreases due to the Gibbs’-Tomson effect, see section 3.2. In the figure, this is represented by the decrease in number densities toward the end of the simulations. Moreover, these three distinct regimes are in agreement with the three regimes displayed in Figure 16, the proposed model predicts an earlier start to the plateaus, which corresponds to an earlier stop to nucleation. This is also the case with coarsening, as the proposed model predicts an earlier start to the decrease in number density, in Figure 16, as well as an earlier increase in the mean radius following the plateau, in Figure 14.

Modelling of precipitation in an Fe-C-Cr alloy