Laves phase formation in Fe-based alloys from strengthening particle to self-healing agent: a review

In this study, were extensively reviewed the hardening and self-healing properties of Laves-phase in Fe-based alloys. First, the microstructural features of different polytypes of the Laves-phase, focusing on the thermodynamics and kinetics of formation in ferritic and martensitic steels were revised. C14 was identified as the dominant polytype in steels, providing strengthening by precipitation, anchoring of dislocation, and interphase boundaries, thereby increasing the creep resistance. Although the Laves phase is widely known as a reinforcement particle (or even a detrimental phase in some systems) in martensitic/ferritic and ferritic steels, recent findings have uncovered a promising property. Particles with self-healing characteristics provide creep resistance by delaying creep cavities formation. In this regard, different elements such as tungsten and molybdenum are known to provide this feature to binary and tertiary ferrous alloys due to their ability to diffuse into the creep cavities and form Laves-phase Fe(Mo,W)2. To date, self-healing by precipitation has only been reported in commercial stainless steel AISI 312, 347, and 304 modified with boron, nevertheless with a little contribution to creep rupture life. Although, commercial computational tools with thermodynamic and kinetic databases are available for researchers, to tackle the self-healing process with exactitude, genetic algorithms arise as a new tool for computational design. The two properties of Laves phase reported in the literature, precipitation hardening and self-healing agent, is a mix that can bring out a new research field. Therefore, it is not unreasonable to think of tailor-made high chromium creep-resistant steels reinforced by Laves-phase coupled with self-healing properties. However, owing to the characteristic of Laves-phase seems to be a complex challenge, mainly due to the crystallographic features of this phase in comparison with the host matrix, available computational tools, and databases.

Mole fraction of W and Fe in sublattice 1 and 2 x i Mole fraction of component i. x

Introduction
Ferritic and martensitic 9%-12% Cr steels are commonly used for high-temperature applications (500 °C-620 °C) due to their excellent creep resistance based on a combination of solid solution, grain boundaries, dislocations, and precipitation hardening.In martensitic/ferritic steels, M C 23 6 and MX carbides are the main reinforcement particles, that prevent the degradation of tempered martensite [1,2].It is well known the anchoring effect over grain and sub-grain boundaries promoted by the strengthening particles, which causes a reduction in the creep strain rate [3,4].Essentially, the reduction in creep resistance is associated with the evolution of tempered martensite because of the coarsening and dissolution of secondary phases (M C 23 6 and MX ) that reduce or eliminates the anchoring effect [5,6].Another factor that restricts the creep resistance is the nucleation of the Laves phase in regions close to M 23 C 6 carbides.While, in 9%-12%Cr steels, the precipitation of Laves consumes alloying elements that are in solid solution, it is not certain that the phenomenon is sufficient to have a significant effect on the high-temperature properties of the alloy.Moreover, depending on the chemical composition of steel, highly dense Laves-phase precipitation can achieve, thus stabilizing the martenstitic/ferritic matrix in the early stage of creep [7][8][9].
Laves phase is frequently identified with degradation of creep properties in high chromium steels.Even though these precipitates reinforce the matrix initially, their fast growth can promote long-term creep deformation [10,11].If the precipitate size is stabilized, Laves-phase is considered one of the most effective reinforcements by precipitation, which has led to the development of ferritic steels with chromium content higher than 13%wt.In these cases, Laves-phase acts by anchoring dislocations and grain boundaries [10,12].Recent studies of the heterogeneous nucleation of Laves phase in creep cavities have been demonstrated that its growth can increase the lifetime of metals under creep conditions by preventing coarsening, agglomeration, and crack propagation.These alloys are considered self-healing by Laves phase precipitation [13].The self-healing phenomenon is initiated by the segregation of the healing agents towards the cavity, which causes the nucleation of Laves phase in the damaged area, then its growth produces the complete or partial filling of the cavity, thus, improving the resistance to creep [14,15].Some evidence has been shown in 347 stainless steels modified with B, which shows self-healing properties [13].To date, there is a limited number of studies focusing on the autonomous filling of the creep cavities in steels.Despite this, it is known that supersaturated ferrous alloys of Au, W, Mo, B, or N show self-healing behavior [13].
Therefore, the purpose of this study is to extensively review the Laves phase features, such as the thermodynamic, precipitation kinetics, and self-healing property, to provide some insights and guidelines for future research work focusing on the development of precipitation-hardened high chromium steels with selfhealing capabilities by Laves precipitation.

Crystallographic description
Laves-phase is an intermetallic compound found in many materials.Its formation has been reported in more than 900 systems, some of which are presented in table 1.It has been found that Laves-phase main properties such as superconductivity, hardness, and wear resistance are rather dependent on the chemical composition.Considering the different alloy systems presented in table 1, it is possible to identify AB 2 as the main stoichiometry, where A are atoms with a larger atomic radius than B atoms.The crystallographic structure of Laves-phase corresponds to a Topologically Close-Packed structure (TCP), which can arrange in different polytypes [16].
The main difference between polytypes of the Laves-phase is the stacking sequence, like the differences between face-centered cubic (fcc) and hexagonal close-packed structures (hcp).The stacking sequence can be explained based on the formation of a quadruple layer as shown schematically in figure 1.Each quadrupole layer (X, Y, Z, X', Y' and Z') fit together to keep a close-packed structure.The stacking principle considers that each of the quadrupole layers can be arranged in a way that X, Y and Z follow the sequence X-Y-Z-X, and X', Y', and Z' are stacked following the sequence X'-Z'-Y'-X'.Ensuing these rules, different combinations can be formed and identified as long and short ranges.Those stacks formed by an even number of quadrupole layers show hexagonal stacking, while those comprising the uneven number of quadrupole layers are identified as rhombohedral stacking (except the three layers of stacking which correspond to cubic stacking).The most frequent short-range polytypes are C14, C15 and C36 (figure 2), considering 2, 3 and 4 quadrupole layers, respectively [17].
Regardless of the polytype, the formation of Laves-phase is promoted for an atomic-radius ratio r r A B ( ) within the range of 1.05 to 1.68 (r A and r B are the atomic radii of atoms A and B, respectively).Thus, the stabilization of Laves-phase depends on the ability of different atoms to shrink or expand [18,19] to form a closed-packed microstructure (stacking factor equal to 0.78).Besides, the abundance of Laves structures among metallic compounds has been attributed to principles governing the geometric arrangement of atoms on ordered lattice sites [18].Although the geometrical features of the unit cell provide some insight into the formation of the Laves   phase, the stability is mainly related to the polytype, and the valence electron concentration per atom within the unit cell, e.g., the C15 polytype in Mg-based alloy is stable for e/a>2.3 and C14 e/a<0.73(e: electrons, a: lattice parameter, figure 3).There are alloy systems that fulfill the radius ratio requirements, but the Laves phase formation is not observed due to the high valence electron concentration, such as the AB 2 compound (A = Ti, V, Nb, Ta, Mo, W and B = Ni, Co) with Si addition, which reduces the electron concentration and thus stabilizes the C14 polytype [18].Thus, different Laves-phase polytypes can be stabilized by incorporation of impurity or defects in the crystal lattice, as has been verified by quenching of alloy systems with high Cr Laves-phase, as well as Laves precipitates formed by rare earth elements (MCo 2 ; M = rare earth), in which the heat treatment stabilizes the hexagonal closed-packed polytypes (C14 and C36) [18].Conversely, it has been observed in high Cr alloy systems that NbCr 2 (with Ti Zr Hf Nb Ta Cr , , , Laves phase polytype) transforms into C15 from C14 polytype during solidification.In contrast, the rest of the precipitates are transformed during prolonged exposure to annealing temperature [17].
Unlike C15 and C36 polytypes, the C14 structure can be distorted, making room for impurities more easily.Therefore, when B atoms are replaced in AB 2 compounds with polytype C15 (in ternary systems), a transformation of the microstructure towards C14 is achieved, obtaining a deviated stoichiometry of the compounds [20].When the stoichiometry of the intermetallic moves away from AB , 2 changes are generated in its properties, e.g., in systems where it is possible to find UFe 2 type Laves phase, an excess of Fe of 0.3% by weight increases the magnetic properties and, along with this, the Curie temperature from 160 K to 220 K [21,22].
Deviation from stoichiometric composition (A 2 B) can affect the electrical properties at room temperature.Through measurements on single crystalline MgZn 2 , it was found that electrical resistivity can increase up to 20% due to chemical gradients.In general, Laves-phase has physical properties close to its former elements.Indeed, the Debye temperature and the thermal expansion coefficient of the intermetallic α(AB 2 ) have values between pure α(A) and α(B) [21].Turning now to the mechanical properties of Laves phases, stoichiometric deviations can influence the yield stress, and it is optimized when the Laves phase composition is close to AB 2 formula.Therefore, hardening by cold-working cannot be achieved in these systems, except in the case of Mg-rich alloy [21].Furthermore, in Ti35Zr5Fe6Mn and CrFeNiNb alloys, it was observed that precipitates with a composition close to AB 2 stoichiometry are responsible of the precipitation hardening [23,24].Likewise, in Mg-Al-Ca alloys (with three stable short-range polytypes), the C14 structure presents higher ductility than C15 and C36.On the other hand, the C15 polytype is the one with more remarkable plasticity [25].

Thermodynamic description
Based on experimental investigations and quantum calculations, the CALPHAD approach provides parameterized expressions that can describe the Gibbs free energy of a phase in a multi-component system.Consequently, it is possible to evaluate the formation of any phase by phase-diagram calculation and equilibrium determination.Although thermodynamic databases are actualized year by year, the method considers the same formalism.The approach consists of the parametrization of Gibbs free energy of a phase as a function of temperature (T), pressure (P), and composition (x) together with experimentally determined parameters related to a specific system (see equation (1)).This approach is summarized and shown in figure 4.
However, the input data must be critically evaluated to consider the most reliable data during parametrization.The model parameter is determined by error minimization procedures [26].Following the procedure in figure 4, to calculate phase and property diagrams an optimized parameter are used, which are compared with the input data.If the result is contradictory or unsatisfactory it is necessary to re-evaluate selected data.If not, the results obtained is incorporated to the database, which allows the prediction of thermodynamic equilibrium in higher order systems and macroscopic features (microstructural, anticorrosive, or mechanical properties) using additional information [26].
The purpose of the CALPHAD method is to describe the Gibbs free energy for each phase involved in a thermodynamic system.Thus, the energy function for an α phase is expressed as follows: Where srf G M α is the surface-free energy relative to other phase, ctg G M α is the solution-free energy, E G M α is the excess Gibbs energy, and phy G M α is the energy contribution associated with specific physical phenomena such as ferromagnetism [27].Additionally, the excess Gibbs energy depends on chemical composition and temperature, according to equation (3).
Where x i is the molar fraction of component i and L ij is the interaction parameter.Usually, L ij is modeled according to the Redlich-Kister polynomial, defined in equation (4) [28].
The procedure is generic for any thermodynamic system.However, for phases which deviate from the ideal stoichiometry, such as Laves phase, it is necessary to use the Compound Energy Formalism (CEF).The compound energy formalism was developed to describe a phase thermodynamically considering that atoms occupy specific sites in a crystal lattice that is constructed by two or more sublattices.The model can consider the effect of composition variations in the interaction energy between atoms (sub-regular model).The structure of a phase is represented by the formula, e.g.(A,B) k (D,E,F) l where A and B mix on the first sublattice and D, E and F mix on the second one.The coefficients k and l are the stoichiometric coefficients, and one mole of formula units thus contains k+l moles of atoms [29].In the case of C14 polytype AB 2, there are three sublattices, each one of them with different coordinates for the atomic positions, as shown in table 2 [30].
The sublattices model adopted by the Laves phase polytypes shows possible arrangements associated with the end-members compounds.Considering that the stochiometric deviation is mainly due to the presence of structural defects, vacancies are included as a constituent of the sublattices without mass or molar fraction and with an equilibrium potential close to zero.Therefore, the molar fraction of component i, and the constituent fraction can be described according to equation (5).
Where b ij is the stoichiometric factor of component i in the constituent j and a s is the number of sites in the sublattice.Since the molar fraction of the component is not sufficient to describe the phase in several sublattices, a new term y js is added to describe the fraction of the sublattice s that is occupied by the constituent j [27].Finally, equations (6) and (7) are derived using CEF, which are subsequently replaced in equation (2).
Where 'i' is the end-member with only one constituent in each sublattice, Π s (y iòIs ) is the product of a constituent fraction in a specific sublattice according to I, and °GI is the Gibbs free energy associated with the formation of the end-member i [27].Following this methodology, Aurelie et al [31] proposed the Gibbs energy function for the formation of Laves Phase Fe 2 W, which adopts the C14 polytype.Two sublattices are identified Fe W , ( ) ) in which the sublattices are commonly occupied by the element shown in bold print.However, every element mentioned in the sublattice description can adopt a position in any of the sublattices.Thus, the following end-member are identified: It is important to mention that only the second end-member from the list (Fe 2 W) corresponds to a stable phase, which can be evaluated through experimental data.The rest of the end-member in the list correspond to metastable or unstable phases, therefore, the energy for these compounds can only be estimated or calculated by ab-initio methods.Thus, the expression for the Gibbs energy of Fe W 2 phase is given by equation (8) [31].
Where x W Fe , 1,2 is the molar fraction of W and Fe in the sublattices 1 and 2, G W Fe : 0 is the formation enthalpy of formation at 298 K. Interaction parameters were not included in the description of this phase ( G M C E 14 ).It is noteworthy that according to table 2, there are three sublattices for the C14 polytype.However, in Aurelie et al [31] the description of the Fe 2 W Laves phase follows the formalism previously discussed in order to maintain simplicity and consistency with the database.Although the incorporation of the three lattices in the model led to a gain in accuracy, the increase in the number of parameters to be optimized with a three-lattice description does not justify the use of this model.

Precipitation kinetics
Laves phase precipitation in high chromium steels occurs in the heterogeneous regions along the matrix lattice, such as dislocations, grain boundaries, sub-grain boundaries, martensite lath boundaries and prior austenite grain boundaries, according to the microstructure of the alloy (ferritic or martensitic/ferritic matrix).Nucleation and growth theories are used to describe the evolution of the microstructure; therefore, the change in Gibbs free energy due to a nucleus formation is defined as follows: where V and A is the volume and the surface area of the precipitate, respectively.ΔG v is the driving force per unit of volume, γ is the matrix/precipitate interfacial energy, ΔG s is the misfit strain energy, and ΔG d is the energy associated to heterogenous nucleation sites [32,33].Before precipitation begins it is required that atoms diffuse through the supersaturated matrix to nucleation sites until the concentration of the 'i' component (x β i ) is locally increased, leading to a decrease in the Gibbs free energy.Then, the atoms are rearranged in the new crystal structure, producing an increase in the energy associated with the process.This energy is released by forming nuclei of the precipitates, denominated as the driving force for the nucleation as defined in equations ( 10) and (11).and b V m is the molar volume of b phase.If the values for the driving force are high enough due to a large undercooling, then the nucleation of Laves phase is promoted on dislocations in the matrix, assisted by the pipe-diffusion of the elements present in the precipitate.Thus, if the dislocation density increases, the number of critical nuclei formed will increase over a fixed time interval.Despite this, if the driving force is not high enough, Laves-phase will be formed at the grain and sub-grain boundaries.Considering that the precipitate adopts a spherical cap form, then the activation energy required for a nucleus to form at grain boundaries can be described as follows: where r * is the critical radius (equation ( 13)) and q is the wetting angle between Laves and the grain boundary [34,35].At grain boundaries DG s is very small and can be neglected.Similarly, the nucleation rate in grain boundaries is expressed by equation (14), where k and h are Boltzmann and Planck constants, respectively.
If the grain size is small, then the total area of the grain boundaries is large, and thus the regions for heterogeneous nucleation are larger (N 0 ).In turn, the activation energy for the diffusion of the atoms (Q) through the grain boundaries, which promotes the formation of Laves, is lower, considering an initial concentration c ¯ [35].
The tendency of the Laves phase to present a heterogeneous nucleation is associated with the crystal structure of the different polytypes.Indeed, an orientation relationship between the Laves phase and some crystalline phases exists, such as in the case of Cu-rich precipitates.It has been suggested that Cu-rich precipitates act as a nucleation site due to the setting of a high-coherence interface between the matching planes, like the case of austenite and Mo-rich BCC precipitate (as shown in table 3) with the C14 polytype [36][37][38].Li Peng et al [39] have identified an orientation relationship between the C15 polytype and the BCC matrix in Crbased alloys, and the mismatch between planes (equation ( 15)) depends on the chromium content in the alloy for a linear behavior of the thermal expansion coefficient (in both phases).
Likewise, if the mismatch is less than 25%, it will form a low energy interface (coherent or semi-coherent), and if the effect of the shape of the nucleus is neglected and assuming an isotropic matrix, the elastic deformation energy (DG s ) can be assumed as proportional to the square of the mismatch d .( ) Otherwise, it is described by equation ( 16) [33,40].Nabarro has proposed that when a precipitate nucleates with an incoherent interface, the elastic strain energy (equation ( 16)) depends on the difference between the shear modules of the precipitate and the matrix (Δ μ), the volumetric mismatch between the available space in the matrix and the precipitate ), and the shape adopted by the nucleus, which is reflected by the shape factor ( / f c a ( )).As the Laves phase approaches the AB 2 stoichiometry, there is an increase in the shear modulus difference between both phases, which produces a dependency between the mechanical properties and the chemical composition of the precipitate, given by the associated increase in the strain energy elastic promoting the formation of dislocations at the interface [40].
The interface between the matrix and the Laves phase is characterized by its anisotropic nature, with areas in which the interface energy reaches low values (coherent or semi-coherent) and others that present high values (incoherent interface).Usually, the growth of the Laves phase occurs in the direction where is formed a highmobile interface (incoherent), and the mean radius r as function of the time (t) can be described by equation (17).Furthermore, growth is also depending on the supersaturation (Ω) together with the diffusion coefficient of the elements that promote the formation of the Laves phase (D , i the diffusion coefficient for element i).In turn, the supersaturation (equation ( 18)) is influenced by the content of the Laves-phase components in the alloy (c ¯) and the equilibrium concentrations ( b c and a c in the precipitate and the matrix, respectively).Therefore, higher contents of elements such as tungsten, molybdenum, and niobium, among others, not only promote nucleation but also the growth of Laves phase particles in high-chromium steels [35,41,42].
Finally, when nucleation and growth are completed, the volumetric fraction of the precipitate becomes constant, followed by the coarsening of particles defined by the Ostwald-ripening equation (equation (19)).
Where m = 3 if volumetric diffusion dominates, r o and r are the initial average radius and the radius at time t, K p is the coarsening constant calculated by equation (20), b V m is the molar volume of the precipitate, D i is the coefficient diffusion of element i in the matrix, b x i is the fraction of element i in the precipitate and / a b x i is the molar fraction at the interface precipitate/matrix [43].
4.1.Formation of Laves phase in 9%-12%Cr martensitic/ferritic steels According to figures 5, 9%-12%Cr steels present a tempered martensite microstructure obtained from normalizing and subsequent tempering treatment in thin-walled component.During tempering it gives rise to the formation of sub-grains structure and the nucleation of M 23 C 6 carbides and MX carbonitrides which stabilized the tempered martensite matrix at high temperatures.Due to the thermodynamic and crystallographic nature of M 23 C 6 carbides its formation occurs mainly at prior austenite grain boundaries, while MX carbonitrides present an heterogenous nucleation in sub-grain and lath boundaries.Also, alloying elements in solid solution, such as tungsten and molybdenum, provide solid solution strengthening, thus increasing the Young modulus of the matrix.Furthermore, solid solution elements can decrease the volume diffusion, avoiding the recuperation of the sub-grain structure by dislocation climbing [44].Although, MX carbonitrides are the main reinforcement particle that retard the recuperation of the tempered martensite by anchoring the sub-grain and lath boundaries during the creep, the precipitation of secondary phases such as Laves phase can occur, which consumes the molybdenum and tungsten from the matrix, reducing the mechanical properties at high temperatures [45][46][47].
Several studies have identified that the C14 polytype Laves phase forms in service at 600 °C with a composition close to (Fe, Cr) 2 (W, Mo) for longer exposure times than those necessary for the appearance of M 23 C 6 carbides.These precipitates tend to nucleate in heterogeneous zones such as prior austenite grain boundaries and adjacent to M 23 C 6 carbides.On the other hand, evidence has been found that nucleation of the Laves phase inside the martensite laths can occur by the presence of dislocations, which act as nucleation sites, this phenomenon can be enhanced by pipe-diffusion through the sub-grain network [47][48][49][50].
The nucleation process of the Laves-phase starts with the segregation of elements such as molybdenum, silicon, and phosphorus towards micro-grain boundaries [51,52].Hence, Laves-phase precipitates at the grain boundaries maintaining a semi-coherent interface with one of the adjacent grains.On the other hand, in the surroundings of M C 23 6 carbides, chromium-depleted zones are generated promoting the enrichment of tungsten and molybdenum, thus enhancing the nucleation rate of Laves phase [53].In both cases, precipitation is controlled by the diffusion of elements that are part of the precipitate phase.This suggests that there are better conditions for precipitation at the grain boundaries compared to the rest of the interfaces (sub-grain boundaries, martensite lath or block boundaries).However, observations indicate that Laves phase nucleates primarily in the regions surrounding the carbides [54].
As inferred from equation (17), the growth of the precipitates is largely controlled by volume diffusion, in this case, of elements such as tungsten, molybdenum, and silicon.However, when M 23 C 6 carbides are surrounded by Laves-phase another mechanism needs to be considered.As shown in figure 6, the precipitates that initially nucleated, assisted by chromium-depleted zones adjacent to the carbides, grow at the expense of tungsten by short-circuit diffusion mechanism, i.e., diffusion across the interface Matrix/Laves phase and grain boundary [55,56].Up to this point, it can be understood that a higher concentration of the mentioned elements would facilitate the nucleation and growth processes of the Laves phase.Therefore, lower Mo contents are  considered in the design of alloys in such a way that a higher concentration of silicon in the matrix achieves precipitation or, instead, has a higher concentration of W [57].However, an increase of 2-3 wt.% of tungsten would tend to increase the precipitate/matrix interfacial energy, resulting in greater coarsening rate due to the high mobility of incoherent interfaces.A similar case is that of rhenium, which has been shown that, although the interfacial energy increases, at the same time, the diffusion rate of tungsten, molybdenum, and chromium decreases, having the opposite effect on coarsening [58,59].Another element that is important to consider is cobalt.There is evidence that content of 3% reduces the coarsening rate by up to four times.Despite this, to date, it is not certain whether the effect is purely on the diffusion of the elements or because it reduces the solubility of tungsten in the matrix.From the latter, it follows that modelling software does not assertively represent the effect of cobalt on the Laves phase [56,58].

Laves phase formation in high-chromium ferritic steels
In ferritic steels with chromium content higher than 12 wt.%Laves phases tend to form in dislocation and grain boundaries.In this system, there are two types of intermetallic, called as low-temperature (LT Laves-phase) and the high-temperature Laves-phase (HT Laves-phase), denoting the temperature range of thermodynamic stability.The former is stable within 550 °C-650 °C, whereas the latter is stable up to 1000 °C [60,61].Although both phases precipitate with a C14 polytype crystal structure, the main difference between them is the chemical composition.During heat treatment, the LT Laves phase precipitate at 600 °C with a niobium-rich composition, followed by the precipitation of the HT Laves phase at a higher temperature with a tungsten-rich composition, giving rise to a dual-phase precipitate with a core of LT Laves phase and a shell of HT Laves phase.This phenomenon is attributed to the faster diffusion of niobium compared to tungsten at 600 °C, as well as the accelerated diffusion of silicon [61,62].One of the factors that affect the nucleation kinetics of Laves-phase is the presence of heterogeneous nucleation sites, such as gran boundaries and dislocations.Thus, an enhancement in deformation led to an increase in the nucleation rate during ageing treatment due to the higher dislocation density, therefore, an increase in nucleation sites (N 0 ), as well as a reduction in the activation energy for nucleation (equation ( 12)).Likewise, the thermomechanical treatment led to the nucleation of fine precipitate due to the fast nucleation, which is promoted by a low strain rate [62][63][64].
On the other hand, regarding the chemical composition of high chromium ferritic alloys, elements such as tantalum and zirconium promotes the precipitation of Laves-phase, in the same way molybdenum, niobium, and tungsten, which have a significant influence on the nucleation rate.Besides, tantalum causes a homogeneous distribution of Fe Mo Ta , 2 ( )particles [64-66].Although the presence of some elements may have a positive effect on the kinetic of Laves, like silicon promoting nucleation, some other elements are known to hinder the precipitation of the intermetallic phase.For instance, yttrium acts by reducing the diffusion of Laves-phase forming elements (such as molybdenum and niobium), leading to a low density and small size of particles [67,68].It is important to consider that the composition of these alloys promotes the formation of other phases like Chi, Sigma and Nb 2 C. Therefore, the evolution of Laves-phase is not just related to growth and coarsening assisted by the fast diffusion of elements through crystalline defects, but also due to the tendency to form sigma, caused by the high chromium content near the precipitates.Conversely, Nb 2 C tend to transform into Lavesphase, releasing carbon to the surroundings, causing Lüders deformation due to Cottrell atmosphere formation [69][70][71].It has been identified that during high-temperature exposure to 20% Cr ferritic alloys, Laves-phase nuclei possess a spherical shape with low interfacial energy due to the orientation relationship between the ferritic matrix and the C14 polytype.During microstructural evolution, Laves-phase precipitates adopt different morphologies from faceted ellipsoidal to the faceted plate, with a lateral growth over the planes with larger lattice mismatch [72,73].It demonstrates that the growth is significantly influenced by the orientation relationship, with a major impact on the nucleation kinetics and the diffusion of Laves-phase promoting elements, especially the combined effect of silicon and molybdenum as C14 polytype stabilizers.Particularly, silicon adopts the position of iron in the crystal structure, which tends to balance the radius ratio to 1.22 [73].

Computational modelling of precipitation kinetics
Precipitation of phases is usually simulated by the Kampmann-Wagner (MKW) model, which allows to calculate the parameters associated with the nucleation and growth process in multi-component systems.The model assumes that each phase in the system is spherical.Thus, phases are classified at every time interval based on the chemical composition, stoichiometry, and size.It is assumed that the phase composition is constant at every position and is time-independent, while the nominal composition and the matrix composition are independent of the precipitate type.Following these assumptions, the numeric balance of the model is described by the partial differential equation expressed in equation (21).
Where f represents the distribution density function, v is the growth rate, S is the nucleation rate, r is the radius of the precipitates, and t is the time associated with the precipitation [74].
As discussed previously, the nucleation of Laves-phase is primarily heterogeneous.However, for the development of the MKW model, it is assumed that the nucleation is homogeneous, and the misfit strain energy is zero (DG s ).Therefore, the critical radius (equation ( 22)) and the activation energy (equation ( 23)) are obtained by deriving the Gibbs free energy for nucleation process (equation ( 9)), considering the model proposed (spherical precipitates and homogeneous nucleation) [75]. 23 The events are studied in time intervals for each precipitate and classified according to its radius.Thus, it is necessary to consider the time-dependent nucleation rate (equation ( 24)) as a function of the nucleation sites, the probability of a supercritical nucleus to dissolve (Z, Zeldovich factor), the frequency at which an atom or molecule contributes to the growth of the precipitate (equation ( 25)), and the incubation time of the precipitate (equation ( 26)), where the parameter j varies on each derivation [76].
Currently, computational tools for alloy modeling, such as TC-Prisma and MatCalc, follow the MKW model.Regarding the precipitation simulation, the main difference is how the driving force for the nucleation is calculated.TC-Prisma uses the parallel tangent method to obtain the maximum driving force for the transformation (equation ( 9)), considering that the nucleus presents a chemical composition close to equilibrium.On the other hand, MatCalc calculates the lowest energy barrier for the nucleation by the determination of the saddle point of a function dependent on the chemical composition and the nucleus size.It is considered the activation energy that maximizes the precipitation probability (equation ( 27)).Additionally, it provides values for the driving force, which is dependent on the nucleus composition.Similarly, MatCalc considers the activation energy within the modeling so that it maximizes the nucleation rate.In either case, the software obtains the minimum value for the activation energy, which is a combination of the driving force and the interfacial energy [77].
There are some discrepancies between the theory and the model description.In TC-Prisma the interfacial energy is assumed to be constant during the phase transformation process and is approximated according to the Becker model (equation ( 28)) before beginning the simulation, where n s is the number of atoms per unit of area, z s is the number of atoms bonded through the interphase, z 1 is the coordination number of the atoms in the matrix, N A is Avogadro's number and DE s is the solution enthalpy in multi-component system.The approximation for the interfacial energy in MatCalc (equation ( 29)) is similar to the one implemented in TC-Prisma but differs from an additional factor as a function of the precipitate size (a r ( )), which increases from 0 to 1 as the radius of the nucleus increases and a function / f T T c ( )that represents the influence of the diffuse interface [77].
To simulate the growth in multi-component systems, the simple model adopted by TC-Prisma considers that the chemical potential at the interface is only affected by the Gibbs-Thompson effect (equation ( 30)).Thus, the growth rate (equation ( 31)) can be obtained along with the chemical concentration of the interface by solving the numerical methods.
Where / m a b and / m b a are the chemical potential in the matrix and the precipitate at the interface, respectively.K is a factor obtained from the intersection of the chemical parameters in the numerical derivation [78].In turn, the growth modeling in MatCalc considers that the diffusion of atoms that are part of the spherical precipitate is equal in every direction.It is well known that this process produces a reduction in the free energy of the system that is induced by the evolution of the precipitates due to the supersaturation in the matrix.In this context, the software models the energy dissipation that originated due to this change.There are three types of energy dissipation associated with homogeneous nucleation, interface displacement (Q 1 ), diffusion inside the precipitate Q , 2 ( ) and diffusion in the matrix Q , 3 ( ) which are described by equations ( 32)-( 34), respectively.
Where M k is the interface mobility, C ki is the concentration in the precipitate, C i 0 is the concentration in the matrix,  J ki is the diffusive flux in the precipitate, J ki ̈is the diffusive flux in the matrix, D ki and D i 0 is the diffusion coefficient of the element i in the precipitate k and in the matrix, respectively.Thus, the growth parameters can be obtained by adding the total dissipation 3 ) to the thermodynamic principle [79].Each software considers different parameters to model the nucleation and growth.Besides, the theoretical and practical features differ up to some point in the description of the Laves-phase [80,81].Sanhueza et al [80] have simulated the precipitation of the Laves-phase, considering that the nucleation is controlled by alloying elements such as tungsten, molybdenum, and silicon, since they have low mobility in the matrix, then assuming faster mobility through the grain boundaries and a constant incoherent interface.From a different perspective, MatCalc modeling considers the change of the interfacial energy during the transformation process.Thus, to approach the orientation relationship between the precipitate and the matrix, simulations are performed considering the assumption that the interface is totally coherent, providing high stability and a high number of precipitates due to the addition of tungsten in P92 steels [81].

Laves-phase as reinforcing precipitate at high temperature
There are different reinforcing mechanisms that affect the mechanical properties of metals and alloys.It is possible to identify two types of reinforcement associated with alloying elements: solid solution and precipitation hardening.Both mechanisms are effective in slugging the dislocation motion at low temperatures, however, at high temperatures due to the higher atomic mobility dislocation can easily surpass these obstacles through climbing [82].In martensitic/ferritic steels the alloying elements have a fundamental contribution by stabilizing the tempered martensite at high temperatures and avoiding its recovery [82,83].Indeed, this is achieved mainly by the fine precipitation of MX carbonitrides and M 23 C 6 carbides along boundaries and subboundaries (prior austenite boundaries, block boundaries, lath boundaries and sub-grain boundaries).However, depending on the chemical composition of the alloy, Laves-phase can precipitate finely dispersed and contribute as a reinforcement particle of the matrix, with similar characteristics demonstrated in highchromium ferritic steels [84,85].
To study the reinforcement of the matrix due to the precipitation of the Laves-phase, it was considered two nucleation sites, along grain boundaries and sub-grain boundaries (assisted by dislocations), therefore the reinforcement has an intergranular and transgranular effect [86].Inside the grain, the Laves-phase precipitates anchoring the dislocations motion and sub-grain boundaries, thus preventing dislocation glide and sub-grain recovery.At first approximation, the effect of Laves-phase precipitation can be quantified by the Orowan stress t , or ( ) i.e., the maximum shear stress needed for dislocation surpasses a field of finely dispersed precipitates and can be described by equation (35).25 36

¯( )
Where G is the shear modulus, b is the Burgers vector, l p is the spacing between precipitates (as described by equation (36)), N is the number of particles per unit area and d ¯is the average diameter of precipitates [86].
During high-temperature exposure, the precipitates will increase their size over time.Thus, the Orowan stress required for the displacement of dislocation decreases, reducing the anchoring effect, allowing the dislocation to pass through the Laves-phase precipitates [85,87].It is well known that the main contribution of precipitates over the mechanical properties is the anchoring of dislocation.Despite this, at high temperatures, the particles that nucleate on high-angle grain boundaries are relevant for enhancing the creep resistance [88].Sun et al [89] have shown that Laves-phase stabilizes the microstructure of ferritic steels due to the anchoring effect that inhibits the nucleation and growth of new recrystallized grains.Based on the study the pinning pressure provided by Laves-phase (P z ) over the grain boundaries and sub-grain boundaries (equation ( 37)), which are larger than those required for the recrystallization (P r ) and growth (P g ), as expressed in equations (37), (38), and (39), respectively.
Where (g ) is the interfacial energy, ( f ) is the volume fraction of precipitate, (l p ) is the average spacing between precipitates placed at the interface, ( r D Dn ) is the dislocation density difference between the recrystallized and the non-recrystallized zone, and (d g ) the grain diameter.
Most of the factors influencing precipitation strengthening depend on the particle size obtained during the nucleation and growth stages [90].On the other hand, during exposure to high temperatures under constant load, high chromium ferritic steels have been found to have high intermetallic stability, unlike the rapid growth observed in 9%-12%Cr ferritic/martensitic steels.In both cases, the creep conditions tend to increase the diffusion of the elements that form Laves-phase, affecting the growth and coarsening of the particulate, decreasing the anchoring on the high-angle grain boundaries [90,91].Initially, the fine Laves-phase adopt the shape of a needle, maintaining a semi-coherent (or coherent) interface with the matrix.This acts by anchoring the dislocations without causing embrittlement [92,93].Following the evolution of the precipitate, it loses coherence as it grows, adopting the shape of a rod and then a block, but reinforcement is not achieved to a great extent due to the size reached by the particles [92,94].Currently, there are high chromium ferritic steels strengthened by Laves-phase precipitation.Lopez et al [95] have proposed a Fe23Cr2.5W0.57NbTitype alloy resistant to creep, in which the Laves phase maintains a low growth kinetics during service.However, the alloy fails because a precipitate-free zones are generated.On the other hand, in commercial ferritic-martensitic steels, Laves-phase largely prevent recrystallization during early creep stages.However, when it exceeds the critical size, it promotes the formation of creep cavities in the recrystallized zones because the precipitate acts as a stress concentrator, which is attributed to the loss of coherence in the Laves/Matrix interface [96][97][98].

Self-healing in ferrous alloys by Laves phase precipitation
The term self-healing is relatively new in the material science and engineering field.It consists not only of the ability to resist extreme service conditions but also the ability to heal after the damage is produced during service [99].The self-healing process can be promoted either by the nature of the environment or because of an external agent (such as thermal, mechanical, electrical, or magnetic stimulus).The former is associated with intrinsic selfhealing by precipitation, which occurs under creep conditions in ferrous alloys [100].In other words, when a cavity is formed, the supersaturated solutes atoms of a healing agent diffuse toward the damaged zone to heal it by precipitation of a newly formed phase until the cavity is completely repaired.It is necessary that the healing agent segregates near the cavity, promoting nucleation and growth before the damaged region reaches a considerable size.The temperature considered optimum for self-healing is in the range -T 0.4 0.6 m ( ) [100].The solute atoms that satisfy the requirements to be considered a healing agent in ferrous alloys are boron, copper, molybdenum, tungsten, and gold, the latter the one with the most promising performance [100].Although most of the healing agents have been studied in binary alloys, austenitic stainless steel AISI 314, 347, and 304 modified with boron have been demonstrated to be self-healing by precipitation of BN under creep conditions.Similarly, Inconel alloys have shown self-healing properties by segregation of boron, carbon, and sulfur, which promotes the precipitation of M 23 C 6 on creep cavities [101,102].

Self-healing mechanisms
Ferrous alloys exposed to creep conditions produce microstructural changes that are dependent on the temperature and the applied stress, such as recrystallization, formation and displacement of dislocations, along with the formation and growth of creep cavities in grain boundaries or secondary phases (figure 7).These cavities are small and initially separated, but over prolonged exposure, these tend to grow due to self-diffusion, as shown in figure 7. If the growth rate is large enough to promote the coarsening of the cavities, the formation of cracks is likely, which leads to the failure of the component [103].
Following the mechanism previously discussed, the self-healing process plays a role during early-stage cavity formation.The healing agent atoms diffuse toward the damaged zone, and subsequently, the nucleation and growth of precipitates take place so that the cavity is filled before the coarsening process.It is necessary that the energy barrier for the nucleation of a precipitate is lower than in the matrix for self-healing to occur.Precipitates with incoherent interfaces tend to show better healing properties.This means that elements with a larger radius ratio relative to the matrix elements could be potential healing agents.For instance, between gold and iron there is a radius ratio of r Au /r Fe = 1.13, which is large enough to avoid precipitation of gold-rich precipitates in the ferritic matrix [104][105][106].Regardless of the features of the healing agents, self-healing is mainly limited to the rate at which the cavities are filled as compared to the growth rate of the cavities.Thus, atomic mobility, along with selective nucleation, is considered the key factor for the self-healing property.The largest size of a cavity that can be healed is given by Dt 2 ,where D is the diffusion coefficient, and t is the time that it takes to self-heal (it is estimated that an acceptable time for the process is 10 5 s).It is noteworthy mentioning that the presence of microstructural defects tends to promote self-healing due to facilitated diffusion [104,107,108].
The driving force for self-healing depends on the potential chemical difference due to supersaturation, the lattice strain in the vicinity of the damaged grains, and the surface energy of the cavity [109].Therefore, the healing of the damaged regions will occur only if the cavity reaches a critical volume, which depends on the local condition of the grain boundaries associated with the crystallographic orientation relative to the applied stress.In binary and ternary self-healing alloys (Fe-Cu, Fe-Au, Fe-Mo, Fe-W, and Fe-Au-W), the chemical composition reaches a 1% of supersaturation of the healing agent to promote the segregation in the cavities [110,111].
Regarding the filling mechanism, Fu et al [112] identified four stages for the self-healing of a Fe-Au-W alloy (figure 8).Initially, the solid solution is supersaturated in healing atoms (Stage I).When exposed to high temperature, there is a segregation of healing atoms toward the grain boundaries promoting the nucleation of Laves phase Fe W 2 and gold-rich precipitates, which are also found to precipitate on dislocations (Stage II).If a cavity is formed, then gold atoms diffuse to the grain boundaries and then toward the cavity.Gold atoms in the grain boundaries are consumed to repair the cavity, followed by the diffusion from precipitated formed on dislocations (Stage III).Finally, if a cavity is formed in a depleted region, the healing process occurs due to the diffusion of tungsten and nucleation of Laves-phase at the cavity (Stage IV).It is noticeable, that if the precipitate does not have enough time to grow and fill the cavity, the self-healing property cannot be achieved.
The self-healing phenomena is a time dependent process that it is controlled by the supersaturation level.Then, it is possible to estimate the healed damage by a volumetric fraction balance in the cavities ( f D ) as compared to the maximum volumetric fraction that solute atoms can fill ( f H ), resulting in a cumulative fraction of the healing agent (equation ( 40)).
Where d g is the grain size, x m and x 0 are the nominal solute concentration and equilibrium concentration, respectively, d c is the cavity size, N c is the amount of cavity, and R is the scaling factor for the effective volume occupied by the solute atom in the precipitate compared to its volume occupied in the solute.In Fe-Mo alloys with 3.7%wt molybdenum, 3%wt supersaturation, 20 mm grain size, and cavity size of 1 mm, then 450 cavities can be healed before the solute reservoir are empty, considering y as the geometrical factor associated to the aperture angle of the cavity [113].Moreover, the filling of the cavities under creep conditions depends on the activation of short-circuit paths, i.e., grain boundary diffusion surpasses lattice diffusion.Then, the maximum volumetric fraction f H kin that can be healed is strongly dependent on the temperature and grain size (equation ( 41)).If the mean free path reached by diffusion of solute atoms through the lattice (2√Dt) is greater than the grain size, the entire solute reserve contributes to the self-healing process, in which case = f f H H kin [113].

Computational modeling of the self-healing process
According to the mathematical model proposed by Versteylen et al [114], under secondary creep conditions, the vacancy concentration at the grain boundaries (x v GB ) increases due to the interaction with dislocations.However, if a maximum concentration is reached at the grain boundaries, then a repulsive barrier to the displacement of dislocations will appear (equation ( 42)).This barrier is negligible if there is a flux of vacancies (J v ) to the cavity through the grain boundaries (with a diffusivity D v GB ) due to a potential gradient between the surface area of the cavity and another region far from the cavity with a distance λ.The potential gradient increases if there is stress applied to a vacancy in the crystal lattice (Ω v ), which is considered equal to the stress (σ).
Since the growth of the cavity is a result of the vacancies flux (figure 9), the volume of the cavity will increase only if there is a continuous generation of vacancies at the grain boundary.It means that the growth rate of the cavity (  V ) is limited by the formation of vacancies, which depends on the dislocation motion.The latter determines the strain rate by creep ( e).It is considered that the growth rate and the strain rate are proportional to each other.In particular, the creep controlled by dislocation climb is related to equation (43).
Where b is the Burgers vector, ρ Dn is the dislocation density, υ t is the climbing rate, d g is the grain size, and 4λ 2 is the area in contact between the cavity and grain boundary [114].On the surface of the cavity, precipitation of particles may occur only if supersaturated solute atoms segregate close to the cavity.Due to potential gradients between solute atoms in the cavity and in the matrix ( m  sol ), the flux of solute atoms initiates to promote the growth of precipitate (equation ( 44)), thus, decreasing the surface energy as well as reducing the stress levels in the cavity.At this point, exist a flux of solute atoms and vacancies in opposite directions, the so-called Kirkendall effect.The difference in diffusivities of the host (D host ) and substitutional solute (D sol ) causes a net flux of vacancies in the opposite direction (equation ( 45)) to the specie with the fastest diffusivity (it is assumed that the solute concentration gradient is x sol = Δx sol GB /λ, where Δx sol GB is the concentration profile of the supersaturated solute on the grain boundaries) [114].

( ) ( )
As shown in figure 9, after a cavity nucleates, a flux of vacancies through the grain boundary toward the cavity promotes the growth of the cavity, followed by a solute flux in the same direction, which contributes to the precipitate nucleation and growth.Therefore, a net flux of vacancies arises in the opposite direction (figure 9(b)).Although the driving force for the nucleation of precipitates is the potential solute gradient, the self-healing process could also be limited by the available supersaturation during service.Besides, the flux of vacancies outward must be larger than the flux of vacancies toward the cavity (equation ( 46)) so that the precipitate growth rate is faster than the cavity growth rate [114].

( )
One condition for the healing process to occur is that opposite fluxes are at least equal, then it is possible to describe the critical stress for the healing process (s crit ) (equation ( 47)), which provides maximum efficiency (equation ( 48)), which means that the alloy can completely heal the creep cavities.If the applied stress exceeds the critical stress, the self-healing efficiency progressively decreases until it reaches zero for stresses that are too high.Despite this, the strain rate in the second creep stage (equation ( 49)) is lower in self-healing steels than in steels without this property [114].
Up to this point, the proposed model considers that the process is controlled only by diffusion through the grain boundaries, but usually, the flux of vacancies is due to volume diffusion.Versteylen et al [115] have shown by finite element modeling that if the intercavity spacing is much larger than the cavity radius, and the diffusivity at grain boundaries (D GB ) is greater than the diffusivity at the matrix (D M ), then the time of filling depends only on D GB and the width of the grain boundaries (diffusion regime in 2D).Similarly, if D GB = D M the solute flux depends on both (3D diffusion regime).On the other hand, if the intercavity spacing is small, the creep cavities are close to each other, the solute in the grain boundary is insufficient, and its flow is through the matrix (diffusion regime in 1D).It should be noted that the precipitates in the cavity can grow only by diffusion in 2D or else in 1D when the cavities have already reached a considerable size [115].
Regarding the design of self-healing alloys by precipitation, the proposed mathematical model can be complementary to conventional methodologies for the development of new alloys.Yu et al [116] have proposed a computational method using genetic algorithms (GA) for modeling self-healing phenomena.Genetic algorithms are based on evolutionary processes in nature, where possible solutions to a problem evolve until optimal performance is achieved [117].
The initial development of the algorithm considers the ranges for the different alloying elements according to the desired properties (figure 10).Possible candidates are randomly generated, assigning them a binary chain (chromosomes) linked to the concentration of each alloying element in base 2 (genes).Thus, the lower limit is represented as 00000 and the upper limit as 11111, obtaining a maximum of 2 5 possible compositions for each alloying element.Therefore, a universe of 32 var alloys can be tested, with var being the number of variables involved in the design.Considering the micro-GA technique, initially, ten possible candidates are generated, which are tested by evaluating the thermodynamic parameters that depend on their composition (fitness) [117].
During fitness, the required properties are associated with the microstructure and composition using established metallurgical principles.For each possible alloy, those that had the best behavior are selected to contribute to the following generations (figure 10).Then, each chromosome is assigned a probability of reproduction so that only a few are selected for the crossover and mutation operation.As indicated in figure 10, the new generations are tested again until an optimal solution to the problem is found that corresponds to an alloy design that satisfies the proposed conditions for service [117,118].
6.3.Self-healing phenomena in high-chromium heat-resistant steels the next step: genetic algorithmsupported modeling 6.  4).Extensive research over almost 40 years resulted in the development of commercial steels like X20, TAF 650, T/P91, T/P92 and T/P93.However, the high-temperature resistance of all these steels is based on traditional metallurgical concepts such as solid solution, precipitation, and dislocation hardening [121][122][123].Indeed, recent efforts are focused on increasing the high-temperature resistance by controlling the precipitation kinetics of nitride and intermetallic phases.Z-ultra project funding by the Fraunhofer Institute for Mechanics of Materials developed 12%Cr creep resistant steels for ultra-supercritical power plants strengthened by Cr 2 N nitride (Z-phase) [124,125].The project finalized in 2016 with more than 30 publications and with prototype tubes into boilers of two Ukrainian power plants for demonstration purposes [126][127][128][129][130][131].On the other hand, Laves-phase strengthened steels seems to be a complex challenge at least in martensitic/ferritic steels, mainly due to the crystallographic features of this phase in comparison with the host matrix [41,81].Laves phase has a partially semi-coherent heterogeneous nucleation in grain boundaries [10,11].Nevertheless, researchers have found that the addition of rare earths promotes a high fraction of fine Laves phase particles, therefore increasing its effectiveness as a strengthening particle [59,132].
6.3.2.Self-healing in high-chromium heat resistant steels: just starting Self-healing phenomena are recent (see table 4), which changed the tailor-made materials paradigm from 'damage prevention' to 'damage management', i.e., high-temperature materials can be designed with an in-built capability to repair local damage during service autonomously [100].Indeed, the Dutch government funded a 20-million-euro research program on Self-Healing Materials in 2006.The research was not restricted to one material class or one particular healing approach.The program was finalized with investigations of self-healing properties in ternary metallic systems.
To date, the self-healing phenomenon has been identified in iron-based alloys with two and three components (Fe-Au, Fe-Cu, Fe-Mo, Fe-W, and Fe-Au-W), and in commercial stainless steel modified with boron (AISI 312, 347, and 304), nevertheless with a little contribution to creep rupture life [].Modeling the selfhealing process in a multicomponent system can be complex.Yu et al [117], proposed a genetic algorithm for modeling self-healing phenomena in martensitic/ferritic alloy reinforced by M 23 C 6 carbides, and Laves-phase as healing particle.The following requirements are established as border conditions: (1) After homogenization and quenching, the volumetric fraction of ferrite must be greater than 99%.
(2) After quenching and tempering, the maximum level of carbides must be at most 0.5% by volume.
(3) During service the maximum volume of detrimental phases must be less than 1%.
(4) There must be a sufficient Cr content in the matrix to maintain resistance to oxidation.Afterward, the optimization criteria are calculated from equation (50) and equation (51) for every alloy that meets the mentioned requirements.
where K p is the coarsening factor, x i m is the molar fraction at the equilibrium of every element in the matrix, C i is the solid solution hardening coefficient and a T is a scalar dependent on the temperature.The parameters required for this, such as the driving force for nucleation, the diffusion coefficients, and the molar volume of the precipitate, are obtained from the available database in Thermocalc [116].At this point, every alloy that meets the mechanical requirements have been identified.However, self-healing should be considered a new property.It has been proposed by Yu et al [116] that a criterion to evaluate the self-healing property is to consider that the volumetric fraction of the Laves phase that can form in the cavities must be greater than 1%.In addition, the times associated with the nucleation processes of the cavities and precipitates should be considered as a restriction.

High chromium steel production
High chromium steels are heat treated after casting and forming, following a solubilization, cooling, and tempering sequence [133].During normalizing, the soaking time and temperature become critical factors in obtaining homogeneous properties in the final microstructure of the steel.It is recommended that the holding time must be according to the thickness of the component to be treated (one hour per inch of thickness) [134].
During the final tempering, the arrangement of dislocations forms the subgrain network, and precipitation of fundamental particles which stabilizes the tempered martensite at high temperatures occurs.To achieve optimal high-temperature properties the control of heat treatment parameters is primal, due to the attained microstructure being very sensitive to this stage.A high normalizing temperature and a low tempering temperature produce a high dislocation density.On the other hand, a high tempering temperature may lead to a low dislocation density as well as an enlargement of martensite lath along with the growth of precipitates, which is highly dependent on the heat treatment time [134,135].
During the manufacturing of steels, every relevant microstructural feature is altered, such as dislocations, grain size, and precipitates.These can be controlled by the combined operation of forming and heat treatment, the so-called thermomechanical treatment [136,137].After thermomechanical treatment, martensite adopts similar characteristics as the martensite obtained by water-quenching.This is because a fully austenitized steel has a high dislocation density in the matrix, which promotes fine martensite formation as well as influences nucleation reaction [138].
Compared to traditional processes, steels processed by thermomechanical treatment have a martensitic microstructure decorated with fine and homogeneously distributed precipitates.It is due to the increase in the dislocation density that acts as a heterogeneous nucleation center for precipitates and influences the diffusion of precipitates forming elements [139,140].Thus, strain at high temperatures alters the energy level inside the grains and in regions close to the grain boundaries.Besides, it increases the driving force for the transformation of austenite into ferrite [139,140].
According to the above, the thermomechanical treatment has a beneficial effect on the nucleation of Lavesphase since the precipitation on a dislocation decreases the associated free energy (equation ( 52)) due to the destruction of the defect ( mb r 0.4 , 2 with μ the shear modulus and b the Burgers vector), resulting in the reduction of the energy barrier for nucleation (equation ( 53)).In addition to the effect on the diffusion through dislocations, there is a notable increase in the nucleation rate for the Laves phase, so the application of the thermomechanical treatment would result in a finely dispersed Laves phase [141].

Characterization techniques
To study microstructural changes in steels, it is common the use of scanning electron microscopy (SEM) and transmission electron microscopy (TEM) techniques.Although the latter allows obtaining data at the atomic level regarding the microstructure, the use of SEM coupled with back scattered electrons signal provides a variety of information that is more representative of the sample, especially in the study of precipitation, where the Laves phase is identified easily, since it is rich in elements with a high atomic number [142,143].Among the common precipitates in high chromium steels, only Z-phase and M 23 C 6 carbides are distinguishable [142,143].Due to the size of secondary phases, SEM is one of the most used techniques in alloy research to study the microstructure and morphology of precipitates [144,145].On the other hand, the x-ray diffraction (XRD) technique makes it possible to identify the crystallography of precipitates.In addition, through the application of different XRD methodologies, it is possible to obtain data regarding dislocation density, crystalline parameters, and lattice deformation [146,147].
As mentioned in previous chapters, it is necessary to study the precipitation kinetics of Laves-phase to confirm that it satisfies the characteristics of a reinforcing particle and a self-healing precipitate.Recent studies [91,96] proposed the study of microstructural evolution during precipitation annealing at different time intervals, for a given temperature, by means of SEM analysis to acquire data on the width, length, and density of the precipitates.Similarly, the self-healing ability can be identified by SEM observation since the precipitates adopt the shape of the cavities, evidencing partial or fully filled cavities (figure 11).The use of TEM to evaluate the chemical composition of the precipitates present in the cavity allows us to understand the behavior of the healing agent and the self-healing process, and the stages that follow.In this context, it is important to verify if the specimen has the ability to heal, even after creep tests, in which case, the concentration profile of the healing agent in the matrix is studied by means of atomic probe tomography [105,110,112].

Conclusions
In this work, the thermodynamic and kinetic of Laves phase nucleation, in addition to its role as precipitationhardening particle and in self-healing phenomena was reviewed, with the aim of contributing to future research related to the development of high chromium hardened and self-healing steels by Laves phase precipitation.Currently, there is a limited number of studies about the mechanism that leads to the self-healing properties of ferrous alloys.Among the available literature, the studies are mainly focused on binary and ternary systems since the self-healing process is not yet fully understood by the scientific community.Some guidelines for future research work are drawn herein: • C14 is the most common Laves polytype in high chromium steels.
• Impurities are easily introduced in the C14 polytype, which tends to stabilize Laves phase.Thus, stabilizing agents like Si, may play an important role in hardened and self-healing alloys by Laves phase precipitation.
• Usually, the thermodynamic stability of Laves phase is predicted through the Calphad method and considering a sub-lattice model.Although validated commercial databases are currently available for several metallic systems and Fe-based alloys, important parameters affecting Laves phase formation, such as the atomic radius ratio and concentration valency electron, are not considered in most commercial software.
• In martensitic/ferritic steels, Laves-phase has mainly an heterogenous nucleation in prior austenite grain boundaries, however, some special features can be taken in account.Depending on the chemical composition of the alloy, prior to nucleation of Laves phase forming elements segregates to micro-grain boundaries, thus, increasing locally the driving force.Furthermore, M 23 C 6 carbides can act as heterogenous nucleation site, due to the formation of Cr-depleted zones adjacent to the carbides.
• In high-chromium ferritic steels, Laves-phase forms a partially coherent interface with the matrix, therefore, precipitation occurs mainly in grain boundaries and dislocations.Thermomechanical treatment can be an alternative to improve the Laves-phase strengthening in ferritic steels.Plastic deformation prior to the precipitation of Laves phase increases the nucleation rate, this is attributed to a rise in the dislocation density.• Efforts to accurately model the precipitation of Laves phase are reported in the literature, with successful results.However, some restrictions need to be considered due to the special features of Laves phase formation in Ferro-based alloys.
• The three key parameters for self-healing studies are segregation tendency of elements, surface heterogeneous nucleation and creep mechanism.Self-healing behavior has been reported in binary and ternary Fe-based systems.Therefore, naturally arises the necessity to study this property in multicomponent Fe-based systems as commercial 9%-12%Cr creep resistant steels and ferritic stainless steels.
• To produce tailor-made multicomponent alloys with self-healing properties, extra computational tools are needed as machine learning and genetic algorithm, that can cover the limitation of commercial softwares inspired by the Calphad method.The general evaluation must ensure the segregation tendency of the healing agent, the nucleation features of the healing precipitate and the cavity nucleation as a damage mechanism.
• Si tends to promote Laves phase precipitation (C14 polytype) on dislocations and grain boundaries.Although Si is not a healing agent, its capacity to segregate some elements may be of great relevance during the selfhealing process by Laves precipitation.According to this, all the elements than can accepted by Laves phases microstructure and has segregation tendency toward the cavity would promote the autonomous healing.
• Theoretical descriptions play a key role in the research and development process of new alloys, together with the use of modeling tools.

o
Density of heterogeneous nucleation sites (grain boundaries) Molar volume of b-phase a s Sites number in the sublattice 's' b ij Stoichiometric factor of component i in constituent j c ¯Concentration of solute atoms in matrix a Maximum creep cavities volume fraction that can be filled by healing agent f H kin Maximum filling fraction restricted by solute atoms diffusion l p Average distance between particles n s Number of atoms per unit area G I Gibbs free energy of 'I ' end-member a G M phy Gibbs free energy associated with specific physical phenomena * r Critical radius of nuclei r A Atomic radius of 'A' element in compound AB 2 r B Atomic radius of 'B' element in compound AB 2 r o Initial mean radius of particle in coarsening constituent fraction in a specific sublattice according to 'I'

Figure 7 .
Figure 7. Evolution of cavities in steel exposed to creep over its lifetime.Reprinted from [103].
3.1.A brief history of the development of martensitic/ferritic steels: microstructure stabilization as driving force The development of 9%-12% creep resistant steels began at the middle of 1950s, strongly motivated by the development of low-pollution power plants.International projects were carried out by the National Research Institute for Metals of Science and Technology Agency (Japan), the European Cooperation in Science and Technology, and the Electric Power Research Institute (USA) (see table

Figure 11 .
Figure 11.(a) Creep fracture surface in Fe-Au B-N alloy (b) fully and partially filled cavities (c) and (d) magnification of zones 1 and 2 in image (b).Reproduced from [107].CC BY 4.0.

Table 1 .
[16]ples of Laves phase types in different alloy systems and their properties[16].

Table 4 .
[119,national projects of development of 9%-12%Cr creep resistant steel with an overall funding superior to 3.000.000US$[119,120].Also, it is included the project 'Self-Healing Materials, Pioneering research in the Netherland' with a cost of 20.000.000€.