Nucleation and growth of carbo-nitride nanoparticles in α-Fe-based alloys and associated interfacial process

2.1 Nucleation rate

The classical nucleation theory is very important in describing the nucleation stage in solid-state phase transformations. From a general point of view, classical nucleation theory assumes that the system reaches a steady state and that the stable nuclei appear at a rate given by the following well-known relation [39, 40]:

where β* is the absorption frequency of a substitutional atom by the critical nucleus, Z is a dimensionless term often called the Zeldovitch factor, with a magnitude typically close to 10^-1, N₀ is the number of substitutional sites per unit volume in ferrite, and ΔG* is the Gibbs energy barrier. The expression of the nucleation rate initially derived by Volmer and Weber did not contain the Zeldovitch factor and led to an overestimation of the nucleation rate. This Zeldovitch factor was first introduced by Farkas to describe cluster fluctuations around the critical size [37]. Indeed, the nucleation rate would be equal to β∗N0exp(-ΔG∗kBT)f only if every critical nucleus would continue to grow without going back to smaller sizes. However, the rate theory shows that a nucleus close to the critical size has a non-negligible probability to re-dissolve under thermal fluctuations. The actual nucleation rate is therefore reduced and is given by Eq. (1), where the term Z corrects for this effect. The quantity ZN0exp(-ΔG∗kBT) can also be seen as the number of nuclei that reaches a size large enough to continue their growth. In the case of spherical nuclei, the corresponding radius r′ can be given an analytical approximation [24].

The Zeldovitch factor is a function of the second derivative of the cluster formation free energy at the critical size [41]. It can be shown, from an approach due originally to Zeldovich [42], that Z is proportional to

Z∝σ3TΔG∗

where σ is the interfacial energy between precipitates and ferritic matrix.

The absorption frequency β* is the frequency at which an additional atom attaches to a critical cluster. It is proportional to the cluster surface area for the reason that the growth-limiting process is the attachment reaction at the interface. As the growth is usually controlled by the long-range diffusion of solute atoms, the absorption frequency is the product of the atom jump frequency with the number of solute atoms contained in a shell of thickness a (a being the lattice parameter of α-Fe) around a spherical cluster:

β∗=4πr∗2DXαCXαa4,

where r* is the critical radius, DXα and CXα are the diffusivity and concentration of solute atom in α-Fe, respectively.

Eq. (1) is often used in a different form that takes into account the so-called incubation time. During incubation, clusters grow from their initial size distribution to a final steady-state distribution. Approximate solutions of the time-dependent nucleation equation discussed by Christian [41] indicate that the time-dependent nucleation rate for a single component system may be approximated by

where dNdt|Inc is the final quasi-steady-state rate and τ is the incubation time. It is interesting to note that, recently, kinetic Monte Carlo simulation has been used to test the classical nucleation theory in the case of homogenous precipitation of niobium carbide (NbC) in α-Fe [30]. The atomistic approach used is well adapted for studying the very first moments of precipitation including the nucleation step (see an example in Figure 1). Surprisingly, the best fit is provided by a slightly different expression of dNdt|Inc than Eq. (2):

Figure 1:

An example of Monte Carlo simulation of NbC precipitation in α-Fe at a given temperature (adapted from [30]). The niobium and carbon atoms are represented in red and gray, respectively, and the corresponding time is given in the bottom left of each snapshot. The density of NbC nuclei can be followed at different times and temperatures.

In all cases, a good estimate for the magnitude of τ can be obtained using an argument introduced initially by Russell [40]. Indeed, the incubation time can also be seen as the time to form a significant number of supercritical nuclei. The latter is approximately the time required for clusters to reach size r* by absorption and emission of solute atoms at rate β*. By analogy with the random walk for atomic diffusion along the distance δ=1/Z at a jump frequency β*, the following approximation can be derived:

τ≈δ22β∗.

A specific Taylor expansion of ΔG shows that [43]

2.2 Thermodynamic aspects

In the nucleation regime, the system evolves through localized fluctuations. In a general manner, the Gibbs energy for the formation of nucleus contains two main contributions. The first one is a volume contribution: by forming nuclei of the new phase, the system decreases its free energy. The gain is proportional to the nucleation driving force Δg_v. The second one is the surface contribution: the creation of an interface between the matrix and the nuclei of the new phase has a certain cost that depends on the interfacial energy σ. In the case of the formation of a spherical nucleus of carbo-nitride, the following relation is classically considered:

where Δg_v is the driving energy for nucleation per unit volume (usually called the driving force), r is the radius of the nucleus, and σ is its interface energy with the matrix. An expression for the driving energy Δg_v is needed for the derivation of the size and composition of the critical nucleus.

In the simplest treatment, the Gibbs energy barrier ΔG* [see Eq. (1)] is obtained as the maximum value of the Gibbs energy as a function of nucleus size (Figure 2). The corresponding size is called the critical size r* because the nucleus may be regarded as being in a state of unstable equilibrium with the surrounding parent phase. From a mathematical point of view, this results in the following relations:

Figure 2:

Schematic evolution of the Gibbs energy during the formation of a spherical nucleus as a function of the radius of the particle. Both the surface and volume contributions are shown (adapted from [44]).

The determination of both the driving force and the interface energy is crucial for calculating both the Gibbs energy barrier and, as a result, the nucleation rate [see Eq. (1)].

2.2.1 Driving force for nucleation

Let us imagine the formation of a β phase of formula M(C_yN_1-y) from a solid solution α-Fe. The driving force for nucleation can be regarded as the quantity of the free energy transferred to dn molecules of M(C_yN_1-y) from the supersaturated solid solution to the precipitate, which is in equilibrium with the α-Fe solid solution. In the simple case of a binary compound of formula AB, the driving force for nucleation Δg_v is schematically represented in Figure 3.

Figure 3:

Schematic representation of the driving force for nucleation Δg_v from the Gibbs energy diagram in a model A–B binary alloys. In the simplest approach, the product V_mΔg_v, V_m being the molar volume of the precipitate β, is determined from the common tangent (in dashed line) to G^α and G^β and the tangent to G^α passing through the nominal composition xB0 of the given alloy [45].

By generalizing the simple case of a binary compound to a ternary compound of formula M(C_yN_1-y), one can write

(7)VmΔgv=[xMβμMβ(xMβ)+xCβμCβ(xCβ)+xNβμNβ(xNβ)]-[xMβμMα(xM0)+xCβμCα(xC0)+xNβμNα(xN0)], (7)

where μ is the chemical potential. The equilibrium between the β phase and α-Fe imposes the classical following relation:

μiβ(xiβ)=μiα(xiα,e),

where xiα,e is the equilibrium composition of the i element in α-Fe.

Therefore, relation (7) becomes

VmΔgv=[xMβμMα(xMα,e)+xCβμCα(xCα,e)+xNβμNα(xNα,e)]-[xMβμMα(xM0)+xCβμCα(xC0)+xNβμNα(xN0)].

By taking into account the stoichiometry of precipitates, we obtain the following equation:

VmΔgv=[12μMα(xMα,e)+12yμCα(xCα,e)+12(1-y)μNα(xNα,e)]-[12μMα(xM0)+12yμCα(xC0)+12(1-y)μNα(xN0)].

The chemical potentials are thus developed as a function of the molar fractions according to the ideal solid solution assumption, and the concentrations in solid solution xiss are also introduced. This leads to the following expression [22, 24, 46]:

(8)Δgv=-RT2Vm[lnxMssxMα,e+ylnxCssxCα,e+(1-y)lnxNssxNα,e]. (8)

2.2.2 Composition of the critical nucleus

Relation (8) clearly shows that, contrary to the case of pure carbide or pure nitride, the driving force for nucleation of a carbo-nitride depends on its stoichiometry parameter y, which gives the carbon and nitrogen contents in the precipitate. There is no objective reason to consider a prioriy as a constant during nucleation and growth of the precipitate.

Eq. (8) uses the equilibrium composition of each element i in α-Fe xiα,e. Strictly speaking, their determination requires a complex thermodynamic calculation. However, it is possible to overcome this difficulty, as shown in [24, 47]. Indeed, the equilibrium between the ferrite matrix and the carbo-nitride M(C_yN_1-y) can be described by the following mass action law under the assumption of ideal solution:

(9)RT[lnxMα,e+ylnxCα,e+(1-y)lnxNα,e]=ΔGM(CyN1-y)0, (9)

where ΔGM(CyN1-y)0 is the Gibbs energy of formation of the carbo-nitride. The carbide MC and the nitride MN are often both of face-centered cubic NaCl-type crystal structure with very similar molar volumes. We thus consider the carbo-nitride as an ideal mix of MC and MN, and following Hillert and Staffansson [48], we can write

(10)ΔGM(CyN1−y)0=yΔGMC0+(1-y)ΔGMN0+RT[ylny+(1-y)ln(1-y)], (10)

where ΔGMC0 and ΔGMN0 are the Gibbs energy of formation of MC and MN, respectively. Eq. (10) gives a thermodynamic representation of a complex carbo-nitride from the thermodynamic of corresponding pure carbide and nitride. However, Eq. (10) does not account for a possible interaction between MC and MN. In other words, the excess molar free energy of mixing of MC and MN is neglected.

The thermodynamic functions ΔGMC0 and ΔGMN0 are related to the solubility products for the individual MC and MN compounds in equilibrium with the matrix by the following classical relations:

(11){ln(xMα,e. xCα,e)=ΔGMC0RTln(xMα,e. xNα,e)=ΔGMN0RT (11)

By substituting Eq. (10) into Eq. (9), it is easy to show according to [49, 50] that

(12){yKMC=xMα,e. xCα,e(1-y)KMN=xMα,e. xNα,e, (12)

where K_MC and K_MN are the product solubility of carbide MC and nitride MN, respectively.

Using Eq. (12), the unknown equilibrium composition in solid solution xiα,e can be replaced by the solubility products of carbide K_MC and of nitride K_MN into Eq. (8):

(13)Δgv=-RT2Vmln[(xMss)(xCss)y (xNss)1-y(yKMC)y [(1-y)KMN]1-y]. (13)

It is thus stated that the nuclei that appear in the matrix are those that produce the maximum variation of Gibbs energy during their formation. Accordingly, the composition of the critical nucleus is defined as the value of y that minimizes the driving force for nucleation in Eq. (13). The expression is found to be

(14)ynucl=[1+xNssxCssKMCKMN]-1. (14)

This equation shows that the composition y_nucl of the critical nucleus does not depend on the metallic atoms concentration M in the solid solution and depends exclusively on the temperature, via the ratio of the solubility products, and on the C and N composition of the solid solution.

If carbon and nitrogen in a solid solution are of equal orders of magnitude, the nucleus will be very rich in nitrogen because the nitride of a given material is much more stable thermodynamically than the corresponding carbide, as shown by the temperature evolution of solubility product of some nitrides and carbides given in Figure 4 [2, 51].

Figure 4:

Solubility data. (A) VC and VN, (B) NbC and NbN [51].

Obviously, the composition of both C and N in nuclei is expected to evolve with time at a given temperature because the compositions in solid solution decrease during the precipitation process. Accessing composition information of such nanoprecipitates can be achieved through techniques combining analytical capabilities with sub-nanometer-scale spatial resolutions. TEM remains the most widely used high-resolution technique (in imaging and analytical modes). One of the most challenging problems that has to be addressed for TEM/STEM chemical characterization of carbo-nitrides is specimen preparation. Indeed, specimens have to be electron transparent, i.e. must have a thickness smaller than 100 nm. Even for thin specimens, the thickness of the foil is significantly larger than the particle diameter. Conducting an analytical investigation of such embedded precipitates will lead inevitably to a mixed matrix-precipitates analysis. Only when the elements of interest in the precipitate are completely depleted in the matrix can a quantitative chemical analysis of the particle be achieved. An alternative specimen preparation has been developed, namely extraction replicas. It consists of dissolving the surrounding matrix after deposition of an electron transparent film. Most widely used films are amorphous carbon, but for evident reasons, the presence of carbon in the film has to be ruled out. Therefore, alternative aluminum oxide-based films are preferred. Using extraction replicas based on either sub-stoichiometric radiofrequency (RF)-sputtered amorphous Al_xO_y (x≈y≈1) films or DC-sputtered bilayer Al₂O₃/Al films, Scott and Drillet [52] have shown that as long as no residual C or N signal was originating from the oxide films, C sensitivity as low as 0.04 wt% can be achieved [52]. Using this approach, quantitative determination of absolute C and N content in nanoprecipitates could be obtained [17, 51, 53]. In particular, regarding the evolution of carbo-nitride composition, as shown in Figure 5, Scott has demonstrated a clear evolution of the N/(C+N) ratio as a function of carbo-nitride size. As the precipitate size increases with ageing time, this evolution can also be regarded as an evolution of their composition with time.

Figure 5:

(A) PEELS measurements of precipitate nitrogen to carbon atomic ratios in proeutectoid ferrite of a ferrito-pearlitic microalloyed vanadium steel Fe-0.38C-0.107V-0.010Ti-0.026Al-0.015 N. The steel had been austenitized at 1250°C for 1 min, deformed above 1000°C, and air cooled to 20°C at 1000°C/h. (B) The corresponding vanadium-to-titanium atomic ratios as measured by EDX [54].

2.2.3 Particular sequence of precipitation

A particular nucleation mechanism has recently been shown in model FeNb(CN) alloys. Indeed, a combined field ion microscopy (FIM), APT, and high-resolution electron microscopy (HRTEM) investigation of the early stages of precipitation has demonstrated the existence of Guinier Preston (GP) zones. GP zones, which are well known and documented in the precipitation sequence of aluminum-based alloys [55], are particular nuclei of a second phase that adopt the same structure as the parent matrix. In most cases, they adopt a particular platelet shape, with one or several atomic planes in thickness. Such platelets have been observed not only in model FeNb(CN) alloys, as shown in Figure 6, but also in nitrided FeTi [57] and FeCr [58] alloys and in industrial steels [59, 60], leading to high strength increase.

Figure 6:

Atomic-scale evidence of GP zones in FeNbCN model steel. (A) HRTEM, (B) FIM, and (C) APT [56].

2.3 Interfacial energy

The interface energy σ between particles and α-Fe matrix plays a key role on both nucleation and growth steps. It is clear through Eqs. (1) and (6) that a small change in σ can lead to a dramatic change in the calculated nucleation rate. As a consequence, the knowledge of σ is a prime necessity for studying the precipitation of carbo-nitrides in ferrite, even if its experimental determination is a tedious process [61] and a wide range of values has been reported [22, 23, 26, 30, 62]. Also, it is important to recognize that the most common approach is to consider the interface energy σ as a fitting parameter that is, for the sake of simplicity, often considered as a constant value [23, 26, 28, 30, 62]. Obviously, there are some clear limitations to this approach as discussed below.

In a general manner, and from a theoretical point of view, the interfacial energy can be calculated from the difference in cohesive energies between the interfacial region and the bulk:

where E_α-Fe/M(C,N) is the sum of bonding energies across the α-Fe/M(C,N) carbo-nitride interface and E_α-Fe/α-Fe and E_{M(C,N)/M(C,N)} are the sum of the bonding energies across the planes parallel to the interface in the ferrite matrix and M(C,N) carbo-nitride, respectively [63]. A value of interface energy of 493 mJ/m² was calculated for coherent α-Fe/TiC interface [63]. This value can be compared with the value of 1.22 J/m² for coherent α-Fe/NbC interface adjusted from Monte Carlo calculations [30] and with the value of 0.78 J/m² for α-Fe/Nb(C_yN_1-y) adjusted from the time evolution of the measured chemical composition of carbo-nitrides [25]. These values of interfacial energy are not directly calculated but fitted using a mean field model for precipitation. Therefore, they are source of possible error.

The interface energy σ given in Eq. (15) includes a contribution from both the chemical bonding energy and the structural strain energy [63, 64].

This would suggest that the interfacial energy depends on the degree of interface coherency. Accordingly, it is often proposed to express the interfacial energy as [65]

where σ₀ is the interfacial energy of a coherent interface, C₁ is a coefficient related to dislocation density, δ is the misfit between the two lattices, and ε is the strain at the interface (ε=0 for an incoherent interface and ε=δ for a coherent interface). Therefore, σ varies from low values for coherent interfaces to high values for incoherent interfaces, and thus, nuclei with coherent interfaces precipitate preferentially [66].

Most of the carbides and carbo-nitrides [TiC, NbC, V(C,N), etc.] exhibit a face-centered cubic crystal structure form with the NaCl structure and have a Baker-Nutting orientation relationship with the ferrite matrix [11, 30, 67, 68].

A schematic illustration of the atomic configuration at the interface between bcc-Fe and MC carbide having the Baker-Nutting orientation relationship ((100)_α//(100)_M(C,N) [010]_α//[011]_M(C,N)) is given in Figure 7.

Figure 7:

Schematic illustration of atomic arrangement of a coherent α-Fe/TiC interface with the Baker-Nutting orientation relationship.

The lattice parameters of most carbides and nitrides are between 0.4 and 0.48 nm [69, 70]. The lattice misfit between ferrite and carbide or carbo-nitride, defined as δ=aM(C,N)-2aα-FeaM(C,N), is thus between 3% and 15% (see Figure 8).

Figure 8:

Lattice misfit of carbides and nitrides with α-Fe matrix [67].

As a consequence, a contraction of the M(C,N) lattice is necessary to maintain the coherency at the interface with the ferrite matrix. This leads to two main remarks. First, the elastic strain energy associated to the precipitation of coherent M(C,N) in α-Fe can be significant and should not be considered as negligible with respect to the driving force for nucleation [71, 72]. Second, for this large atomic misfit, it would be energetically favorable to replace the coherent interface with a semi-coherent interface and even an incoherent interface. This operates through dislocations that are introduced in the interface to accommodate the misfit between the two phases. Therefore, the result is a coherency loss, and the interfacial energy is expected to evolve with time during the precipitation process. In general, σ ranges between 200 and 500 mJ/m² during nucleation (for a semi-coherent interface) and may reach up to 1 J/m² during growth and coarsening (for an incoherent interface). This aspect is often neglected in many models of precipitation of carbo-nitride in ferrite [23, 24, 26, 62]. Furthermore, as noted by [30, 67], the chemical interface energy between Baker-Nutting-oriented ferrite and carbo-nitrides is likely to be highly anisotropic, and as a consequence, the assumption of spherical nuclei is certainly questionable in many cases.

3.1 Laws for growth: case of the mono-atomic precipitate

In this section, we deal with the laws of growth of precipitates. The nature of the matrix does not modify the analytical form of the laws to be discussed here, only the values of the constants to be used. Let us then consider one precipitate observed at the end of its homogeneous nucleation process. It is embedded in a supersaturated α-Fe matrix, the composition of which is supposed to be uniform at this stage. As a result, the precipitate will spontaneously grow by incorporating solute atoms. This process involves three physical phenomena that will be analyzed in sequence:

1. local equilibrium at the precipitate-matrix interface,

2. solute transport toward the interface,

3. precipitate-matrix interface motion.

For the sake of simplicity, the inherent simplifications in the mean field modeling are discussed from the simple case of mono-atomic precipitates (for example, the case of precipitation of copper in binary Fe-Cu). Most of the assumptions remain valid for the growth of bi-atomic MC or MN or tri-atomic M(C,N). This is justified by realizing that the kinetics of growth of a carbide or carbo-nitride are limited by the diffusion of the slow substitutional atom (M), with the interstitial atoms (C and N) being very fast diffusers. In the following paragraphs, we will introduce the necessary refinements when extending the growth laws to the bi-atomic MC and to M(C,N) precipitates.

3.1.5 Effect of spatial distribution

As a result of the nucleation stage, or as a result of the fabrication process, the spatial distribution of the particles will rarely be uniform. On the contrary, some of the particles will be clustered, whereas some regions will look particle-free, at least in a two-dimensional (2D) section. During growth, when the diffusion length scale reaches the average distance between the particles, the growth kinetics of a given particle will be influenced by its neighboring particles (effect called soft impingement). Knowing this, the question is whether the time evolution of the average radius of the particles is accurately predictable.

A three-dimensional (3D) model has been used and is illustrated in Figure 10A. Precipitates of the same radius r₀ were randomly distributed over space, and the growth of each particle has been calculated through a 3D finite volume scheme [30]. We observe that clustered particles, having to share the solute atoms with one another, grow slower that the isolated ones. However, it appears that the average radius <r> of the distribution obeys a kinetics that is exactly that of a single representative particle of initial size r₀ (Figure 10B). In other words, the mean field approximation is fully justified here.

Figure 10:

Effect of the spatial distribution of nuclei on the growth kinetics. (A) Initial state of the finite volume calculations (2D section, precipitates in red). (B) Time evolution of the mean radius compared to the analytical model [Eq. (19)] [30].

3.2 Bi-atomic precipitate

The growth of a bi-atomic precipitate such as NbC or NbN precipitate can be treated similarly to the case of a mono-atomic precipitate: as diffusion of the metallic atoms will be rate limiting, allowing for the use of a variant of the Zener equation. However, coupling of the local equilibrium condition with the local flux compatibility at the interface has to be taken into account to evaluate the interfacial concentration.

During the growth of a carbide MC, for example (the treatment would be the same for a pure nitride MN), a gradient in M concentration builds up around the precipitate. As carbon atoms diffuse very much faster, the concentration profile of carbon remains almost flat, assuming no carbon-metal interaction. The Zener equation is then written as, with obvious notations,

In this equation, the use of atomic fractions makes the ratios V_Fe/V_MC appear, where V_MC is the volume of 1 mol of MC. In the ternary Fe-M-C phase diagram, the local equilibrium at the precipitate-matrix interface takes the form of a solubility product, modified by the Gibbs-Thomson effect:

where K_MC is the solubility product for MC in the matrix. This equation is an approximation that holds only in the case of diluted solid solution and stoichiometric precipitates. For the general case of multicomponent precipitates and non-dilute solid solution, the reader can refer to [78]. Eq. (23) is not enough to determine the interfacial M concentration in the matrix XMα/β, and a compatibility equation has to be used. In the general case, the compatibility condition originates from the equality of the growth rates of Eq. (22), whichever solute element of reference is used (C or M). The simplest approach, valid as long as the matrix is dilute compared to the precipitate composition, is to write the flux compatibility condition: the incoming fluxes of C and M have to be equal at the interface for the stoichiometry of MC to be respected. In the invariant field approximation, this is written as

Eq. (24) defines an almost vertical straight line in the (x_C, x_M) isotherm graph (Figure 11). Graphically, the solution of the set of Eqs. (23) and (24) is the intersection between the equilibrium isotherm and the flux compatibility line. During an isothermal precipitation treatment, the matrix composition decreases along the stoichiometry line, and consequently, the M interfacial composition XMα/β increases from an initial value to its equilibrium value.

Figure 11:

Time evolution of the matrix and interfacial compositions during an isothermal heat treatment. For clarity, the Gibbs-Thomson effect has been neglected here.

3.3 Growth of carbo-nitride M(C,N): the core-shell model

A major difference with the previous case is that a carbo-nitride M(C_y,N_1-y) allows for a wide range of composition on the interstitial sub-lattice: in fact, possible carbo-nitrides of general formula M(C_yN_1-y) range from MC (y=1) to MN (y=0), and the composition of a given particle will gradually change with time during its growth. Consequently, the composition distribution has to be taken into account together with the size distribution of the particles. The mathematical treatment developed in [24] is summarized in the following.

The kinetic equation for growth is analogous to that for MC and is written as

where VM(Cy, N1−y) is the volume of 1 mol of M(C_yN_1-y). Growth occurs by accretion of successive shells of variable composition y. The calculation of the shell composition is done together with the calculation of the matrix composition in the vicinity of the interface by the simultaneous resolution of the local equilibrium and the flux compatibility conditions, Maugis and Gouné [24] showed that a very good approximations is given by

It appears that, for each time interval, the composition of the shell y_g during growth is the same as that of the critical nucleus [see Eq. (14)] and is only driven by the composition of carbon and nitrogen in the solid solution, irrespective of the diffusion coefficients of M, C, and N.

The model allows for describing the evolution of the precipitation state of M(C_y,N_1-y) particles in a ferritic matrix, as shown in Figure 12.

Figure 12:

Time evolution of the precipitation state in a Fe-Nb-C-N alloy aged at 750°C and 650°C. (A) Comparison between the calculated and the measured mean radius at both 650°C and 750°C. (B) Comparison between the calculated and measured nitrogen and niobium precipitate elements at 750°C. (C) Calculated mean composition of precipitates at 650°C and 750°C. The experimental data are given at 650°C. Here, the ratio C/N in the alloy is close to 1, and the nucleating particles are nitrogen-enriched and tend to the alloy stoichiometry only in the late coarsening stage.

In the core-shell model used here, carbon-nitrogen interdiffusion inside the particle is supposed to be very slow compared to the diffusion in the matrix. The alternative hypothesis of infinitely fast intraparticle diffusion could have been used, which would have led to slightly different results: in particular, under this second hypothesis, the precipitate equilibrium composition would be reached at the end of the growth regime, which is contrary to our experimental measurements (see Figure 12).

3.4 Coupling nucleation and growth

Nucleation and growth are not successive stages, but are actually overlapping and hence are strongly coupled. In the mean field approximation, this coupling can be introduced mathematically through a nucleation and growth equation: during a heat treatment, the average radius r evolves not only due to growth of all existing precipitates, but also due to the nucleation of new precipitates of radius r′. In terms of average growth rate, the coupling is simply additive and is rendered by the following equation [29]:

where the first term in the right-hand side is the growth rate of a precipitate of radius r and the second term is the effect of nucleation. In cases of isothermal heat treatments or moderate heating rates, this procedure has proved to be an excellent approximation relative to a Kampmann and Wagner numerical resolution or by comparison with more sophisticated models [62].

The coupling between nucleation and growth thus allows for describing the evolution of precipitation state of M(C_y, N_1-y) in the ferritic matrix, as shown in Figure 12.