Coupling Inward Diffusion and Precipitation Kinetics; the Case of Nitriding Iron-Based Alloys

The inward diffusion of I into the alloy-sheet substrate (one-dimensional diffusion) is given by Fick’s second law:where C _I is the concentration of I in the alloy matrix at depth x and time t. D _I is the intrinsic diffusion coefficient of I in the alloy matrix. Considering the chemical potential gradient as driving force for diffusion, D _I can be expressed as:[32]where D _I ^* is the self-diffusion coefficient of I and ϕ _I is the thermodynamic factor of I with the activity a _I and the activity coefficient γ _I of I in the alloy matrix. X _I is the mole fraction of component I.² For a dilute ternary system, γ _I can be described by:[33]where γ _I ⁰ is the activity coefficient of I for infinite dilution and ɛ _I ^I and ɛ _I ^M are the interaction coefficients for I-I and I-M in the matrix, respectively (the activity coefficient of M, γ _M, can be expressed analogously.). X _M is the mole fraction of M in the matrix.

For solving the diffusion equation, the boundary conditions (BCs) are required. The flux of I provided at the surface of the substrate by the medium surrounding the substrate, J _I,s, can (for gaseous nitriding) be described by:[34,35]where β is the rate constant for the reaction by which I atoms are delivered to the surface of the substrate, C _I,eq represents the concentration of dissolved I in the solid substrate (at its surface) in equilibrium with the surrounding medium that provides I, and C _I,s indicates the actual concentration of dissolved I at the surface of the substrate. The flux J _I,s is equal to the flux of I through the surface described by Fick’s first law. Thus, the BC at the surface is given by:

In view of the symmetry of the sheet substrate, the (net) flux of I through the center plane of the sheet is zero. Thus, the BC at the center is given by:

The nucleation rate is taken to comply with classic nucleation theory,[36] and the particle growth rate is assumed to be controlled by long-range diffusion. The nucleation and growth rates are coupled by their dependency on the mean solute concentration in the matrix, as in well-known models for precipitation in a homogeneous alloy.[24] The PSD is represented by discrete particle size classes, each of a certain population (i.e., number) density. In the present case, for class management, a Lagrange-like multiclass approach[22] is used, where the radius evolution of each class is tracked while its population density remains constant. Classes with a particle radius smaller than a certain cut-off radius are removed from the PSD. The evolution of the PSD with time is computed by numerical integration of the nucleation and growth rates:

The total N concentration (C _N ^tot ) of internally nitrided Fe-M alloys can be given as \( C_{\text{N}}^{\text{tot}} = C_{\text{N}}^{ 0} + C_{\text{N}}^{{{\text{M}}_{y} {\text{I}}_{z} }} + C_{\text{N}}^{\text{str}} + C_{\text{N}}^{\text{int}} \), where C _N ⁰ is the concentration of the N dissolved in unstrained ferrite, \( C_{\text{N}}^{{{\text{M}}_{y} {\text{I}}_{z} }} \) is the concentration of N corresponding with precipitation of M _y N _z , C _N ^str is the concentration of the (mobile[13]) N dissolved additionally to C _N ⁰ in the strained ferrite matrix as a consequence of elastic accommodation of the precipitate/matrix misfit,[28] and C _N ^int is the concentration of (immobile[13]) N adsorbed at the precipitate/matrix interface (see Section I).

For the case of spherical M _y N _z particles, the value of C _N ^int , which depends on particle size, can be given by (cf. Eqs. [7] and [8]):where B is the number of adsorbed N atoms per unit area of the precipitate/matrix interface.

The mobile excess N concentration C _N ^str is part of the total concentration of the N dissolved in the strained ferrite matrix; i.e., C _N = C _N ⁰  + C _N ^str . The enhanced N concentration in the strained matrix is accounted for by introducing a concentration value C _N,eq for the BC at the surface in the inward-diffusion part of the model (see the earlier Section II–A), which is larger than the concentration value C _N,Eq ⁰ of the unstrained matrix. The M _y N _z precipitate developing in the matrix shows a positive misfit with the matrix (ɛ > 0). In the case of a fully elastic accommodation of the particle/matrix misfit, the increased solubility C _N,eq ⁵ is given by:[28,29]where \( \bar{V}_{\text{N}} \) is the partial atomic volume of N in the matrix and σ is the hydrostatic component of the matrix-stress field caused by all existing particles. Introducing the total volume fraction φ of the M _y N _z precipitate particles, σ is given by:[28,42,43]

To account for the additional misfit introduced by the (particle-size dependent) amount of excess N adsorbed at the particle-matrix interface, Eq. [20a] can be modified to:where misfit parameter ɛ _i ^′ and volume fraction φ _i ^′ of each particle size class i including the N adsorbed at the precipitate/matrix interface are given by:

The parameter f describes the extent to which the full misfit due to building out of the lattice of the M _y N _z particle by the adsorbed N atoms is experienced.[13,14]

To solve the diffusion and precipitation (nucleation + growth/coarsening) equations, the finite difference method (FDM) was employed with the equations discretized in time by using an adaptive time interval Δt (see below) and in space with a depth interval Δx. For solving the diffusion equation by FDM, the Crank-Nicolson method for partial differential equations[49] and the tridiagonal matrix algorithm[50] were adopted. For solving the precipitation equations by FDM, the Euler method for ordinary differential equations[50] was used.

At the jth time step, the concentration of dissolved N in the ferrite matrix (increased, as compared with the j − 1th time step, by inward diffusion of N, excluding the precipitation at the current time step) at every depth step k, (C _N) _k ^j,diff is calculated by Eqs. [1] through [6]. Then the nucleation and growth (including coarsening) of M _y N _z (i.e., (N _i ) _k ^j and (R _i ) _k ^j for each class i of M _y N _z at every depth) are calculated as described in Eqs. [7] through [17]). Classes with (R _i ) _k ^j less than the radius corresponding to the size of one M _y N _z molecular unit are removed from the PSD and the M and N components are added to the solutes content of the matrix. From the calculated (N _i ) _k ^j and (R _i ) _k ^j , the nitride particle number density N _k ^j , the mean radius \( \bar{R}_{k}^{j} \), and the volume fraction \( (\varphi_{{{\text{M}}_{y} {\text{N}}_{z} }} )_{k}^{j} \) are calculated as follows:

(C _N ^int ) _k ^j is calculated by using Eq. [18] and \( (C_{\text{N}}^{{{\text{M}}_{y} {\text{N}}_{\text{z}} }} )_{k}^{j} \) is calculated from:where \( V_{{{\text{M}}_{y} {\text{N}}_{z} }} \) is the molecular volume of M _y N _z . The concentrations of dissolved N (C _N) _k ^j and dissolved alloying element (C _M) _k ^j after inward diffusion of N and realization of nitride precipitation are updated by application of mass balances for the N and the alloying element at depth step k as follows: where \( \Delta \left( {C_{\text{N}}^{{{\text{M}}_{y} {\text{N}}_{z} }} } \right)_{k}^{j} \), \( \Delta (C_{\text{M}}^{{{\text{M}}_{y} {\text{N}}_{\text{z}} }} )_{k}^{j} \), and Δ(C _N ^int ) _k ^j indicate the changes in concentration of N and the alloying element caused by precipitation of M _y N _z and the change in N concentration due to N adsorption at the precipitate/matrix interface during the jth time step at the kth depth step, respectively. The value of C _N,eq for the surface BC in the N diffusion model (Eq. [5]) is updated by Eqs. [19] through [22]. The total N concentration at depth step k, (C _N ^tot ) _k ^j , is calculated by:

At the (j+1)th time step, N diffusion and nitride precipitation calculations are performed with the parameters updated at the jth time step, and so on. The discussed scheme of calculating N diffusion and nitride precipitation is illustrated in Figure 1.

The shorter the time interval Δt, the better the accuracy of the calculation. However, a too small Δt increases the calculation time without gain in accuracy. Therefore, Δt is adapted during the sequential calculations as follows: Δt is increased by factor 1.1 (Δt → 1.1Δt) if the critical radius R ^* updated at the current time step differs less than 1 pct from the value updated at the previous time step for all depths where the precipitation occurred. If this condition is not satisfied, the calculation for the current time step is repeated with a smaller time interval (Δt → 0.5Δt) with the parameters updated at the previous time step.

The inward diffusion and precipitation-coupled model developed in this study was applied to the gaseous nitriding of Fe-2.23 at. pct V alloy specimens, with a thickness of 1 mm, at 793 K, 823 K, and 853 K (520 °C, 550 °C, and 580 °C) for 10 hours by using a nitriding potential (r _N) of 0.103 atm^−1/2, i.e., the same conditions as applied in the experiments of Reference 14, so that experiment and simulation results can be compared. Under these nitriding conditions, no iron nitrides can be formed[14] and vanadium nitrides are precipitated in the form of VN.[14,31] This was shown for several Fe-V alloys, where after so-called de-nitriding of specimens previously nitrided until saturation (i.e., no further nitrogen uptake occurs), the amount of remaining, strongly bonded N is at the level of the amount of V in the specimen, indicating that the precipitate composition can be described as VN.[31] Gaseous nitriding of iron-based alloys in a flowing NH₃/H₂ gas mixture allows precise control of the applied chemical potential of N by the so-called nitriding potential r _N (=\( {{p_{{{\text{NH}}_{ 3} }} } \mathord{\left/ {\vphantom {{p_{{{\text{NH}}_{ 3} }} } {p_{{{\text{H}}_{ 2} }}^{3/2} }}} \right. \kern-0pt} {p_{{{\text{H}}_{ 2} }}^{3/2} }} \), p is the partial pressure)[51] and of the kinetics of the dissociation of ammonia at the iron surface.[27,34,35,52]

The diffusion and precipitation-coupled model including the effect of excess N and surface reaction kinetics is termed full model. Simulations were also performed with the model in a less “complete” state, to expose the role of various, individual processes integrated in the full model: (1) To illustrate the importance of considering the role of excess N, simulations neglecting the role of excess N have been performed; the model in that fashion is called the basic model. (2) To investigate the effect of the time (and temperature) dependence of the nitride-precipitation kinetics, simulations were also performed under the assumption of instantaneous establishment of local thermodynamic equilibrium in the diffusion zone, implying instantaneous nitride precipitation once the solubility is (locally) surpassed. This is called the “equilibrium approach” (EA) (see Section I). (3) To illustrate the effect of neglecting the kinetics of N absorption at the surface, simulations were carried out assuming instantaneous establishment of local equilibrium with the NH₃/H₂ gas mixture at the surface of specimen; i.e., the following BC was used: C _N| _x=0 = C _N,eq.

The simulations were performed with a depth interval Δx of 10 µm (and with a smaller Δx of 1 µm especially for the time evolutions of the precipitation state according to the basic model and for the EA model, which both are numerically unstable for large Δx) and an initial time interval Δt of 0.01 seconds.

Thermodynamic data for γ _N, γ _V, and K _sp for VN, which were used in the present simulations, were obtained by using Thermo-Calc software and the TCFE7 database,[53] and they have been listed in Table I. Other used input data are presented in Table II. The temperature dependence of the bulk modulus of VN was assumed to be identical to that of TiN for which literature data are available.[62] The value of B (cf. Eq. [18]) was obtained by using the measured value of C _N ^int (0.40 at. pct) for homogeneously, through nitrided Fe-2.23 at. pct V thin foil (thickness of 0.2 mm) specimen, nitrided for 24 hours at 853 K (580 °C) at r _N = 0.103 atm^−1/2.[31] The value of B was taken as temperature-independent.

Results of the basic model, for the case of nitriding Fe-2 at. pct V alloy at 853 K (580 °C) for 10 hours using a nitriding potential of 0.1 atm^−1/2, are shown in Figure 2 in terms of the contents of dissolved N and V in the matrix, the content of N present in the form of VN-particles, and the volume fraction, the number density, and the mean radius of the VN-particles, all as a function of depth below the surface. For this nitriding time, up to a depth of 250 µm, nearly all V has precipitated as VN (Figure 2(a)), which is expected due to the negligible solubility of V in the ferrite matrix at the employed nitriding conditions. The small negative depth gradient of the total N content in the diffusion zone (at x < ~250 µm) originates from the depth gradient of N dissolved in the matrix. In accordance with a constant solubility product of nitride, the content of V dissolved in the matrix shows a positive depth gradient (see the inset of Figure 2(a)); i.e., the dissolved V content increases with decreasing dissolved N content.

Within the region of (practically) completed VN precipitation (x < ~250 µm), an interesting variation of particle number density and mean particle size as a function of depth occurs (Figure 2(c)): The number density of VN particles is very high at the surface, but it decreases rapidly with increasing depth (note the logarithmic scale in Figure 2(c)). In accordance with the practically constant volume fraction of VN within this region (Figure 2(b)), the mean radius of the VN particles shows an opposite trend. These observations indicate a strong variation of the rates of nucleation and growth as a function of depth. Due to the continued, fast provision of N by the decomposition of ammonia at the surface, the region adjacent to the surface of the specimen is exposed to a higher flux of N than at distinct depths, where N has to be provided by solid state diffusion of N from the surface. The inward flux is additionally reduced by the precipitation of VN in regions closer to the surface. Consequently, the driving force for nucleation in the region adjacent to the surface is higher than in the subsurface regions, leading to a higher number density and smaller size of VN particles in the surface-adjacent regions. This phenomenon was indeed experimentally observed in transmission electron microscopy (TEM) investigations carried out on nitrided Fe-Cr[63] and Fe-Cr-Al[64] alloys.

The earlier discussed effects are even more dramatically illustrated considering the time evolutions of the number density, size, and volume fraction of VN at the surface (depth = 0 µm) and at the depth of 200 µm below the surface (Figure 3).

At the surface, the number density of VN particles increases rapidly after the onset of nitriding (Figure 3(a)). The volume fraction of VN, however, is still low (Figure 3(c)) due to the very small size of the VN particles (Figure 3(b)). With increasing nitriding time, the number density of VN particles remains almost constant (after about 6 seconds), but a gradual increase in mean radius and amount of V precipitated as VN occurs due to continued particle growth (almost) up to the equilibrium fraction (at about 900 seconds). Hence, two different stages of the precipitation reaction at the surface can be distinguished: an initial stage dominated by rapid nucleation (stage I), followed by a stage of particle growth with negligible nucleation, i.e., almost pure growth (stage II). This behavior is caused by the high sensitivity of the nucleation rate on the degree of supersaturation, which is determined by the contents of dissolved N (given by competing rates of its provision at the surface and its consumption by the precipitation) and dissolved V in the matrix. In stage II, a low level of supersaturation remains due to the relatively high consumption rate of N by growth of the large number of existing particles. With increasing time, in the surface-adjacent region, no significant change in the number density of particles and the mean radius is observed indicating negligibly slow coarsening kinetics (stage III): The coarsening rate is very low due to the extreme depletion of V in the matrix and the continuous supply of N, which, in view of the solubility product of VN, means that the rates of both positive growth of larger particles and negative growth of smaller particles are very slow (Eqs. [14] and [17]). The evolution of the PSD of VN at the surface is illustrated in Figure 4; after 1.5 seconds (in stage I), after 20 seconds (in stage II), after 2000 seconds (in stage III), and after 36,000 seconds, i.e., after the total nitriding process time of 10 hours. The results shown reflect the earlier interpretation of the evolution of the precipitation state in the surface-adjacent region: The particle number density increases in stage I (Figures 4(a) and (b)); the particle size increases in stages I and II (Figures 4(a) through (c)); and then the PSD broadens in stage III (Figures 4(c) and (d)).

At the depth of 200 µm, the evolution of the state of precipitation can also be subdivided into three stages, as shown in Figure 3. Stages II and III are similar to those described for precipitation in the surface-adjacent regions, i.e., almost pure growth in stage II and slow coarsening in stage III. However, stage I differs strongly from that observed for the surface-adjacent region: the mean radius initially increases continuously with time, but this is then followed by a decrease toward the end of the stage I, in contrast with the continuous increase in mean radius occurring at the surface until the end of stage I. This phenomenon can be explained as follows:

Due to the much lower flux of N as compared with the surface-adjacent region, the degree of supersaturation attained at the moment of VN precipitation and, thus, the driving force for VN precipitation is pronouncedly smaller at a distinct depth in the diffusion zone than in the surface-adjacent region. Thereby a much smaller number of particles (Figure 3(a)) of larger size nucleate and then grow (Figure 3(b)) in the early stage of the VN precipitation, in association with a relatively low consumption rate of N as compared with the surface-adjacent region. With increasing nitriding time, upon (virtual) completion of the VN precipitation in the surface-adjacent region, at a depth of 200 µm, the provision rate of N by inward diffusion gets larger than its consumption rate by VN precipitation, leading to an increase of the dissolved N content in the matrix; the driving force for nucleation at this depth can then increase despite the already reduced dissolved V content in the matrix. This results in the nucleation of a relatively large number of particles (Figure 3(a)) of size smaller than the momentary mean radius, leading to a decrease of the mean radius (Figure 3(b)). After this more or less abrupt increase of the nucleation rate, at a next moment of time at this depth, the nucleation rate decreases due to a decrease of the driving force for nucleation by the consumption of the solutes V and N, and particle growth becomes relatively dominant causing a visible increase of mean radius: Stage II of almost pure growth is entered.

According to the EA model, at any location all N above the equilibrium content corresponding to the solubility product of the nitride is consumed for instantaneous precipitation of nitrides. As follows from the results shown in Figure 5(a), the EA approach, as expected, leads to a sharp transition between the nitrided case and the unreacted core, whereas incorporating nitride-precipitation kinetics (the basic model) leads to a more gradual decrease in the total N content with a depth between the fully precipitated case and the unreacted core. As shown in Figure 5(b), in the basic model, the overall evolution rate of the amount of VN is much lower. This allows a larger inward diffusion flux of N into deeper regions, which results in an earlier onset of precipitation in these deeper regions than for the EA model.

Disregarding the ammonia dissociation kinetics at the surface implies. C _N| _x=0 = C _N,eq for t > 0. Consequently, a larger inward flux of dissolved N occurs (Eq. [5]), especially in the early stage of nitriding. This leads to larger values of C _N ^tot at every depth and, in particular, to a more pronounced nitriding depth as shown in Figure 6. In reality, as indicated by the results obtained from the basic model, C _N at the surface only equals C _N,eq after through thickness, homogeneous nitriding of the specimen.

The N content-depth profiles of Fe-2.23 at. pct V alloy nitrided with r _N = 0.103 atm^−1/2 at three different temperatures [793 K, 823 K, and 853 K (520 °C, 550 °C, and 580 °C)) for 10 hours, as measured[14] and as simulated according to the full model and the basic model, are shown in Figure 7. Evidently, incorporating the role of excess N (mobile and immobile) is essential to arrive at results compatible with experimental observations: The results obtained by the basic model show significant disagreement with the measured data, whereas the results obtained by the full model match fairly well with the experimental data at all temperatures.

Accounting for the effect of excess N greatly influences the predicted nitriding kinetics: The mobile (=dissolved in the matrix) excess N, C _N ^str , mainly induces an increase of the nitriding depth and slightly contributes to an increase of the total N content, C _N ^tot , at every depth where nitride precipitation occurs[13] by enhancing the flux of N diffusion into the bulk. (Eqs. [5] and [19] through [22]). The immobile (=adsorbed at the precipitate/matrix interface) excess N, C _N ^int , primarily causes an increase of C _N ^tot at every depth where nitride precipitation occurs, and it induces a small decrease of nitriding depth[13] by consumption of N in the region of precipitation.

The results of the full model and the basic model for the nitriding at 853 K (580 °C) in terms of VN number density, VN mean radius, VN volume fraction, V content dissolved in the matrix, N content dissolved in the matrix (C _N = C _N ⁰  + C _N ^str ), N content consumed for precipitation of VN, and N content adsorbed at the precipitate/matrix interface (C _N ^int ) are shown as a function of depth in Figure 8. The trends as a function of depth according to the full model as shown in Figures 8(a) through (f) are similar to those observed for the basic model (see our earlier discussion). However, the quantitative results are different, as a consequence of the incorporation of the role of excess N: Compared with the basic model, the full model predicts larger particle density and smaller particle size. This is due to the larger driving force for nucleation caused by the larger N flux from the surface into the bulk, which is induced by a higher value of C _N,eq (Eqs. [5] and [19] through [22]) due to the misfit-strain field incorporated in the full model. The particle size calculated with the full model (the mean radius, in the region from the surface to the depth of 200 µm, equals 1.2 to 1.9 nm; the corresponding average particle volume varies from 7.2 to 28.7 nm³) is compatible with the very tiny VN particles (5 nm long and one- to two-monolayer-thick platelets, with a corresponding average particle volume of 5.2 to 10.4 nm³) as observed in TEM investigations performed on the same alloy specimens of different thicknesses (200 µm) nitrided under identical nitriding conditions.[65]

It is of interest to note that C _N ^int decreases with increasing depth in the precipitated zone (Figure 8(g)) despite a practically constant VN fraction in this region (0 to 380 µm), where nearly all V has precipitated as VN. This is a consequence of the decrease of precipitate/matrix interface area (per unit volume of VN) due to the increase in mean particle radius with increasing depth.