Autonomous Filling of Grain-Boundary Cavities during Creep Loading in Fe-Mo Alloys

The creep results for the as-quenched (solutionized) Fe-Mo alloy loaded at different constant stress levels at a fixed temperature of 823 K (550 °C) are presented in Figure 1. The strain–time curves in Figure 1(a) indicate that this alloy shows a considerable ductility, which leads to high creep strains (>50 pct) upon fracture. The corresponding strain rate as a function of strain (Figure 1(b)) indicates a short primary creep regime where the creep rate rapidly decreases, until a more steady-state creep starts at a strain of about 0.08. From the strain versus time (Figure 1(a)) and the strain rate versus strain (Figure 1(b)), it can be seen the primary creep stage extends until a strain of about 0.08 is reached. The primary creep is followed by an oscillating creep rate before the minimum strain rate is observed. The strain rate oscillation is attributed to continuous dynamic recrystallization.[25‐27] After the steady-state creep regime, a gradual increase in creep rate takes place, which marks the start of the tertiary creep regime. The fraction of the creep lifetime occupied by the quasi steady-state creep regime decreases at higher stress levels.

The quasi steady-state strain rate and the creep lifetime as a function of the applied stress at a temperature of 823 K (550 °C) are shown in Figures 1(c) and (d). As expected, the creep rate increases with increasing stress. The stress exponent is deduced from the Sherby-Dorn equation for the steady-state strain rate[28]:where A is a structure-dependent constant, Q is an apparent activation energy, d is the grain size, m is the grain size exponent, σ is the stress, and n is the stress exponent. The deduced stress exponent of n ≈ 15 is relatively high and is related to the interaction between the solute Mo atoms and the high dislocation density generated at these high-strain levels.[9,29] The creep lifetime decreases for increasing stress with a maximum lifetime until fracture of 849 hours at the lowest applied stress of 180 MPa. In Figures 1(e) and (f), the corresponding area fraction of Fe₂Mo Laves phase precipitates, obtained from an image analysis of the SEM data (described in the “Section III–B”), is shown as a function of stress and lifetime, respectively. The precipitate fraction increases with the creep life time and is still relatively low compared to the equilibrium fraction of 9.8 vol. pct Fe₂Mo.

The results of temperature-dependent creep tests on the solutionized Fe-Mo alloy performed at a stress of 200 MPa and temperatures of 813 K, 823 K, and 838 K (540 °C, 550 °C, and 565 °C) are shown in Figure 2. Using Eq. [1], an experimental activation energy of Q = 319 ± 14 kJ/mol was obtained, which is consistent with the activation energy for the diffusion of Mo in bcc iron.[30] In Figures 2(e) and (f), the corresponding area fraction of Fe₂Mo Laves phase precipitates, obtained from an image analysis of the SEM data (described in the “Section III–B”), is shown as a function of temperature and lifetime, respectively. The precipitate fraction increases with temperature.

An additional set of creep tests on the same alloy, but with a different average grain size (d = 28 μm), was conducted at a temperature of 838 K (565 °C) for an applied stress of 160, 200, and 237 MPa. From these experiments, the same stress exponent of n ≈ 15 was obtained. In Table II, the quantitative results of the creep tests have been summarized.

The fracture surface after creep failure for a stress of 200 MPa and a temperature of 823 K (550 °C) is shown in Figure 3 and indicates a ductile fracture mode. Mo-rich precipitates are observed on the fracture surface, as indicated by the arrows. The phase composition was analyzed by EDS (a quantitative analysis of the phase composition is presented in the section describing the TEM and atom probe measurements).

Figure 4 shows the SEM morphology of the Fe-Mo alloy after creep failure. In Figures 4(a) and (b), the microstructure after creep at a stress of 180 MPa and a temperature of 823 K (550 °C) is shown. The direction of the applied stress is indicated in the upper left hand corner. In the SEM picture of Figure 4(a), a majority of grains are elongated along the stress direction. Such a grain elongation is as expected for a fracture strain of about 60 pct. Mo-rich deposits are formed heterogeneously along the grain boundaries. The largest precipitates are formed at triple junctions, which are also the preferential nucleation sites for creep cavities. At high magnification (Figure 4(b)), the microstructure shows a partial filling of a microscopic defect site (1), as well as irregular-shaped precipitates that are not connected to a pore, marked by (2) and (3). These observations are explained by a preferential Mo segregation on the creep cavity surface and a continuous growth of the formed precipitates until a complete filling of the creep cavities is achieved. Figures 4(c) and (d) shows the microstructure after creep at a higher temperature of 838 K (565 °C) for a stress of 200 MPa. Again, heterogeneous precipitation along grain boundaries is clearly visible. Creep cavities are mainly formed along grain boundaries perpendicular to the stress direction, which are the favorite sites for cavity nucleation. However, the growth and coalescence of these cavities are relatively fast, so that the Mo may not have had sufficient time to precipitate at the cavity surface and fill the damage site completely.

In order to observe a more pronounced interaction between the creep damage and the site-selective Mo precipitation at the free cavity surface, a lower load of 160 MPa was applied at the higher temperature of 838 K (565 °C). The strain at rupture reaches a relatively high value of about 60 pct for which damage is observed at (i) grain boundaries oriented perpendicular to the loading direction, (ii) triple points, and (iii) sliding grain boundaries oriented at a shallow angle to the loading direction. Figure 5 depicts the SEM images of creep damage and site-selective Mo precipitation along parallel grain boundaries. Examples of (i) unfilled cavities, (ii) partially filled cavities, and (iii) fully filled cavities can clearly be observed. Creep damage and site-selective Mo precipitation are not only restricted to parallel grain boundaries, but also seem to occur in some localized deformation bands within the grains, as marked by the white arrows in Figure 5(a).

In Figure 6(a), the SEM image illustrates the creep damage and Mo precipitation observed at grain boundaries oriented perpendicular to the loading direction. The partially filled triple junction reveals a complex precipitate phase comprised of several individual crystallites. By combining a series of SEM images, the area fraction of Fe₂Mo precipitates was obtained for the studied creep conditions. The obtained results are shown in Figures 1 and 2 and summarized in Table II.

To obtain more insight into the microstructure evolution during creep, EBSD experiments were performed. In Figure 7, orientation maps for the solutionized Fe-Mo alloy are shown (a) before and (b) after creep at a stress of 160 MPa and a temperature of 838 K (565 °C). In contrast to earlier measurements on creep-loaded solutionized Fe-Au alloys,[22] no indications for an extensive subgrain formation were observed. However, the average grain size for the Fe-rich bcc matrix phase derived from the EPMA figures was reduced from 33 μm before creep (Figure 7(a)) to 22 μm after creep (Figure 7(b)) (in accordance with the dynamic recrystallization assumed on the basis of the oscillating creep rate). Note that the value of the grain size before creep is close to the statistically more accurately value of 28 μm, obtained by analyzing a series of SEM images of the same alloy. A comparison of the orientation maps before and after creep suggests that some 〈111〉 micro-texture has formed during the creep tests. The white areas in the creep-failed sample correspond to the Laves phase Fe₂Mo.

TEM measurements were performed on the Fe-Mo alloy after creep at 180 MPa and 823 K (550 °C) to characterize the structure and composition of the precipitates formed in the grain interior and the precipitates formed at the grain-boundary cavities. As shown in Figure 8, nanoscale precipitates are formed in the matrix after creep. A high dislocation density is present due to the extensive plastic deformation during creep (Figures 8(a) and (b)). Previous TEM studies on similar model alloys (Fe-Cu, Fe-Cu-B-N, Fe-Au, and Fe-Au-B-N) did not reveal a high dislocation density for undeformed specimens that were heated to 823 K (550 °C).[16‐18,22]

Precipitates are also present along grain boundaries, as shown in Figure 8(c). Some grain boundaries are completely covered with a thin film of the Fe₂Mo phase (see Figure 8(d)). The nature of the precipitates and film was identified by EDS and by lattice imaging, respectively. For example, the Fe and Mo compositions were measured with the electron beam on four different positions on the precipitates in Figure 8(c) (see numbers) as 33 (±1), 33 (±1), 32 (±2), and 31 (±2) at. pct Mo, consistent with a Fe₂Mo-type precipitate. Several lattice images were obtained from different precipitates that showed lattice spacings and lattice-plane angles consistent with a hexagonal Fe₂Mo lattice structure with a = 0.472 nm and c = 0.772 nm. In thermodynamic equilibrium at a temperature of 823 K (550 °C), the presence of the Fe₂Mo phase in combination with the Fe-rich bcc matrix phase is expected for our binary Fe-Mo alloy.[31] One of the lattice images covers an area of Figure 8(c) that contains precipitate number 4 and the neighboring matrix. The associated Fourier transform in Figure 8(g) shows that the electron beam was aligned along the [0001] axis of the hexagonal structure for precipitate number 4. Part of the same lattice image of the matrix neighboring precipitate number 4 is shown in Figure 8(f). The associated Fourier transform (of a larger area) in Figure 8(h) shows the electron beam direction in the Fe matrix is [100]. The indices in the Fourier transforms further indicate that within a few degrees the orientation relation \( (2\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{2} 00)_{{{\text{Fe}}_{2} {\text{Mo}}}} \, || \, (110)_{\text{Fe}} \) is found. Hence, for this precipitate, an orientation relationship with one of the Fe grains is approximately: Other orientation relationships exist as well, as is to be expected for precipitates at a grain boundary.

APT was employed to quantify grain-boundary segregation, the chemical composition of the Mo-rich precipitates at the grain boundaries, the matrix composition in direct contact to the grain-boundary precipitates, and the matrix composition far away from the grain boundaries. In total, six tips with a total volume of 3.5 × 10⁶ nm³ were measured. The tips were prepared from two lift-outs, one at the grain boundary and the other in the matrix.

According to thermodynamic calculations, two phases are in equilibrium at 823 K (550 °C) in the Fe-Mo alloy investigated here: the Laves phase Fe₂Mo with 33 at. pct Mo and the α-iron matrix with 0.8 at. pct Mo. Our APT results summarized in Figure 9 indicate that the Mo concentration in the grain-boundary precipitates in the α-iron matrix close to the grain-boundary precipitates and in the α-iron matrix far away from the grain boundaries amount to 36, 2.0, and 3.5 at. pct Mo, respectively. The measured composition of the precipitates and the adjacent matrix is in close agreement with the predicted equilibrium values. However, the composition of the matrix further away from the grain boundaries is still close to the nominal alloy composition of 3.7 at. pct Mo, and thus far from thermodynamic equilibrium (this was also found from EDS in TEM). It can be concluded that although the system is close to local equilibrium at the precipitates, the overall system is still far away from thermodynamic equilibrium. Mo is thus still supersaturated in the matrix and available for further healing of creep defects.

Two different grain boundaries were measured and the amount of segregated solutes was analyzed by means of a 1D concentration profile and ladder diagrams.[32] As indicated in Figure 10, the crystallographic character of the interfaces was analyzed based on the crystallographic information contained in the detector event histograms of the atom probe runs by means of atom probe crystallography.[33] Both measured grain boundaries were identified as high-angle grain boundaries. At a grain boundary with a disorientation angle of about 16 deg, a Mo solute excess of 3.15 at/nm² was measured (Figures 9(a) and (b)), while at a grain boundary with a disorientation angle of about 35 deg a Mo excess value of 3.37 at/nm² was measured (Figure 10). Such a variation in the solute excess between two different grain boundaries is normal, as the enrichment of solutes at grain boundaries is highly dependent on their crystallographic character.[34,35]

The fractographs of the creep rupture surface in Figure 3 indicate that intergranular failure is the dominant failure mode for this material. As indicated in Figures 1 and 2, creep failure occurs at a relatively high strain of about 60 pct. The electron microscopy images of Figures 4, 5, and 6 indicate that creep damage forms at (i) grain boundaries oriented perpendicular to the loading direction, (ii) grain boundaries oriented along the loading direction, (iii) grain-boundary triple points and finally, to a lower extent, at (iv) localized deformation bands inside the grains. The shape of the larger Mo-rich deposits clearly reflects the morphology of the creep cavities, demonstrating that site-selective deposition of the Mo-rich phase at free creep cavity surfaces.

In order to evaluate the kinetics of the self-healing process, it is essential to obtain the effective grain-boundary and volume diffusion coefficient for Mo in the Fe matrix. As no information is available for the grain-boundary diffusion coefficient of Mo in pure iron, the Fe self diffusion along grain boundaries is applied. At 823 K (550 °C), which is in the middle of the explored temperature regime, the grain-boundary diffusivity in iron amounts to D _GB = 2.4 × 10⁻¹⁴ m² s⁻¹,[36] while the volume diffusivity of Mo in iron is estimated to be D _V = 2.8 × 10⁻²¹ m² s⁻¹[30] Using the literature value for the volume diffusion coefficient of Mo in an iron matrix at 823 K (550 °C) and a time to failure of 800 hours the estimated diffusion length amounts to \( 2\sqrt {D_{\text{V}} t} \) ≈ 0.2 μm for volume diffusion and \( 2\sqrt {D_{\text{GB}} t} \) ≈ 500 μm for grain-boundary diffusion. The large difference between the diffusion length and the Mo-depletion distance indicates that only a thin layer close to the grain boundaries will be solute-depleted. This is supported by the current atom probe results which indicate that the α-Fe matrix further away from the grain boundaries still has the nominal composition. In previous experiments on Fe-Au alloys,[21,22] it was found that the value of the effective volume diffusion coefficient was three orders of magnitude larger than expected as a result of an extensive subgrain formation during creep loading. For the Fe-Mo alloy, the EBSD experiments did not show experimental evidence for extensive subgrain formation (Figure 7). The observed distance between the deformation bands in the Fe-Mo alloy is of the order of the grain size (see Figure 4), and therefore, the generated diffusion pathways are macroscopically spaced in the Fe-Mo alloy, while the diffusion pathways in the Fe-Au alloy are spaced microscopically. As a result, these deformation bands are not expected to contribute significantly to the effective volume diffusion and the solute-depleted regions remain relatively narrow in the vicinity of the grain boundaries that act as fast diffusion pathways.

Our current experiments on the self-healing behavior of creep damage in Fe-Mo alloys are in qualitative agreement with those obtained for the Fe-Au alloy system.[21,22] From these studies, the following requirements to achieve an efficient autonomous repair of creep damage can be formulated[37]:

If the filling (i.e., autonomous repair) of creep cavities is purely driven by the supersaturation of the solute Mo atoms, then it can easily be estimated how much damage can potentially be healed by balancing the volume fraction of grain-boundary creep cavities f _D, with the volume fraction that can maximally be filled by the deposit formed by the supersaturated solute f _H.

The volume fraction of creep damage can be estimated as f _D = N _C V _C /V _G, where N _C is the average number of grain-boundary creep cavities per grain, V _C is the average volume of a single creep cavity, and V _G is the average grain volume. For a lens-shaped creep cavity, one finds V _C = πψd _C ³ /6, where d _C is the cavity diameter and ψ = (2-3cosα + cos³α)/2sin³α ≈ 0.69 a geometrical factor that is defined by the opening angle α ≈ 75 deg of the creep cavity.[38] Assuming a spherical grain shape, the grain volume amounts to V _G = πd _G ³ /6, where d _G is the grain diameter. Combining these relations leads to

The volume fraction of creep damage that can maximally be healed by the supersaturated solute f _H can be evaluated from the initially homogeneously distributed nominal solute concentration x ₀ (=3.7 at. pct Mo for this alloy), the equilibrium solute concentration in the matrix x _m (=0.8 at. pct Mo at T = 823 K (550 °C)) and a scaling factor R ≈ 1 for the effective volume occupied by the solute atom in the precipitate compared to its volume occupied in the solute:

The scaling factor R can be approximated by the difference in atomic volume occupied by the solute Mo relative to Fe: R ≈ v _Mo/v _Fe = 1.30. The maximum phase fraction of grain-boundary precipitates depends on the solute concentration in the precipitate x _p (=33.3 at. pct Mo for the Fe₂Mo precipitates) and amounts to f _p = (x ₀ − x _m)/x _p = 9.3 pct.

The consumed fraction of the healing reservoir χ can now be defined as follows:

Initially there is no damage, and therefore χ = 0. During creep, this value increases until the reservoir of supersaturated solute is depleted when a critical value of χ = 1 is reached. From Eq. [4], it can be seen that for a given type of healing precipitate, the critical value for which the solute reservoir is depleted is reached later when (i) the grains are larger, (ii) the supersaturation is higher, (iii) the creep cavities are smaller, and (iv) the number of creep cavities per grain is lower. As expected, the grain size and cavity size are the dominant contributions. For an average grain size of d _G = 20 μm, a cavity size of d _C = 1 μm and a supersaturation of Δx = x ₀ − x _m = 3 at. pct Mo about 450 cavities (N _C) can be filled at the grain boundary of a single grain before the supersaturation is exhausted (χ = 1).

It should however be noted that the above discussion is only valid for long time scales. For creep experiments with a finite time, not all supersaturated solute may have had the time to diffuse toward the grain boundaries and contribute to the filling of the creep cavities. As discussed before, the estimated diffusion length within the grain amounts to \( 2\sqrt {D_{V} t} \). The diffusion-limited volume fraction that can maximally be healed f _H ^kin is reduced to the fraction of the grain that is located within a diffusion length of the grain boundary:

Only when the diffusion length \( 2\sqrt {D_{\text{V}} t} \) is larger than the grain radius d _G/2, all available solute reserves can contribute to pore filling, resulting in f _H ^kin = f _H. This situation is reached after a characteristic time of t ₀ = d _G ² /16D _V . Using the literature value for volume diffusion coefficient of Mo in bcc Fe of D _V = 2.8×10⁻²¹ m² s⁻¹ at 823 K (550 °C)[30] and a time to failure of 800 hours, the estimated diffusion length amounts to \( 2\sqrt {D_{\text{V}} t} \) ≈ 0.2 μm. For a grain size of d _G = 20 μm, the available solute is reduced by a factor f _H ^kin /f _H ≈ \( 12\sqrt {D_{\text{V}} t} /d_{\text{G}} \) = 6 pct, which means that the quoted maximum number of cavities per grain (N _C) that can be healed reduces from 450 to about 30 under the given conditions.

For the kinetically limited regime, the maximum number of cavities that can be healed now scales as \( N_{\text{C}}^{\hbox{max} } \left( t \right) \propto \sqrt t \). For steady-state creep, the number of cavities per grain \( N_{\text{C}} \left( t \right) = \dot{N}_{\text{C}} t \) however scales linearly with time. As a result, the part of the reservoir accessible by diffusion may potentially run out (when t ₁ < t ₀) before all supersaturated solute is consumed, which then provides a limit on the maximum allowed strain rate to provide efficient filling of the cavities.

The previously developed model for the filling of creep cavities[22] will be used to estimate the average filling ratio η of creep cavities as a function of (increasing) applied stress (or decreasing failure time). It is assumed that the site-selective precipitation at the cavities starts immediately after the nucleation of creep cavities. The lifetime generally scales with the applied stress σ as: t _f  = kσ ⁻ⁿ , where k is a temperature-dependent constant and, as discussed in “Section III,” the stress exponent amounts to n ≈ 15 for the Mo alloy.

The healing time t _h can be evaluated by assuming a time-dependent creep cavity volume of V _c(t) = V ^* + aσt and a precipitate volume of V _p(t) = bt. The time to fill an individual creep cavity t _h = (V ^*/b)/{1 − σ/σ _c} now strongly depends on the applied stress, where σ _c = b/a is the critical stress beyond which the cavity grows faster than the filling takes place.[22] The corresponding volume of the filled cavity is V _h = bt _h = V ^*/{1 − σ/σ _c }. The fraction of filled cavities now amounts to

It is not straightforward to quantify the effect of healing on the overall creep behavior experimentally. The best would be to have a reference sample that is not self-healing, but has comparable creep properties. In a previous paper, we established that self-healing extends the lifetime by using both solutionized and solution-depleted Fe-Cu alloys as reference system for the studied Fe-Au alloys.[21,22] In these experiments, it was found that the filling of creep cavities significantly reduces the strain rate. The same mechanisms are expected to apply for creep in Fe-Mo alloys. More experimental and modeling studies are required to establish the link between the microscopic creep cavity filling and the macroscopic creep rate in detail.