Computational Modeling for Coarsening of (Fe,Cr)2B in Borated Stainless Steel
2019 Volume 60 Issue 2 Pages 369-372
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2019 Volume 60 Issue 2 Pages 369-372
The microstructure of borated stainless steels subjected to high temperature annealing process near solidus was confirmed using scanning electron microscopy. The spheroidization and coarsening of (Fe,Cr)2B occurred significantly during annealing at 1200°C. The coarsening rate was much faster than that predicted by existing models due to the anisotropy of the precipitate, i.e., its non-spherical shape. We simulated the coarsening behavior in a multi-component diffusional simulation including the anisotropic effect with respect to annealing time. The model illustrated the coarsening behavior of (Fe,Cr)2B well in borated stainless steels, and the interfacial energy between the precipitate and the austenite matrix was estimated to be 1.8 J·m−2.
Fig. 5 Measured and simulated particle size as a function of time. (a) Blank squares represent measured particle size of B18 specimen. Dashed, dot, double dot and solid lines represent particle radii predicted by Robino’s model, LSW model, model-I and model-II, respectively. (b) Blank squares, circles and upper triangles represent measured particle size of B02, B08 and B18 specimen, respectively. Dot, dashed and solid line represent particle radii predicted by model-II for B02, B08 and B18 alloy, respectively.
Borated stainless steel (BSS) is an austenitic stainless steel containing a significant amount (0.2 to 2.25 mass%) of natural or enriched boron.1) Because it has high thermal neutron absorbability, BSS is mainly used to store the nuclear fuel spent. It is inevitably produced in nuclear power generation and emits a large amount of thermal neutrons. In order to prevent these thermal neutrons from being released into the environment, where they may remain for a long period of time, it is necessary to develop a structural material having high neutron absorbability and corrosion resistance. Neutrons are absorbed by a process in which the 10B isotope, which comprises about 20% of natural boron, is converted to 11B when it meets the neutron. However, the high boron content induces a large amount of boride formation, which causes deterioration of the hot-workability and mechanical properties. The mechanical properties of BSS are known to be mainly influenced by the volume fraction and morphology of the borides.2,3) Therefore, the mechanical properties need to be improved by controlling the size and morphology of boride through appropriate heat-treatment.
Coarsening is observed in various precipitates during heat-treatment of steel at high temperature.4,5) In coarsening, larger particles are more energetically favored over smaller particles, and so the number of particles decreases and the average particle size increases, because the overall process uses up the driving force of the capillary effect. Lifshitz-Slyozov and Wagner (LSW) performed mathematical formulation of the coarsening process of spherical particles by assuming that bulk diffusion of alloying elements is the rate controlling process.6,7) In LSW theory, the average particle radius can be expressed as
| \begin{equation} r(t)^{3} - r(0)^{3} = kt,\quad k = (8\sigma V_{m}^{2}DC_{e})/(9RT), \end{equation} | (1) |
The LSW theory has been combined with the multi-component diffusional transformation method, and the coarsening was well simulated in certain systems.8,9) However, LSW theory needs to be modified because anisotropic precipitates, i.e., non-spherical shape, have high coarsening driving force due to their small curvature radius. Pressure change, ΔP, due to capillarity at the interface of the spherical precipitate is represented as 2σ(1/r); this can be compared to the pressure at the interface, given by principal radii of curvature r1 and r2, which can be evaluated as σ(1/r1 + 1/r2).10) Therefore, capillarity change must be included in order to simulate the coarsening of anisotropic precipitates. This study quantified the size of precipitates during the annealing process of BSS produced by ingot-metallurgy. Based on the measured precipitate size, we propose a method using multi-component diffusional transformation to simulate the coarsening behavior of anisotropic borides in BSS.
Based on type 304 stainless steel, three alloys were designed with various B contents; the chemical composition of the alloys is shown in Table 1. Alloys were labeled B02, B08, and B18 according to the B content. The 50 kg ingots were reheated at 1150°C for 1 hour and hot-rolled into plates with thickness of 40 mm. And, the as-received specimens were prepared by heat treatment at 1050°C for 30 minutes, followed by cooling in air. Cylindrical specimens with radius of 10 mm and length of 20 mm were cut and sealed in vacuum quartz tubes, which were water-quenched after annealing at 1200 ± 20°C for 1, 2, 4, 8, 24, 48, 96, and 192 hours using a box furnace.
Microstructures were observed by scanning electron microscopy (SEM). After polishing with a diamond suspension of 1 µm, backscattered electron (BSE) images were obtained using a JEOL IT-3000 SEM with energy-dispersive X-ray spectroscopy (EDS) in large current mode. The volume fraction, average mean diameter, and aspect ratio were quantified by image analysis software (Image-Pro Analyzer 7.0; Media Cybernetics, Rockville, MD) for each image. The diameter of each precipitate was evaluated by the diameter of a circle equivalent to the area of the precipitate, and the aspect ratio was the ratio between the major and minor axes of an ellipse equivalent to the precipitate. In each image, more than 1000 precipitates were obtained so as to allow an evaluation of the average of diameter and aspect ratio. Five images of different locations on each specimen were analyzed and the mean and standard deviation were evaluated.
2.2 Computational methodsUsing ThermoCalc software, the thermodynamic equilibrium of the system was calculated based on the TCFE9.0 database.11) The coarsening of boride was simulated for the 18Cr–12Ni–1.8B mass% system, which is similar to the alloy composition of B18, which has the largest amount of borides. According to the LSW theory, the maximum precipitate size can be assumed to be 1.5 times the average precipitate size. Therefore, it is possible to simulate coarsening using multi-component diffusional modeling implemented in DICTRA software with MobFe4 mobility database and a spherical closed system, as shown in Fig. 1.12) Due to the capillary effect, there is a concentration gradient between the equilibrium concentration in the vicinity of the precipitate, C(rp), and the equilibrium concentration at the boundary of the system, C(rave), in the austenite matrix. With this concentration gradient, precipitation coarsening occurs by elemental diffusion. The average size of the initial precipitates is assumed to be the average size of the specimen after 2 hours annealing, at which time the size distribution became similar to the ideal distribution in the LSW theory. The precipitate and the austenite region were divided into 100 equally spaced linear grid points and the initial composition was set to thermodynamic equilibrium at 1200°C. The total size of the system was set to maintain the volume fraction of precipitate in thermodynamic equilibrium.
Closed system with spherical cell with (Fe,Cr)2B phase particle enclosed in austenite matrix.
The driving force of coarsening is the free energy change (ΔGm) due to capillarity at the interface of a precipitate that has a certain amount of curvature. In the case of model-I, ΔGm = 2σVm/r, where Vm is molar volume corresponding to 1 mole of substitutional elements, was applied assuming a spherical precipitate of radius r. In the case of model-II, we modified the capillary effect by assuming a cigar-shaped ellipsoid (two minor axis with length 2 × l and one major axis with length 2a × l) in which a is the average aspect ratio. In the ellipsoid, the two principal curvature radii at the points on the major axis have a minimum curvature radii of (l/a, l/a) and ΔGm = σVm(a/l + a/l). At the points on the minor axis, the principal curvature radii is (l, a2l), the maximum curvature radii, and ΔGm = σVm(1/l + 1/a2l). In this study, for efficiency of calculation, the capillary effect of an ellipsoid is assumed to be ΔGm = σVm(a + 0.5 + 0.5/a2)/l, which is the average of maximum and minimum. Since the average radius of the precipitate was measured as the radius of the circle equivalent to the area, $l = r/\sqrt{a} $ and $\Delta G_{m} = f(a) \times \sigma V_{m}/r$, where $f(a) = \sqrt{a} \times (a + 0.5 + 0.5/a^{2})$ is an aspect ratio dependent shape factor. Since the aspect ratio, a, is a function of the annealing time, the shape factor is also a function of time and regressed as f(t) = c1 + c2 log(t).
The thermodynamic equilibrium phase diagram can be used to estimate the initial microstructure of each specimen and the phase changes that occur during heat-treatment. Figure 2 shows the calculated equilibrium phase diagram corresponding to 0.07C–18Cr–12Ni–1.5Mn mass% with respect to the B amounts and temperature. This diagram allows us to interpret the phases of BSS because they are in good agreement with the liquidus and solidus temperatures measured.2) The amount of B corresponding to the eutectic point is 2.15 mass% and the temperature is 1256°C. Thus, the three alloys (B02, B08, B18) had the hypo-eutectic composition and the initial microstructure consisted of a mixture of austenite and (Fe,Cr)2B precipitates solidified. Since Cr23C6 is formed at a temperature below 900°C, the annealing condition to control the morphology of the boride was determined in the range of 1000–1250°C. During annealing at 1200°C, the variation of (Fe,Cr)2B morphology can be studied without forming other phases.
Equilibrium phase diagram for borated stainless steel based on TCFE9.0 database. Liquid, FCC, BCC, orthorhombic M2B and Cr23C6 phases are allowed in calculation. Diagram was established on 0.07C–18Cr–12Ni–1.5Mn mass%, corresponding to compositions of type 304 stainless steel.
Figure 3 shows the initial microstructure of the B18 specimen and the microstructural evolution during annealing at 1200°C. Prior to annealing, a plate-like boride was produced through eutectic transformation with austenite. The borides had the anisotropic morphology that might easily cause cracks. The shape of the precipitates became closer to spherical shape and the average size increased with the annealing time. The average size and aspect ratio of the precipitate are summarized, as a function of time, in Fig. 4. As the B content increased, the average precipitate size increased at the same annealing time, and the aspect ratio showed insignificant variation according to the B content. As shown in Fig. 4(c), as-received specimen has more precipitates in a smaller size region than distribution predicted by LSW theory. During annealing, the peak position of the distribution moves closer to the average precipitate size, which is similar to the distribution predicted by the LSW theory.
Scanning electron micrographs of B18 specimen for (a) as-received, and (b–d) annealed specimens. Annealing conditions of specimens are (b) 8 h, (c) 48 h, and (d) 192 h at 1200°C.
Variation of (a) precipitate size and (b) aspect ratio of B02, B08 and B18 specimens, as a function of time, during annealing at 1200°C. Data to the left of the dashed line represent as-received specimen. Blank squares, circles and upper triangles represent B02, B08 and B18 specimens, respectively. (c) Variation of normalized size distribution of B18 specimen for different annealing time. Solid, dash, dot, dash dot and dash double dot line represent for 0 h, 2 h, 8 h, 48 h and 192 h annealing, respectively.
Figure 5(a) indicates the coarsening of precipitate calculated from the various models. In Robino’s study, $R(t)^{3} - R(0)^{3} = 0.182 \times t$ (R in µm, t in h) was derived based on observations that the mean size increases from 0.5 µm to 1.0 µm over an annealing for 4 hours at 1225°C.2) Despite the different experimental conditions between Robino’s and this study, the measured values were exactly reproduced following the Robino’s model. This is probably associated with a negative contribution to the coarsening rate by the B content (1.14 mass%), which was lower than that of the B18 alloy, and a positive effect due to the higher annealing temperature. Since Robino’s model is derived without physical parameters, its application to the other conditions is limited.
Measured and simulated particle size as a function of time. (a) Blank squares represent measured particle size of B18 specimen. Dashed, dot, double dot and solid lines represent particle radii predicted by Robino’s model, LSW model, model-I and model-II, respectively. (b) Blank squares, circles and upper triangles represent measured particle size of B02, B08 and B18 specimen, respectively. Dot, dashed and solid line represent particle radii predicted by model-II for B02, B08 and B18 alloy, respectively.
The parameters of the LSW theory were determined by calculating the diffusion coefficient and equilibrium concentration of chromium, because the coarsening rate is mainly determined by chromium diffusion.2) The calculated parameters at 1200°C are Ce = 16366 mol·m−3, D = 9.63 × 10−15 m2·s−1, Vm = 2.08 × 10−5 m3·mol−1 and σ = 1.8 J·m−2. To allow comparison with other models, applying the same interfacial energy, a lower coarsening rate was observed than that in the measured results, despite the relatively high interfacial energy. Because the LSW model had the assumption of the diffusion of single alloying elements and spherical shape of precipitates, the modeling results indicated a limit to reproduce the measured precipitate size.
We used multi-component diffusion simulation to overcome the limitation of the LSW theory. Model-I showed a coarsening rate faster than that of the LSW model. In the case of multi-component diffusion simulation, the equilibrium chromium concentration at the interface is lower than that predicted by the LSW theory. Thus, the concentration gradient was larger, the diffusion of chromium in the austenite matrix was higher, and the coarsening rate was faster. Nevertheless, model-I relatively underestimated the measured precipitate size. This is probably due to the underestimated capillary effect, because model-I did not consider anisotropy of the precipitate. In model-II, which assumed anisotropy of an ellipsoid precipitate, we applied the shape factor f(t) = 6.78755 − 0.62837 × log(t) (t in seconds) with correlation coefficient of R2 = 0.92; the model reproduced the measured data well. In particular, model-II shows a faster coarsening rate at initial annealing time than predicted by Robino’s model, because it had a higher aspect ratio. Figure 5(b) shows the results of coarsening simulation of B02 and B08 alloys using the same parameters. Simulation performed well in the early stage of annealing. On the other hand, annealing for 192 h shows overestimation compared to measurements. This may be because the surface morphology of the precipitate changes smoothly during annealing and the capillary effect decreases.
The interfacial energy, 1.8 J·m−2, was relatively high because the morphology of the precipitate was more complicated than that of the ellipsoid precipitate, and the effect of curvature at the interface was included in the interfacial energy. It is assumed that the shape factor is simplified to average the maximum and minimum of the ellipsoid, but that factor could be between the maximum and minimum, depending on the surface shape complexity of the precipitate. Since the observed shape of the precipitate is complicated, it might be biased to a larger value. When there is a high aspect ratio at the beginning of annealing, the difference between the average and the maximum of the capillary effect is about 40%. Since the coarsening rate is controlled by the product of the shape factor and the interfacial energy, the interfacial energy could be lower if we assume a precipitate with a shaped surface more complex than an ellipsoid.
Annealing of borated stainless steels at 1200°C caused spheroidization and coarsening of the eutectic structure. As the annealing proceeded, the average size of the precipitates increased, and the aspect ratio decreased. The coarsening rate was faster than those predicted by existing models, and it was possible to modify this rate using multi-component diffusional simulation and anisotropy of precipitate. Assuming a precipitate of ellipsoid shape in borated stainless steels containing 1.78 mass% B, the interfacial energy was found to be 1.8 J·m−2.
This work was funded by the Fundamental R&D Program of the Korea Institute of Materials Science (KIMS), Grant No. PNK5850. This work was also supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF-2016R1C1B1011593) funded by the Ministry of Science, ICT & Future Planning.
