Investigation of Cu Diffusivity in Fe by a Combination of Atom Probe Experiments and Kinetic Monte Carlo Simulation

Abstract

In the present study, the diffusion coefficient of Cu in Fe was experimentally estimated from the precipitation kinetics down to 390°C. At this temperature, diffusion couples, which is a typical method to obtain diffusion coefficients, cannot be applied. The matrix Cu concentration and the number density of Cu precipitates in Fe–Cu alloy, which were the main parameters used to estimate the diffusion coefficient, were directly obtained using atom probe tomography. The temperature dependency of the diffusion coefficient of Cu in Fe estimated in the present study was more reliable than that obtained in a previous study, which also reported the diffusion coefficient of Cu in Fe from precipitation kinetics. This indicated that our estimation of the diffusion coefficient of Cu in Fe with atom probe tomography measurements yielded greater accuracy. In addition, the estimated diffusion coefficient of Cu in Fe tended to deviate to higher values from the extrapolated diffusion coefficient of Cu in Fe, which was obtained by diffusion couples, with decreasing temperature. This deviation is discussed by employing a kinetic Monte Carlo simulation.

Fig. 2 Arrhenius plot for the experimentally obtained diffusion coefficient of Cu in Fe. D_Cu estimated in this study was closer to the extrapolated values reported in our previous study²³⁾ (dashed line) than D_Cu estimated by Lê et al. (squares),²⁵⁾ and tended to deviate toward higher values from the extrapolated values with decreasing temperature.

1. Introduction

Cu precipitation has significant effects on the mechanical properties of various steels products. One of the most important issues is the origin of neutron irradiation-induced embrittlement of reactor pressure vessel (RPV) steels in first-generation light water reactors during long-term operation.^1–3) Furthermore, Cu precipitation could contribute to strengthening of steels, such as high-strength low-carbon steels and maraging stainless steels.^4,5) Cu precipitation could also be utilized to improve the weldability and atmosphere corrosion resistance of steels, which would make them suitable for applications in shipbuilding.⁶⁾ In addition, as tramp elements in scrap steel, the accumulation of Cu could cause deterioration of the hot workability during casting or hot rolling.⁷⁾ Nevertheless, Cu precipitation-strengthened steel produced from scrap steel is under development.⁸⁾ Therefore, the precipitation of Cu in steels and model alloys has been a focus of many studies.^9–18) As a key parameter to investigate the evolution of Cu precipitation, measurement of the Cu diffusion coefficient (D_Cu) in Fe is essential. However, few experimental studies have focused on the estimation of D_Cu below 550°C, at which Cu precipitation in steels is technologically relevant. For example, Cu precipitates are formed around 300°C in RPV steels, which is the operation temperature of light water reactors.

A typical method to obtain D_Cu is to measure the Cu distribution in Cu–Fe diffusion couples, and several studies have obtained D_Cu by this technique, in which cases Cu distributions were measured by activity measurements,^19,20) electron probe microanalysis,²¹⁾ and laser-induced breakdown spectroscopy.²²⁾ The diffusion temperatures in these studies were typically higher than 700°C because the techniques to measure Cu distributions generally require long-range diffusion at distances of at least several micrometers due to the spatial resolution limits. In our previous study,²³⁾ atom probe tomography (APT) was employed to measure the Cu distribution in Cu–Fe diffusion couples with nanoscale spatial resolution, whereby D_Cu was obtained down to 550°C, which is lower than that in previous studies.^20,21) However, it was difficult to measure the Cu distribution in Cu–Fe diffusion couples at lower temperatures even using APT, because the amount of Cu diffused into Fe crystals decreases to a level lower than the detection limit of APT due to the decreasing solubility limit of Cu in Fe with the decrease in temperature.^23,24)

At lower temperatures, in which diffusion couples cannot be applied, Lê et al. estimated D_Cu from the precipitation kinetics based on a classic growth model that considered the precipitated fraction and the number density of Cu precipitates formed in Cu-supersaturated Fe–Cu alloy in the temperature range of 390 to 500°C.²⁵⁾ The precipitated fraction was obtained from electrical resistivity measurements, where the resistivity of the Fe–Cu alloy was assumed to be a linear function of the residual Cu concentration in matrix. The number density of Cu precipitates was estimated from transmission electron microscopy (TEM) observations. However, the solute precipitated fraction could be not accurately evaluated due to the deviation from Matthiessen’s rule for electrical resistance measurements.²⁶⁾ Consequently, D_Cu reported by Lê et al. shows an unexpected negative dependence on temperature, which is somewhat unnatural.

In the present study, to improve the accuracy of D_Cu estimation, APT was employed to directly measure both the matrix Cu concentration and the number density of Cu precipitates. D_Cu was then estimated in the temperature range of 390 to 600°C from the precipitation kinetics using the same classic growth model as that used by Lê et al.²⁵⁾ In this classic growth model, it is assumed that Cu atoms do not interact with each other during migration. However, for a Cu-supersaturated Fe–Cu alloy, Cu atoms tend to be attracted to each other, which promotes the aggregation of Cu atoms.^11,12,27) The effects of this deficiency in the classic growth model on the estimation of D_Cu was investigated by a kinetic Monte Carlo simulation.

2. Methods

The Fe–Cu alloy used in this study contains 0.89 atom% Cu nominally, which was prepared by arc-melting of 4N Fe and 5N Cu. The ingot of this alloy was cold-rolled and cut into 1 mm thick discs with a radius of 5 mm. The discs were annealed for stress relief at 825°C for 4 h and quenched into ice water before thermal aging, which was conducted at 390°C for 2184 h, 425°C for 310 h, 475°C for 9 h, 550°C for 9 h, and 600°C for 1 h.

The specimens for the APT measurement were fabricated from the alloy discs using a focused-ion beam apparatus. APT measurements were performed using a local-electrode atom probe (LEAP-4000XHR, Ametek-Cameca) at a pulse rate of 200 kHz, a pulse fraction of 20%, an evaporation rate of 0.5% per voltage pulse, a DC voltage typically in the range of 3 to 8 kV, and a specimen temperature of 50 K.

To estimate D_Cu from the precipitation kinetics, the classic growth model used by Lê et al.²⁵⁾ was also used in this study. By solving the definite integral of following equation on an interval between the time 0 and the aging time,   

\begin{equation} \frac{d\xi}{dt} = 3D_{\textit{Cu}}c_{m0}^{\frac{1}{3}}\left(\frac{4\pi N_{d}}{3}\right)^{\frac{2}{3}}\xi^{\frac{1}{3}}(1 - \xi), \end{equation} (1)

D_Cu can be obtained at a certain time for each thermal aging conditions, where c_m0 is the initial matrix Cu concentration (i.e., the total Cu concentration), N_d is the number density of Cu, t is the diffusion time, and ξ represents the precipitated fraction at a certain time, which is defined as:   

\begin{equation} \xi = \frac{c_{m0} - c_{m}}{c_{m0} - c_{s}}, \end{equation} (2)

where c_m is the residual Cu concentration in the Fe matrix, and c_s is the solubility limit of Cu in body-centered cubic (bcc) Fe crystals.

N_d, c_m0 and c_m were directly measured by APT in the temperature range of 390 to 600°C. Regarding c_s, the extrapolated values reported by Salje and Feller-Kniepmeier²¹⁾ were used, which was obtained by log c_s = 4.495 − 4627/T, where T is temperature.

3. Results

Figure 1 shows atom maps for Cu in the Fe–Cu alloy with a slice size of 10 nm. The Cu precipitates were identified by the maximum separation method.^28,29) Cu precipitates formed in Fe–Cu alloy have been reported to be almost pure Cu and coherent with a bcc matrix,^30,31) therefore the size of the Cu precipitates was calculated by converting the average number of Cu atoms detected in the precipitates to the radius of an equivalent sphere.³²⁾ The average number of Cu atoms in the Cu precipitates, the average radius and the number density of Cu precipitates, total Cu concentration and the residual Cu concentration in matrix are listed in Table 1.

Fig. 1

Atom maps for Cu in thermally aged Fe–Cu alloy at (a) 390°C for 2184 h, (b) 425°C for 310 h, (c) 475°C for 9 h, (d) 550°C for 9 h, and (e) 600°C for 1 h. Cu precipitates formed in Fe–Cu alloy after thermal aging.

Table 1 Average number of Cu atoms in Cu precipitates, average radius of Cu precipitates, number density of Cu precipitates, total Cu concentration, and residual Cu concentration in matrix in thermally aged Fe–Cu alloy.

D_Cu for the each aging condition listed in Table 1 were estimated using eq. (1) and shown by the stars in Fig. 2 together with the reported experimental values.^23,25) Notice that the estimation temperature of D_Cu in this study is lower than that reported in our previous study,²³⁾ in which D_Cu was measured by diffusion couples. D_Cu estimated in this study is particularly consistent with the result reported in our previous study²³⁾ at 600°C, and closer to the extrapolated values reported in our previous study²³⁾ (dashed line) than D_Cu estimated by Lê et al. (squares).²⁵⁾ However, D_Cu tended to deviate somewhat to higher values than the extrapolated values of D_Cu reported in our previous study²³⁾ with decreasing temperature. This systematic deviation is further discussed in the following.

Fig. 2

Arrhenius plot for the experimentally obtained diffusion coefficient of Cu in Fe. D_Cu estimated in this study was closer to the extrapolated values reported in our previous study²³⁾ (dashed line) than D_Cu estimated by Lê et al. (squares),²⁵⁾ and tended to deviate toward higher values from the extrapolated values with decreasing temperature.

4. Discussion

In comparison with the study of Lê et al.,²⁵⁾ D_Cu reported by Lê et al.²⁵⁾ shows a negative dependence on temperature, which is somewhat unnatural. Reviewing our present study, the total Cu concentration, the residual matrix Cu concentration, and the number density of Cu precipitates in Fe–Cu alloys, which are main parameters used to estimate D_Cu, were directly and precisely measured using APT. Consequently, D_Cu shows better temperature dependency and linearity than that estimated by Lê et al.²⁵⁾ in the Arrhenius plot. This result suggests the validity of the data estimated in our present study.

Regarding the deviation of D_Cu between our study and the extrapolated values in our previous study,²³⁾ we considered that was caused by a deficiency in the classic growth model, in which Cu atoms are supposed to migrate independently and not to interact with each other. However, for actual Cu-supersaturated Fe–Cu alloys, Cu atoms tend to attract each other, and this attraction promotes the aggregation of Cu atoms.^11,27,33) This is referred to as a Cu–Cu attractive interaction in the following. We then made a hypothesis that D_Cu was overestimated because the Cu–Cu attractive interaction was not considered in this classic growth model.

To demonstrate this hypothesis, we newly developed a kinetic Monte Carlo simulation code that simulates the Cu precipitation in Fe–Cu alloys, including the Cu–Cu attractive interaction. The simulation model is based on the residence time algorithm.³⁴⁾ In this simulation model, Fe and Cu atoms and a single vacancy are placed in a rigid bcc lattice, and the model allows solute atoms to migrate via the vacancy exchange mechanism only towards their first nearest-neighbor position. The jump frequency of an atom to a vacancy position is given by:   

\begin{equation} f = \nu \exp \left(-\frac{E}{k_{B}T}\right), \end{equation} (3)

where ν is the vibration frequency of solute atoms, k_B is the Boltzmann constant, T is temperature, and E is the migration energy for the vacancy jump. In the simulation model, E is dependent on the surrounding configuration of the jumping atom and the vacancy, which was calculated by a rigid lattice model with pair interactions between first- and second-nearest neighbors.^11,35) The pair interaction energies were taken from the study by Soisson and Fu,¹¹⁾ in which they fitted these values using ab initio calculations.

To simulate the Cu–Cu attractive interaction and to allow Cu atoms to precipitate, the binding energy between Cu atoms was introduced by pair interactions.^11,26) For instance, for 2 Cu atoms with a nearest-neighbor relationships to each other, the binding energy is the difference between the sum of pair interactions when Cu atoms are separated and aggregated. As shown in Fig. 3, when 2 Cu atoms are separated, there are 9 Fe–Cu pair interactions ($\epsilon _{\textit{FeCu}}$), and 6 Fe–Fe interactions ($\epsilon _{\textit{FeFe}}$). When Cu atoms are aggregated, there are 1 Cu–Cu pair interaction ($\epsilon _{\textit{CuCu}}$), 7 $\epsilon _{\textit{FeCu}}$, and 7 $\epsilon _{\textit{FeFe}}$. The binding energy between Cu atoms is obtained as $E_{\textit{CuCu}}^{b} = 2\epsilon _{\textit{FeCu}} - \epsilon _{\textit{CuCu}} - \epsilon _{\textit{FeFe}} = 0.12$ eV. The value is positive; therefore, attractive interactions are considered to exist between Cu atoms. The details regarding pair interactions are summarized in the Appendix.

Fig. 3

Example of pair interactions for separated Cu atoms and aggregated Cu atoms. The positive binding energy between the 2 Cu atoms can be obtained from the difference between the sum of pair interactions when Cu atoms are separated and aggregated.

The kinetic Monte Carlo simulations indicated that vacancies frequently exchanged with neighboring Cu atoms and immediately went back to the original state, and that these repeated events did not contribute significantly to the evolution of Cu precipitation, which slowed down the efficiency of the simulation. To avoid this problem, the simulation was accelerated using the method reported by Daniels and Bellon.³⁶⁾

The simulation was performed at 475°C for 9 h. The size of the simulation box was 15 × 15 × 15 nm³, and the concentration of Cu employed was 0.89 atom%, which is the same as the Cu concentration of the Fe–Cu alloy used in this study. Cu atoms were randomly placed in lattice sites for the initial state. To compute the average radius and number density of Cu precipitates, and the matrix Cu concentration in the simulation, Cu precipitates that included more than 10 Cu atoms were considered.

Figure 4(a), (b), and (c) show the evolution of the average radius and the number density of Cu precipitates, and the matrix Cu concentration obtained in the simulation, respectively. The average radius of Cu precipitates increased with time. The number density of Cu precipitates initially increased with time, and then decreased due to precipitate coarsening. The matrix Cu concentration decreased with time. Figure 4 also shows the average radius and the number density of Cu precipitates, and the matrix Cu concentration measured experimentally using APT, together with the experimental results reported by Toyama et al., where these parameters for the same alloy were measured using APT for different aging times.^31,37) As shown in Fig. 4, the simulation results agreed with the experimentally observed precipitation results.

Fig. 4

Monte Carlo simulation of the precipitation kinetics in Fe–0.89 atom% Cu during thermal aging at 475°C. Evolution of (a) average radius of Cu precipitates, (b) number density of Cu precipitates, and (c) matrix Cu concentration. The simulation reproduced the experimentally observed precipitation results.

Although a Cu–Cu attractive interaction was considered in the simulation, it is still possible to calculate the Cu diffusion coefficient in the simulation ($D_{\textit{CGM}}^{\textit{sim}}$) using the same classic growth model as that used in the experiment, while the total Cu concentration, the residual Cu concentration in matrix, and the number density of Cu precipitates were obtained in the simulation, the values of which were 0.89 atom%, 0.27 atom% and 3.5 × 10²⁴ m⁻³, respectively. $D_{\textit{CGM}}^{\textit{sim}}$ is shown in Fig. 5 as a green triangle, and is consistent with D_Cu estimated experimentally at 475°C due the good reproduction of the simulation. On the other hand, because the movement of all Cu atoms in the simulation was recorded, the true diffusion coefficient of Cu in the simulation ($D_{\textit{True}}^{\textit{sim}}$) can also be calculated by the definition of the diffusion coefficient:³⁸⁾   

\begin{equation} D = \frac{\langle R^{2}\rangle}{6t}, \end{equation} (4)

where ⟨R²⟩ is the mean square displacement of Cu atoms, and t is the diffusion time. $D_{\textit{True}}^{\textit{sim}}$ is shown in Fig. 5 as an orange triangle, which is closer to the extrapolated values of D_Cu obtained using the diffusion couples in our previous study,²³⁾ being smaller than $D_{\textit{CGM}}^{\textit{sim}}$.

Fig. 5

Arrhenius plot for diffusion coefficient of Cu in Fe obtained by simulation. $D_{\textit{True}}^{\textit{sim}}$ is closer to the extrapolated values of D_Cu obtained with the diffusion couples in our previous study²³⁾ than $D_{\textit{CGM}}^{\textit{sim}}$.

$D_{\textit{True}}^{\textit{sim}}$ is smaller than $D_{\textit{CGM}}^{\textit{sim}}$. This suggests that the diffusion coefficient obtained by classic growth model could be overestimated for the simulation. The reason could be considered is that the Cu–Cu attractive interaction was not considered in this classic growth model. Consequently, this discussion suggests that D_Cu estimated experimentally was also overestimated by the same reason, which refers to that the Cu–Cu attractive interaction was not considered in the classic growth model.

5. Conclusion

In summary, D_Cu was successfully estimated experimentally using APT combined with the classic growth precipitation model in the temperature range of 390 to 600°C. Although the resultant values appear to be more reliable than the previous study where electrical resistivity measurements and TEM were employed,²⁵⁾ D_Cu tended to deviate toward higher values from the extrapolated values of our previous study²³⁾ with decreasing temperature. A kinetic Monte Carlo simulation was employed to reproduce the precipitation kinetics of Cu in Fe–Cu alloy at 475°C. The simulation results suggested that the deviation of D_Cu caused by a Cu–Cu attractive interaction that was not considered in the classic growth model.

Acknowledgments

This work was supported by KAKENHI (26709073, 17H03517 and 20H02661). This work was performed under the GIMRT program of IMR, Tohoku Univ. (18M0405, 19M0407, and 20M0406).

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Appendix

The pair interactions were fitted at ab initio data using the following relations in the study of Soisson and Fu (F. Soisson and C.-C. Fu, Phys. Rev. B 76, (2007)):   

\begin{align*} &E_{V}^{\textit{for}}(A) = \frac{z_{1}}{2}\epsilon_{AA}^{(1)} - \frac{z_{2}}{2}\epsilon_{AA}^{(2)} + z_{1}\epsilon_{AV}^{(1)} + z_{2}\epsilon_{AV}^{(2)},\\ &E_{B}^{\textit{sol}}(A) = \frac{z_{1}}{2}(\epsilon_{AA}^{(1)} + \epsilon_{BB}^{(1)} - 2\epsilon_{AB}^{(1)})\epsilon_{AA}^{(1)} \\ &\qquad \qquad - \frac{z_{2}}{2}(\epsilon_{AA}^{(2)} + \epsilon_{BB}^{(2)} - 2\epsilon_{AB}^{(2)}),\\ &E_{XY}^{b(n)}(A) = \epsilon_{AY}^{(n)} + \epsilon_{AX}^{(n)} - \epsilon_{XY}^{(n)} - \epsilon_{AA}^{(n)},\\ &E_{A}^{\textit{coh}} = \frac{z_{1}}{2}\epsilon_{AA}^{(1)} - \frac{z_{2}}{2}\epsilon_{AA}^{(2)}, \end{align*}

where $E_{V}^{\textit{for}}(A)$ is the vacancy formation energy in A matrix, $E_{B}^{\textit{sol}}(A)$ is the formation energy of a substitutional impurity B in A matrix, $E_{XY}^{b(n)}(A)$ is the binding energy between X and Y, lying on the n^th nearest neighbor sites in A matrix, $E_{A}^{\textit{coh}}$ is the cohesive energy of pure A; $\epsilon _{AB}^{(n)}$ is the pair interactions between X and Y, lying on the n^th nearest neighbor sites, A, B, X, and Y can be Fe or Cu atoms, as listed in Table A1, V indicates vacancies.

Table A1 First and second nearest-neighbor pair interactions used in the kinetic Monte Carlo simulation.