2.1 Details for IM3D calculations

Primary radiation damages for self-ion irradiation in nickel were predicted by using the recently developed high-efficiency IM3D code with the "QKP" method[18]. IM3D is an open-source parallel 3D MC code with binary collision approximations (BCAs) for rapidly simulating the energetic ion transportation and defect production in materials. Based on fast indexing of scattering integrals and the SRIM stopping power database, IM3D can well reproduce SRIM results for bulk or layered materials and is >10⁴ times faster than SRIM. Thus, IM3D is a more suitable tool to realize the high-throughput calculations in energetic ion–target interaction systems.

As shown in the schematic graph in Fig. 1, the spatial distributions of primary radiation damages for self-ion irradiation in bulk nickel with incident ion energies up to 15 MeV and normal incidence were predicted. The displacement threshold energy of 40 eV for nickel was used during the calculation. For each calculation, 10⁶ nickel ions were averaged to get convergent defect productions. The target was divided into a certain number of bins with each size of 30×30×30 nm³. The parameter settings for ions and targets during the calculations are summarized in Table S1 in the Supplementary Materials.

Fig. 1Full-screen View

Schematic graph for self-ion irradiation in nickel, in which a normal incidence at the center point of bulk nickel is adopted, and the depth direction of the target is along the z-axis.

2.2 Radiation-enhanced diffusion coefficients

The theoretical framework of RED has been successfully developed from the rate theory of defect evolutions[5,13]. The atomic diffusion in materials under irradiation mainly arises from the following three aspects[5,16]:

where the first item D_mix on the right-hand side (RHS) is related to the ballistic mixing[24]. The atomic displacement cascades start immediately after the primary knock-on atom (PKA) event, and propagate for approximately 10 picoseconds until all atoms ballistically come to rest. During the cascade, significant quantities of replacement and displacement of atoms, and recombination and agglomeration of defects happen. The ballistic mixing process dominates at low temperature and is a temperature-independent quantity, which can be described by the Einstein relation

where R is the nearest-neighbor distance, and η is on an order of 10²–10³ from experimental measurements[25]. K is the atomic displacement rate in dpa/s. The displacement rate is considered to be near a constant on the scale of micrometers (even centimeters) under neutron irradiation, while it is a depth-dependent quantity in accelerator-based ion irradiation. The displacement rate in ion irradiation can be estimated by SRIM/IM3D with the "QKP" model[17,18].

The second item D_rad on the RHS of Eq. (1) is related to the long-range transport of atoms. After irradiation, the concentrations of point defects including vacancies and interstitials are much higher than those at the thermal equilibrium condition. These point defects, which have high mobility at elevated temperature, show a long-distance diffusion character. Theoretically, all mobile point defects and defect clusters contribute to D_rad:

where $f_{n\text{v}}$ and $f_{n\text{i}}$ are the geometrical correlation factors for the corresponding defect clusters, and n is the size of the vacancy or interstitial clusters. In this work, only the single vacancies and interstitials are simply considered, and their spatial correlations as well as the defect clusters are neglected.

According to the point defect balance equations[1,15]:

where K₀ is the production rate of MSDs, which is generally 1.5–9% of the dose rate for heavy ion irradiation on metal or alloys[26]. K_iv denotes the rate constants for I–V recombination. K_vs, K_is denotes the loss of vacancies and interstitials at sinks, and C_s is the effective sink concentration.

where r_iv is the radii of vacancy-interstitial recombination, r_vs and r_is are annihilation radii for vacancy and interstitial respectively at sinks, and $\Omega =a_0^3/4$ is the atomic volume of the target materials. The equivalent effective sink concentration C_s is assumed to characterize the defect sinks in materials[5]. Then, the sink strength[7] can be written as

It is recognized that different types of sinks in materials may exhibit preferential attraction for one defect type over another[1]. In this work, we did not make a distinction between vacancies and interstitials, and assumed that C_s,V = C_s,I = C_s, where C_s,V and C_s,I are the effective sink concentrations for vacancies and interstitials, respectively[5].

When a steady state is achieved, $C_{\text{v}}{D}_{\text{v}}=C_{\text{i}}{D}_{\text{i}}$ is assumed for a good approximation. The RED coefficient $D_{\text{rad}}$ for steady-state conditions is calculated to be

where D_v is the diffusion coefficient of vacancy:

where $a_0$ is the lattice constant and ${\nu }_0$ is the attempt frequency for vacancy exchange[1,27], k_B is the Boltzmann’s constant. For an fcc lattice, each atom has 12 nearest neighbors (z =12), and the jump distance is $A{a}_0=\frac{1}{\sqrt{2}}{a}_0$. Then, we have $\alpha =1/6z{A}^2$ = 1 [1].

The parameters discussed above are listed in Table 1, in which the values from the available previous studies and used in this work are also collected.

The third item D_th on the RHS of Eq. (1) is related to the thermal diffusion, which can be expressed by a relation in the form of the Arrhenius equation.

where D₀ is the pre-factor for the diffusion, Q is the activation energy, and k_B is the Boltzmann constant.

3.1 Primary radiation damage calculations

As mentioned above, accurate estimations of the damage rates under ion irradiation is one of the key concerns in calculating the diffusion coefficients. In this subsection, the high-efficiency MC code IM3D was adopted to predict the depth profiles (1D) of primary radiation damages for self-ion irradiation in bulk nickel, as illustrated in Fig. 1. It is known that SRIM/TRIM is the most widely used tool for calculating the spatial distribution of primary radiation damages, including vacancies, interstitials, injected ions, etc., in energetic ion–material interactions. Therefore, we first performed benchmark calculations in the same ion irradiation systems.

The depth profiles of vacancies and implanted ions for self-ion irradiation in nickel calculated by IM3D are shown in Fig. 2 (a) and (b), respectively. Five independent examples with ion energies of 1, 3, 5, 10, 15 MeV and normal incidence are plotted and compared with the predictions from SRIM (see the dot points in black). Good agreements between IM3D-QKP and SRIM-QKP are found in predictions of the depth distributions on vacancies and implanted ions. It can be seen that with an increase in ion energy, the peaks of vacancy concentration (or dose rate) and implanted ion concentrations (or implanted ion rate) move backward correspondingly. Moreover, their peak concentrations gradually shrink with increasing ion energy.

Fig. 2Full-screen View

(Color online) Predictions of primary radiation damages for nickel self-ion irradiation. Comparison of the PRDs including depth profiles of vacancies (a) and implanted ions (b) calculated by IM3D and SRIM, respectively. (c) Comparison of ion projected ranges predicted by IM3D and SRIM with the QKP method. (d) Depth profiles of damage and implanted ions for nickel self-ion irradiation with energy of 10 MeV at various irradiation doses of 10¹⁴, 10¹⁵, 10¹⁶, 10¹⁷ and 5×10¹⁷ #/cm², corresponding to peak damage doses of approximately 0.1, 1, 10, 100 and 500 dpa, respectively. The ion concentration profiles are plotted on the right y-axis for clarity.

The averaged ion projected ranges (Rps) with ion energies up to 15 MeV are also drawn to get a better comparison between IM3D and SRIM. As shown in Fig. 2(c), the ion projected ranges predicted by IM3D (solid black line) are in good conformity with the results of ion stopping and range tables calculated by the SRIM code (red dash-dotted line), especially for the situations of ions with lower energies. The discrepancies between them (absolute errors) are evaluated by

With increasing ion energy, the projection ranges calculated by IM3D go slightly deeper when compared with those predicted by SRIM. The absolute errors are less than approximately 0.13 μm for all ion energies considered (up to 15 MeV) in this work, indicating a good correspondence between the calculations of IM3D and SRIM with the QKP method.

The depth profiles of dose and atomic concentrations [32] for nickel self-irradiation with energy of 10 MeV with different radiation doses are plotted in Fig. 2(d). The damage profiles were predicted by the IM3D-QKP method with various total ion fluencies in a range of 10¹⁴ to 5×10¹⁷ #/cm², corresponding to the peak doses from ~0.1 to 500 dpa. The damage peaks appear at a depth of 2.38 μm, which is about 0.25 μm shallower than the peak position of injected ion concentration (2.63 μm). Such results are also in good agreement with those predicted by SRIM. The longitudinal straggling range for ions goes deeper with increasing damage dose. For the damage dose considered here, the damaged region at levels exceeding 10^-3 dpa is found at a depth less than 3.10 μm for a peak damage dose of 0.1 dpa, whereas it increases to 3.33 μm for a peak damage dose of 500 dpa.

Generally, the concentration of implanted ions in nickel self-ion irradiation is approximately four orders of magnitude smaller than that of vacancies by a rough comparison between Fig. 2 (a) and (b). The implanted ions can be ignored at lower damage doses; for example, the peak concentration of implanted ions is only ~1.5×10^–3 % when the peak damage is ~0.1 dpa. In contrast, it reaches nearly 1.5% when the peak damage dose is up to ~100 dpa (with total ion fluencies of 10¹⁷ #/cm²), which can no longer be readily ignored.

3.2 Radiation-enhanced self-diffusion in nickel

As discussed in subsection 2.2, three main aspects contribute to the diffusion under irradiation: the long-range diffusion D_rad, the cascade mixing D_mix, and the thermal-activated diffusion D_th. The former two items are related to the RED. Various parameters should be carefully considered to get relatively accurate RED behavior: (1) the depth-dependent damage rate under ion irradiation; (2) the effective sink concentration or sink strength for the target materials; and (3) the thermal-related diffusion parameters including the activation energy and the pre-factor.

The displacement rate for ion irradiation can be estimated from the primary radiation damage calculated by MC code SRIM/IM3D, as discussed in subsection 3.1. The surface, dislocation, and grain boundaries in materials, as well as the radiation-induced voids, dislocation loops/networks et al., all behave as sinks for the point defects[1]. Here, the effective sink concentration (C_s) in nickel is estimated from the contributions of dislocations and voids[33]:

where Λ is the dislocation density, C_h is the atomic concentration of voids, r_h is the void radius, and Z is the geometrical factor for defect loss at dislocations as shown in Table 1. This definition may provide a lower-bound estimation of the effective sink concentration in materials. The calculated C_s as a function of temperature are displayed in Fig. 3(a), in which the dislocations and void concentrations are mainly considered. To do this, the material parameters including the dislocation densities and void concentrations from experimental measurements via transmission electron microscopy (TEM) are collected and analyzed, as listed in Table S2[34–36]. These data include the measurements from irradiated bulk nickel under self-ion and neutron irradiations (see Note 1 in the Supplementary Materials)[34–36].

Fig. 3Full-screen View

(Color online) Estimation of effective sink concentration in irradiated nickel as functions of reverse temperature (a) and irradiation dose (b). The dashed lines are plotted for a visual guide and to show the tendency of the relevant quantities. ^[a]The displacement rate 3×10^-3 for nickel self-irradiation is estimated with E_d = 40 eV for nickel[34]. ^[b]The displacement rates 7×10^-2 and 7×10^-4 dpa/s are estimated with E_d = 25 eV[35], corresponding to values of 4.35×10^-2 and 4.35×10^-4 respectively when uses E_d = 40 eV, from rough estimations by the Norgett–Robinson–Torrens (NRT) model[37].

Fig. 3(a) shows the tendency of effective sink concentrations, the pink dashed lines are obtained by numerical fitting on the nickel self-ion irradiation data shown in the figure. The blank dashed lines depict the C_s for lower and higher displacement rates. A general tendency can be found that C_s increases progressively when the dose rate increases. The C_s exhibits relatively lower values at higher temperatures with 10^-7 order of magnitude, and it gradually increases and then decreases slightly when the temperature decreases. The peak C_s shifts to the left (higher temperatures) and higher concentrations with gradually increasing dose rates, exhibiting a similar tendency to that of swelling properties as a function of temperature in structural materials. Details are provided in Note 2 in the Supplementary Materials. The C_s shown in Fig. 3(b) exhibits an effect of defect saturation with increasing dose. In addition, we did further comparison of effective sink concentrations under different irradiation conditions, in which the influence of helium irradiation on C_s is mainly discussed. The detailed plots and discussion are shown in Note 2 in the Supplementary Materials.

The third item is related to the thermal diffusivity of nickel self-diffusion, which can be written in Arrhenius form as shown in Eq. (10). Here, by reviewing the previous theoretical and experimental studies, the pre-exponential factors and activation energies (parameters in Eq. (10)) for nickel self-diffusion are collected and listed in Table 2. The corresponding Arrhenius plots are shown in Fig. 4 (a). It can be seen that the thermal coefficients vary greatly among different methods. By analyzing these data, the experimental values from the work of Maier et al. were adopted to calculate the diffusion coefficients for nickel under ion irradiation[38].

Fig. 4Full-screen View

(Color online) (a) Thermal diffusion coefficients for nickel as a function of reverse temperature available from previous references (details are listed in Table 2). (b) Diffusion coefficients for nickel self-ion irradiation at steady state as a function of reverse temperature with damage rates of 10^-2, 10^-3 and 10^-4 dpa/s. The opened circles and diamonds represent the results of Müller et al.’s experimental investigations on the RED under the equivalent dose rates of 1.2×10^-2 and 1.2×10^-4 dpa/s, respectively[5].

Based on the above discussion, the diffusion coefficients of nickel under self-ion irradiation as a function of temperature and displacement rates are plotted in Fig. 4(b). Here, displacement rates in a range of 10^-4–10^-2 dpa/s are selected, which are commonly used in ion irradiation experiments on nickel-based alloys. Other parameters used for the prediction of the diffusion coefficients are listed in Table 1. In addition, a temperature-dependent C_s was carefully considered (see Fig. 3(a)). Moderate effective sink concentrations C_s indicated by the dashed pink line were assumed for the damage rate of 10^-3 dpa/s, while the scaled C_s according to the general tendency of C_s shown in Fig. 3(a) were used for other damage rates. The estimated diffusion coefficients are in good agreement with the experimental measurements of tracer diffusion in bulk nickel, as the circles and diamonds show (representing the diffusion under the equivalent dose rates of 1.2×10^-4 and 1.2×10^-2 dpa/s, respectively; corresponding data are provided in Table S3 in the Supplementary Materials)[5]. It can be seen that the influence of radiation on atomic diffusions is promoted under higher damage rate.

As shown in Fig. 4(b), the contributions from thermal diffusion (pink dash-dotted line), radiation-enhanced (red dotted line) and ballistic mixing (blue dashed line) are also clearly shown. Let us take the damage rate of K = 10^-3 dpa/s as an example to illustrate the diffusion mechanisms. At very high temperature (>1000 K), the thermal diffusion dominates due to the higher mobility of point defects. The effect of radiation on atomic diffusion gradually increases with decreasing temperature (from 1000 K to 600 K). Previous studies showed that the diffusion coefficients in some parts of this temperature range exhibit a temperature-independent phenomenon due to the higher sink annihilation rate on defects[1,5,42]. However, by considering the temperature dependence of C_s, it can be speculated that diffusion mechanisms are related to both defect annihilation in sinks and defect recombination. In other words, defect recombination is also important due to the relatively lower C_s at higher temperatures estimated from experimental investigations. At lower temperatures (<650 K), the ballistic mixing diffusion D_mix is proportional to the damage rate, and it is estimated to be 1.1×10^-21 m²/s for K = 10^-3 dpa/s. The defect recombination is promoted due to the lower mobility of point defects, whereas the influence of defect sinks is less important. Besides, defect recombination also dominates for higher-dose-rate situations due to the higher concentration of defects.

The depth dependence of diffusion coefficients (D_rad) for nickel under self-ion irradiation conditions was also estimated. A specific temperature range of 900–950 K is chosen, which is around the operating temperature of MSRs (650 ℃, or 923 K)[21,22]. We considered the ion irradiation with various energies (5, 10, and 15 MeV) under ion fluxes of 10¹² and 10¹³ #/cm²/s, which correspond to peak damage rates of ~10^-3 and ~10^-2 dpa/s, respectively, due to the similarity of depth-dependent diffusion coefficients between different ion energies. The depth-dependent diffusion coefficients for 10 and 15 MeV are plotted in Note 3 in the Supplementary Materials. We took the 10 MeV nickel ion irradiation with peak damage rates of 10^-3 dpa/s (Fig. 5) as an example to illustrate the temperature and depth dependence on diffusion coefficients. The peak damage rate for 10 MeV nickel self-ion irradiation was estimated to be 0.9×10^-3 (rounding up to 1×10^-3) dpa/s at a depth of ~2.4 μm. The peak D_rad was estimated to be ~ 0.6×10^-19 (blue dotted line) and ~0.9×10^-19 (pink dotted line) m²/s at 900 and 950 K, respectively, while the total diffusion coefficients D* were calculated to be ~0.7×10^-19 (bold blue line) and ~1.4×10^-19 (bold pink line) m²/s for the temperatures of 900 and 950 K, respectively. It can be seen that both D_rad and D_th increase with increasing temperature. In particular, the ratio of thermal-activated diffusion is significantly increased for total diffusion coefficients at 950 K. It also can be seen that the temperature-independent component D_mix has little influence on the total diffusion coefficients D* at high temperatures. The calculated diffusion coefficients D* were also compared with the previously available experimental investigations on the nickel tracer diffusion under similar self-ion irradiation conditions, showing that they are of the same order of magnitude. According to Eq. (8), D_rad is approximately proportional to the root value of damage rate (K^0.5 or K₀^0.5). The diffusion coefficients are also in coincidence with the results of Doyle et al.’s calculations. They studied the diffusion coefficients of nickel under ion irradiation with a simplified model, in which the depth-dependent RED coefficients are proportional to (F_i/F_max)^E. The exponential scaling factor is in a range of 0.5–1.0, denoting the recombination-dominant and sink-dominant defect evolution mechanisms[14,20]. However, when using a temperature-dependent effective sink concentration, it is shown that both sink-domination and defect recombination are involved for the defect evolutions at the temperature of interest.

Fig. 5Full-screen View

(Color online) Depth dependence of damage rate (left y-axis) and the diffusion coefficients (right y-axis) for nickel under self-ion irradiation at 900 and 950 K. The parameters are as follows: ion energy E = 10 MeV, peak damage rate K_max = ~1.0 ×10^-3 dpa/s with ion flux of 10¹² #/cm²/s.

3.3 Radiation-enhanced diffusion of as-implanted ion profiles

The as-implanted ions in heavy ion irradiation exert a profound effect on the microstructural evolution of target materials. The typical artefact phenomenon is related to the depth-dependent precipitation in materials and the suppression of void swelling when the ions come to rest. Thus, a detailed investigation on the time evolution of the implanted ion profiles in heavy ion irradiation is of significant importance to understand the chemical change, and even the macroscopic properties such as void swelling. In this subsection, the diffusion of implanted ion profiles under irradiation is studied to identify their long-term evolution. Herein, the depth profiles of implanted ions (tracers implanted assumed) in nickel are used as a start to see the RED in nickel self-ion irradiation. The concepts for this method are in successful description of the evolution for depth-dependent atomic profiles under ion irradiation conditions by considering the RED and surface sputtering[13,14,43,44]. Details are provided in Note 3 in the Supplementary Materials.

It is recognized that a higher diffusion coefficient can give rise to a larger broadening of the atomic profile under similar irradiation conditions. The diffusion coefficients are both temperature- and damage-rate-dependent quantities. Thus, relatively higher total damages under fast irradiation (higher damage rates) were selected to see the influence of RED on the broadening of atomic profiles. As shown in Fig. 6(a), we first studied the influence of damage rate on the broadening of atomic profiles under ion irradiation with ion energies of 5, 10, 15 MeV. Two damage rates with peak values of roughly 10^-3 and 10^-2 dpa/s, peak total damage of ~250 dpa and a temperature of 900 K were selected. The depth-dependent diffusion coefficients shown in Fig. S2 were adopted during the calculation. It shows that the total diffusion coefficients for ion energies of 5, 10, 15 MeV at the peak damage position were all calculated to be roughly 2×10^-19 m²/s for the peak damage rate of ~10^-2 dpa/s (see Fig. 4b and Fig. S2). The irradiation time was estimated to be 7 h to get the total damage dose up to 250 dpa. Broadening of the atomic profiles was found for the ion energies of 5, 10, 15 MeV, and the peak concentrations were reduced by approximately 3–5% when compared with those that did not consider the RED effect. Meanwhile, for the situations under the damage rate of 10^–3 dpa/s, the total diffusion coefficients at 900 K were estimated to be approximately 0.7×10^-19 m²/s for ion energies of 5, 10, and 15 MeV. Significant broadening of the atomic profiles was found with peak concentrations reduced by approximately 10–13% during ~70 h irradiation (250 dpa), indicating the consequence of long-term irradiation.

Fig. 6Full-screen View

(Color online) (a) Influences of damage rate on the broadening of implanted ion profiles during ion irradiations. Parameters are as follows: ion energies of 5, 10, and 15 MeV, temperature of 900 K, and peak damage of approximately 250 dpa were adopted. (b) Influences of total damage (or irradiation time) on the diffused profiles of implanted ions under 10 MeV nickel self-ion irradiation. A simplified condition with constant diffusion coefficient is adopted for comparison. Parameters are as follows: peak damage rate K_max ~ 1.0×10^-3 dpa/s, ion flux of 10¹² #/cm²/s, and peak damage rates of 250 and 500 dpa or total ion fluencies of 2.5×10²¹ and 5×10²¹ #/cm², respectively.

Then, the time evolution (or dpa dependence) of atomic profiles for nickel self-ion irradiation with energies of 10–15 MeV was selected to avoid the surface effect on long-term irradiation, in which the same depth-dependent diffusion coefficients shown in Fig. 5 and Fig. S2 were adopted. The evolutions of atomic profiles (regions for peak concentration) under irradiation with 10 and 15 MeV are shown in Fig. 6(b) and Fig. S4, respectively. As seen from the results of 10 MeV irradiation shown in Fig. 6(b), significant broadening is obtained with the total damage reaching 500 dpa, and the peak concentration reduced to ~83% when compared with that the RED effect was not considered. Moreover, a much-simplified model in which we used a constant coefficient in the ion projected range was also considered for comparison. The broadened profiles are plotted in Fig. 6(b) (see the blue dotted lines). It can be seen that using a flat peak diffusion coefficient would not exert a larger difference in broadening of the atomic profiles with the total dose considered in this work.

The two case studies intuitively show the time evolution of the depth-dependent atomic profiles. Significant broadening is found with increasing total damage (or dpa). It is recognized that atomic diffusion in ion irradiation is generally along the ion-bombarding direction (1D diffusion). Thus, a broad beam situation is assumed in which the beam spot size is much larger than the area of interest in the target material, in which the lateral straggling of ions (diffusion in 3D) can be ignored [13,45]. Besides, the sputtering effect from the surface of the target is also neglected during the calculations, as we considered the heavy ion irradiation with relatively higher energies[43], and the depth region of the peak atomic concentration was focused on for the two case studies.