Introduction

Void swelling at intermediate temperatures, occurring at approximately 0.3−0.6 of the melting temperature, is widely observed in structural materials exposed to high levels of particle radiation1,2,3,4. Such a phenomenon leads to significant effects that result in unacceptable volumetric changes, degraded mechanical performance, and a limited lifetime of structural components in nuclear reactors4,5,6,7 and other extreme environments, such as space vehicles8. Furthermore, the underlying cause—defect interaction in materials—is a critical focus in numerous significant fields, such as high entropy alloys9, energy storage materials10, and perovskites11. One critical observation is that swelling rates are generally much higher in fcc materials as compared to bcc materials5,6,12,13, with a few exceptions14. Because of the critical significance of such differences, the underlying reason has received significant attention and generated extensive discussions in recent decades. However, a conclusive explanation is yet to be established15, although multiple mechanisms have been proposed by previous studies.

Among these mechanisms, defect accumulation and the effect of microstructures in irradiated bcc and fcc metals has garnered the most attention16. Previous studies17,18 indicate that to create the same level of defect density, Fe requires a dose that is two to three orders of magnitude higher than Cu. Additionally, the difference in defect clustering during the thermal spike phase between fcc and bcc metals19 has been suggested as a contributing factor, but a fundamental understanding of how these factors influence the swelling behavior has not been established. Some other effects, like the glissile self-interstitial atom (SIA) cluster removal mechanism13,20,21, which affects the survival fraction of defects from displacement cascades22, are also considered. Nevertheless, none of the models are capable of reproducing the experimental values of swelling rates at different temperatures in irradiated materials. Moreover, microstructural features, such as dislocations, act as the sinks of mobile point defects4, and the bias model23,24 has been established to quantify the void swelling. Specifically, dislocation bias has been extensively discussed4,23,24,25,26, but the difference between bias factors obtained in fcc and bcc materials cannot fully explain their swelling rate difference. Nevertheless, it is clear that the effect of microstructural features is the critical factor to investigate these phenomena, and the model to study swelling behaviors must consider the types and densities of the sinks formed during the irradiations. However, due to the complexity of irradiation conditions, identifying the primary underlying cause of the observed difference in swelling behavior between fcc and bcc materials presents a significant challenge.

Among all the major types of microstructural features in fcc Cu, this study identifies SFTs (mostly between 1.5 and 3.5 nm27,28,29) as a significant cause of the swelling rate differences between fcc and bcc materials, based on state-of-the-art mesoscale and atomistic kinetic Monte Carlo (AKMC) simulations30,31. SFTs have been long considered neutral sinks for vacancy and SIA point defect absorption due to the low elastic strain associated with the stair-rod dislocations that comprise the SFT periphery32. We observe both perfect and non-perfect SFTs27 to be non-neutral sinks that prefer to absorb mobile SIAs over mobile vacancies in fcc copper. Such a phenomenon is due to the underlying mechanism of the SFT-point defect interaction, showing strong absorption of mobile SIAs from the faces and vertices of the SFT, and weak absorption of mobile vacancies from the edges of the SFT. This mechanism leads to higher bias values of SFTs compared to other types of sinks in irradiated copper26, including straight dislocations (both edge and screw types), 1/3<111> interstitial loops, and voids. The high bias values of SFTs, coupled with high number densities (higher than 1022 m−3 based on experimental observations under 600 K12,28,29,33,34,35,36), collectively result in the high sink strength of SFTs. We further analyze the effects of each type of sink on void swelling phenomena based on the obtained sink strengths and predicted swelling rates and compare computational results with a large number of neutron- or ion-irradiation experiments, with different irradiation conditions and material microstructures12,29,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49. We demonstrate that SFTs are the primary cause of high swelling rates in irradiated fcc copper and play a decisive role in creating the observed differences in swelling rates between general fcc and bcc structural materials.

These findings challenge traditional beliefs that SFTs do not act as biased sinks in irradiated fcc materials. Moreover, the insights gained from this study represent a significant step towards understanding the roles of different sinks on swelling behavior, which in turn can inform the development of materials with enhanced swelling resistance. For example, one possible approach to achieve improved swelling tolerance could involve reducing the density of small SFTs (≤3.5 nm in size), e.g. through tuning stacking fault energies or compositional complexity. Additionally, increasing the initial dislocation density by cold working will also influence the swelling behavior of fcc materials. These strategies could potentially lead to the development of more robust fcc structural materials capable of withstanding high-dose irradiations and being more resistant to irradiation-induced swelling.

Results and discussion

Mechanisms of SFT-point defect interactions

The mechanism of SFT-point defect interaction was quantified from statistical analysis of the diffusion behavior of mobile point defects near an SFT. Contrasting behavior occurred for SIAs and vacancies due to the varying stress/strain field in the matrix surrounding the SFT. We find that mobile SIAs are absorbed at the faces and vertices of SFTs, corresponding to regions with tensile strains; mobile single vacancies are absorbed at the edges of SFTs, corresponding to regions with compressive strains. The volumetric strain map of a 2 nm SFT is shown in Fig. 1a, b, where regions under tension appear in red and are normal to the faces and around vertices of the SFT, and regions under compression appear in blue and are normal to the SFT edges. Such a pattern persists over long distances (greater than 10 nm) and is indicated in light yellow and light blue. Please note that, as the strain fields are identical on each face of the SFT, only the top face has been chosen to clearly illustrate the 3-D properties. The schematic representation of this is depicted in Fig. 1b (from another angle of Fig. 1a).

Fig. 1: Top view of the SFT strain field and diffusion tendency of SIAs.
figure 1

a Volumetric strain field of a 2 nm SFT (top view). b Another perspective of the SFT strain field. The scheme of the SFT for both (a) and (b) are respectively shown in the top right corner inset figures. Diffusion tendency of an SIA (c) and a vacancy (d) around a perfect 2 nm SFT based on KMC simulations are provided for comparison with (a) from the same orientation. The color represents the expected diffusion direction, and the size of the spheres represents the expected probability of the diffusion path. Complete 3-D rotational movies are provided in the Supplementary Information.

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To clearly represent the varying SFT-point defect interaction, we depict the diffusion tendency of point defects around a 2 nm SFT in Fig. 1c, d for SIAs and vacancies, respectively. Diffusion tendency was obtained based on KMC simulations at 100 K, showing the expected diffusion direction and its probability of a mobile point defect at every atomic position near the SFT. The simulation at 100 K captures the key diffusion direction produced during irradiation at higher temperatures. Here, the expected diffusion direction is represented by the vector sum of the diffusive steps taken from a given atomic position, and the probability corresponds to the length of this resultant vector relative to the combined lengths of all diffusive step vectors. To enhance the clarity of the figure, such vectors are represented using spheres. In Fig. 1c, the attractive (tensile) regions (labeled in yellow and red) show the diffusion direction of SIAs toward the SFT, and the repulsive regions are labeled in green and blue. The size of the sphere represents the probability of an SIA diffusing along that pathway, showing a consistent tendency but a weaker effect over long ranges. The magnitude of the interaction with the SFT is much weaker for vacancies than that for SIAs. Figure 1d shows that the SFT-vacancy interaction is very strong only at very short ranges (1−2 nm). Attractive pathways are found within 1 nm next to the edges of SFTs; conversely, repulsive pathways are predominant at many geometric locations for the SFT-vacancy interaction. This contrasting interaction behavior, along with the high SFT number densities observed in irradiated Cu that shortens the average distance between mobile point defects and SFTs, leads to a high effectiveness of SFTs as a sink for mobile SIAs in copper.

This observed mechanism can explain the results of several previous experimental studies and theoretical calculations. SFTs were originally suggested to collapse by absorbing a sufficient number of SIAs32,50 in irradiated fcc materials. Later, the annihilation of SFTs as a result of irradiation was confirmed by in-situ observations in fcc materials51,52, and the growth/shrinkage mechanisms were attributed to the absorption of mobile point defects27,50. Here, we observed a high preference for SFTs to absorb SIAs over vacancies, which will lead to significant shrinkage of SFTs. This observation further clarifies why SFTs rarely grow beyond 3.5 nm in size as they form in cascades. Additionally, a possible growth mechanism involving migration and coalescence of small SFTs (15-vacancy SFT) was proposed by Martinez et al.53 based on molecular dynamic (MD) simulations between 600 K and 1000 K.

Experimental SFT lattice strain measurement

SFTs appear as bright triangular shapes when imaged along a \(\left\langle 110\right\rangle\) zone axis by low-angle annular dark-field scanning transmission electron microscopy (LAADF-STEM) (Fig. 2a), indicating the possible presence of strain54. Local lattice strain in the plane perpendicular to the incident beam direction can be determined by tracking changes in Bragg disk positions in convergent-beam electron diffraction (CBED) patterns (Fig. 2c) as a function of real space STEM probe position. Two-dimensional maps of strain can therefore be produced by measuring Bragg disk shifts in a rectangular array of positions across the sample. Here, we used a novel 4D-STEM method with exit-wave power-cepstrum (EWPC) analysis55, which is robust to small sample mis-tilts and reciprocal space intensity variations such as those caused by dynamical diffraction, to extract strain information from such an array of CBED patterns (Fig. 2c). As Fig. 2b illustrates, experimental EWPC-STEM data of dual-ion-irradiated copper reveals a distribution of tensile areal strain (approximately \({\varepsilon }_{{xx}}\) + \({\varepsilon }_{{yy}}\)) inside and surrounding an SFT. For comparison with the results of the simulation in Fig. 1, a rotation was applied to match the experimental orientation and averaging the strain within the thickness that encompassed the entire tetrahedral SFT in the \(\left[1\bar{1}0\right]\) direction (Fig. 2f). The resulting areal strain distribution (Fig. 2e) qualitatively matches the experimental strain map (Fig. 2b) fairly well, given the additional experimental noise and sub-nm in-plane spatial resolution, showing higher tensile strain in the upper left SFT vertex than the two others, and reduced stress immediately outside of this vertex. Additionally, the dependence of SFT strain field on size is qualitatively similar, although the predicted spatial extent of the strain field will be lessened for smaller SFTs.

Fig. 2: Comparison of experimental and simulated SFT areal strain.
figure 2

a Atomic-resolution LAADF−STEM image of an SFT in dual-ion-irradiated Cu at 440 °C to 30 dpa and 48 appm He. b Areal strain perpendicular to the beam direction measured by 4D-STEM EWPC method. c Average CBED pattern in the region of the SFT. d Crystal structure of the SFT used for simulation calculation. e Areal strain averaged over the SFT strain field in the \(\left[1\bar{1}0\right]\) direction. f Schematic diagram of the areal strain averaging method used for the simulated results. Faces, edges, and vertices of the SFT are labeled in (a), (b), and (e) for comparison with Fig. 1. Notice that the Edge (vertical) refers to an edge that is vertical to the view plane, and the labels for vertices in (b) and (e) are hidden for a clearer view.

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One difference between the experimental and simulated results is that the two other vertices also exhibit a tensile strain in the experimental results, while they do not in the simulated results (Fig. 2e), which may be due to the finite probe size or differences in averaging mechanisms. Quantitatively, the maximum experimental and simulated tensile areal strains are on the same order of magnitude as well, at approximately 3.5% and 1.5%, respectively. Our simulation did not account for the influence of helium (which can be generated in a real nuclear reactor environment from nuclear transmutation reactions) on the SFT strain field, however, which could potentially increase the strain induced through pressure effects. The differences in strain distribution observed in the experimental and simulated results for the two vertices on the right (Fig. 2b, e) and the overall maximum strain differences may therefore also be related to implantation of helium atoms in the experimental sample. Despite these small differences, EWPC-STEM strain mapping lends support to the findings of the simulation, indicating that SFT’s have a relatively large tensile strain and therefore an affinity for mobile SIAs.

SFT bias and comparison with other sinks

The mechanisms of SFT-point defect interactions lead to the preferential absorption of mobile SIAs over vacancies, which is termed bias. We quantified the bias values of SFTs, dislocation loops, and straight dislocations using the same AKMC approach. Both bias equations and the AKMC framework are elaborated in the Methods Section. The results are shown in Fig. 3 and demonstrate that the bias for both perfect and imperfect SFTs is the highest among all major sinks that exist in irradiated copper.

Fig. 3: Bias of different types of sinks based on theoretical calculations.
figure 3

a SFT bias with different sizes (in red bands) between 573 and 673 K. Non-perfect SFT (32 vacancies, ~2 nm) bias is also included and labeled in a purple band. 1/3<111> interstitial loop bias (10 nm size) is labeled in a blue band. b. Dislocation bias from this study (473–773 K) and the comparison with previous works, mostly between 500 K and 800 K. Edge dislocation bias is labeled with a yellow band, and screw dislocation bias is labeled with a green band. Dislocation bias values calculated by previous works are labeled in filled markers (edge dislocations) and open markers (screw dislocations).

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Since SFT sizes observed in neutron-irradiated copper are mostly between 1.5 nm and 3.5 nm based on experimental observations27,28,29, the bias values of 1.5 nm, 2 nm, and 3.5 nm perfect SFTs (with 21, 36, and 105 vacancies) were calculated and are shown in Fig. 3a. Within the experimentally observed SFT number densities in the void swelling temperature regime (between 1018 m−3 and 1023 m−3,12,28,29,42,56,57), the perfect SFT bias is found between 0.54 and 0.75. The bias decreases with rising temperatures from 573 K to 673 K and decreases with increasing SFT sizes from 1.5 nm to 3.5 nm. Such size dependence is also supported by a previous binding energy calculation58. For non-perfect SFTs, we select an intermediate size of approximately 2 nm (with 32 vacancies) for our calculation. The bias values of non-perfect SFTs are found to be between 0.36 and 0.38 at the same temperature and density range as the perfect SFTs, shown in Fig. 3a. Our results reveal a significantly lower bias of non-perfect SFTs compared to perfect SFTs at the same size, temperature, and number density. This phenomenon can be attributed to the inability of non-perfect SFTs to form four vertices as perfect SFTs do. Since this study has demonstrated that vertices play a critical role in absorbing mobile SIAs and repulsing mobile vacancies, the absorption region of mobile SIAs is thus much smaller in non-perfect SFTs than in perfect ones.

The bias of a 1/3<111 > {111} interstitial loop was calculated to be around 0.1 between 573 K and 673 K as shown in Fig. 3a. Since the typical diameter of loops is larger than 5 nm and increases up to ~30 nm with increasing irradiation doses in experiments29,33,59,60, a 10 nm loop was employed in this study. Similarly, the dislocation bias values of both edge and screw types were found to be much lower than those of SFTs, labeled in yellow and green bands in Fig. 3b. Edge dislocation bias is shown between 0.08 and 0.1 with the commonly reported dislocation density between 1012 m−2 and 1013 m−2 in experiments12,29,34,35,36,41,42,43,44,61. Plus, it is not sensitive to the temperature change from 473 K to 773 K, during which the swelling phenomenon occurs. The obtained bias for edge dislocations is in good agreement with previous works, including atomistic simulations by Chang et al.62, elasticity theory calculations by Skinner et al.63 and Fisher et al.64, and the production bias model by Golubov et al.59. Alternatively, Wolfer20, Seif et al.65, and Sivak et al.66 reported relatively high dislocation bias values around 0.3. However, the highest edge dislocation bias listed above is still lower than that of SFTs. In comparison, screw dislocation bias exhibits much lower bias values, ranging from −0.04 to 0.02 as the temperature is raised from 473 K to 773 K. The negative values were also reported by Sivak et al.66, suggesting that screw dislocations in fcc copper are less effective at absorbing mobile SIAs compared to edge dislocations. Detailed bias values from 473 K to 773 K for all major sinks are provided in Supplementary Fig. 7 to 12.

Swelling rates and effects of major sinks

The swelling rate in irradiated copper is determined by calculating the sink strength of all major sinks, as elaborated in the Methods Section. The obtained results are compared with experimentally reported swelling rates under irradiations12,29,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49, as shown in Fig. 4, where they are plotted as a function of irradiation temperature. These experimental results cover a wide range of conditions, including irradiation doses from 0.001 dpa to 150 dpa, and temperatures between 473 K and 802 K. We examine the contributions of each type of sink to the swelling phenomena and subsequently identify three primary scenarios. The violet curve represents the SFT-dominant scenario (i), the blue curve stands for the loop-dominant scenario (ii), and the green curve corresponds to the dislocation-dominant scenario (iii). The predicted swelling rates in these scenarios are calculated based on the sink strength of SFTs, loops, dislocations, and voids. The equations for sink strengths and swelling rates are provided in the Methods Section. Each scenario illustrated in Fig. 4 is derived based on the fitted sink density of experimental observations12,29,33,34,35,36,42,43,44,47,57,61. These fitted sink densities are represented by dotted lines in Fig. 5a–c for SFTs, loops, and dislocations, respectively. The fitted void density and size are provided in Supplementary Figs. 5 and 6.

Fig. 4: Comparison between predicted swelling rates and experimental observations.
figure 4

Three scenarios are provided. SFT-dominant scenario (i) is labeled by the violet curve, loop-dominant scenario (ii) is labeled by the blue curve, and dislocation-dominant scenario (iii) is labeled by the green curve. Edge dislocations and screw dislocations are considered 50%–50% in scenario (iii). For experimentally reported data, colors from black to white represent different doses from <0.01 dpa to 150 dpa (shown in log scale). Darker shades denote lower doses, which typically correspond to lower irradiation temperatures.

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Fig. 5: Calculated sink strengths of SFTs, dislocation loops, and straight dislocations.
figure 5

Results in this figure are obtained based on experimentally observed densities and temperatures12,29,33,34,35,36,42,43,44,47,49,57,61 using the equation provided in the Methods Section for a SFTs, b dislocation loops, and c dislocations. Dislocation loop type is assumed to be 1/3<111>{111} interstitial and straight dislocation in (c) is 50% edge type and 50% screw type (both dissociates to two Shockley 1/6<112> partials). The shaded band shows the root mean square error (RMSE) of the fitting. Note that the data of “non-SFT clusters” provided by Singh and Zinkle57 is assumed as representing 50% interstitial loops for estimations in (b). Dislocation density reported in two of the experimental studies by Singh et al.35,42 is not considered for the fitting process.

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In Scenario (i), SFTs serve as the dominant sink, corresponding to the upper limit of swelling rates (~1% dpa-1) in neutron-irradiated copper. This scenario typically occurs under 600 K, at which the high densities of SFTs are observed in experiments (around 1022 m-3 or more)12,28,29,33,34,35,36, as shown in Fig. 4. In this scenario, the sink strengths of SFTs for SIAs are mostly higher than 1013 m-2 (Fig. 5a), which are higher than those of straight dislocations (Fig. 5c). Such high SFT sink strengths are due to the calculated high bias values and the experimentally observed high SFT number densities under 600 K (>1021 m-3). The high number densities reduce the average distance between mobile point defects and SFTs, thereby enabling SFTs to absorb mobile SIAs more efficiently. This remains true even though the impact range and sink strength of a single SFT are smaller than those of a loop or a straight dislocation (details in Supplementary Fig. 1). The high density of SFTs can be attributed to high probability for direct formation within energetic displacement cascades. This is generally considered a result of the low stacking fault energy (SFE) in fcc copper27,35. With increasing temperatures, SFTs will annihilate due to their interaction with mobile SIAs or thermally evaporations50. This leads to the experimentally observed decrease of SFT densities with increasing temperatures57. Additionally, the impact of non-perfect SFTs on the predicted swelling rate is also estimated and the bias values are found to be close to those of perfect SFTs, which are substantially higher than those of other types of sinks.

The swelling rates in scenario (ii) exhibit a close agreement with the average values of experimental observations (from around 0.1 to 0.5% dpa-1)12,29,33,34,35,36,37,38,39,40,41,42,43,44, as demonstrated by the blue curve in Fig. 4. The 1/3 < 111 > {111} interstitial loop is assumed as the governing type of sink in this scenario, the sink strengths of which are normally higher than those of straight dislocations, as demonstrated in Fig. 5b. The presence of loops, compared with scenario (i), reduces the swelling rate under 700 K. For instance, the swelling rates observed by Zinkle et al.29 and Singh et al.33,34,35 under 600 K in Fig. 4 (circled in orange) are lower than the predicted swelling rates that only consider SFTs. Here, the collective effect of the low bias dislocation loops with their number densities (generally higher than 1020 m-3 under 600 K29,42,57) is evident in experimental observations. Notably, the number density of both SFTs and dislocation loops decreases with increasing temperature, leading to a weaker effect of scenarios (i) and (ii) at high temperatures under a void swelling regime.

In scenario (iii), the contribution of straight dislocations on the swelling rates prevails, especially at high temperatures (approximately above 700 K). This dislocation-dominant scenario represents the predicted lower limit of the swelling rates of irradiated copper. The sink strengths of dislocations decrease not as much as those of SFTs or loops with increasing temperatures (as shown in Fig. 4c). These temperature-insensitive sink strengths can be attributed to the stable dislocation densities observed in experiments, mostly ranging between 1012 and 1013 m-2, 12,29,34,35,36,41,42, with only a few experiments reporting higher dislocation densities (over 1013 m-2)43,44,61. Comparatively, the number densities and sink strengths of SFTs and loops decrease significantly with increasing temperatures based on experimental observations29,41,57,61. Thus, the role of dislocations in the swelling phenomena observed in fcc copper is more consistent than that of SFTs and loops. Furthermore, since dislocation bias values alone cannot fully explain the observed differences in swelling rates between bcc and fcc materials, it is possible that SFTs are the primary factor determining these swelling rate differences.

Additionally, at such high temperatures, vacancy evaporations impact swelling phenomena significantly22,67,68, which takes place from the void surface. When the temperature is lower than 548 K, no more than 10% of the vacancies in voids will evaporate; yet this ratio reaches 60% when the temperature is 771 K. The equations for evaporation rate calculations are provided in Supplementary Note 1 and Supplementary Fig. 4. Additional experimental data is summarized in Supplementary Figs. 2 and 3 as a reference.

Methods

Construction of sinks and strain calculations

The construction of the sinks was performed using the LAMMPS molecular dynamics code69 with interatomic potential developed by Mishin et al.70. Perfect SFTs with 1.5 nm, 2.0 nm, and 3.5 nm in size were formed by relaxing a triangular 1/3<111> vacancy loop containing 21, 36, and 105 vacancies, respectively. A non-perfect SFT with a size of ~2 nm (containing 32 vacancies) was generated from a perfect 2 nm SFT (containing 36 vacancies) by absorbing four mobile SIAs. The 1/3<111> interstitial dislocation loop of 10 nm size was set up with 223 inserted SIAs on the <111> plane. The 10 nm void was built up by removing 44115 atoms in a spherical shape. The coordinates were \([1\bar{1}0]\), \([11\bar{2}]\), and \([111]\) for the simulation boxes of sinks above, with periodic boundary conditions in all directions. Both the edge and screw dislocations were 1/2<110>{111} type, which dissociated into two 1/6<112> partial dislocations. The strain fields were calculated using OVITO71 visualization and analysis code.

Atomistic kinetic Monte Carlo (AKMC)

The bias of different types of sinks was calculated using a recently developed AKMC approach30,31, the scheme of which is shown in Fig. 6. The KMC processes of point defect diffusion starts from the spherical surface of the simulation system, where the distance to the center of the biased sink remains the same. The lifetime of a point defect, which is defined as the time required to diffuse from an initial position to a sink, was then calculated based on the KMC simulation. The time of point defect diffusion to the first nearest neighbor depends on the migration energy barriers (MEBs), which are mostly calculated using SEAKMC72,73. Dipole tensor method65,74 was employed to calculate the MEBs far from the sink30. Detailed approaches for the MEB calculation are provided in Supplementary Note 2 and the results are shown from Supplementary Figs. 13–28. The bias based on the lifetime is derived,

$${Bias}=\frac{{Z}_{i}-{Z}_{v}}{{Z}_{v}}=\frac{\frac{{\tau }_{i}^{r}}{{\tau }_{i}}-\frac{{\tau }_{v}^{r}}{{\tau }_{v}}}{\frac{{\tau }_{v}^{r}}{{\tau }_{v}}}=\frac{{{\tau }_{i}^{r}\tau }_{v}}{{{\tau }_{v}^{r}\tau }_{i}}-1,$$
(1)

where \({\tau }_{i}\) and \({\tau }_{v}\) represents the lifetime of SIAs and vacancies, and the superscript \(r\) means random walk. \({Z}_{i}\) and \({Z}_{v}\) stand for capture efficiencies of SIAs and vacancies to the sink22. Plus, the sink strength in the physical system75 is calculated using

$${k}_{\alpha }^{2}={{Z}_{\alpha }k}^{2},$$
(2)

where \({{{{{\rm{\alpha }}}}}}\) is either \({{{{{\rm{i}}}}}}\) or \({{{{{\rm{v}}}}}}\), representing interstitials and vacancies, respectively. The sink strength without the influence of the interaction between the sink and the point defect, \({k}^{2}\), is obtained during AKMC calculations,

$${k}^{2}=\frac{2n}{{d}_{j}^{2}\langle {n}_{j}\rangle },$$
(3)

where n is the dimensionality (n = 3 in this study), \(\langle {n}_{j}\rangle\) is the average number of jumps of PDs, and \({d}_{j}\) is the jump distance76,77. Notably, the sink strength for voids is obtained based on rate theory22,

$${k}^{2}=4\pi \left\langle R\right\rangle \rho ,$$
(4)

where \(\left\langle R\right\rangle\) represents the average void radius and \(\rho\) is the void density. The reason to employ this equation is the significant impact of both void size and density on the sink strength. However, it is challenging to obtain accurate void sink strength considering a wide range of sizes, especially considering the evidence that small voids are biased sinks for mobile point defects4,78,79,80,81,82.

Fig. 6: Scheme of the simulation system.
figure 6

Point defects (green spheres) start their diffusions from the edge of the system and end up being absorbed by the SFT (the blue tetrahedron). MEBs in the core region are calculated using SEAKMC.

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Swelling rate calculation

Swelling rate calculation was derived22,30,83 and calculated based on the sink strength for comparison with experimental results. When considering only one type of sink, taking SFT as an example (scenario (i)),

$${Swelling}\,{rate}= \varepsilon *\frac{{k}_{{Void},v}^{2}({k}_{{Void},i}^{2}+{k}_{{SFT},i}^{2})-{k}_{{Void},i}^{2}({k}_{{Void},v}^{2}+{k}_{{SFT},v}^{2})}{\left({k}_{{Void},v}^{2}+{k}_{{SFT},v}^{2}\right)\left({k}_{{Void},i}^{2}+{k}_{{SFT},i}^{2}\right)}\\ * \left(1-\frac{{J}_{v}^{{em}}}{N}\right).$$
(5)

The term \({k}_{{sink},\alpha }^{2}\) represents the sink strength of a specific sink (void or SFT in this equation) in the ideal system. Subscript \(\alpha\) can represent either interstitials with \(i\) or vacancies with \(v\), respectively. Survival fraction \(\varepsilon\) is selected to be 0.1 based on recent analyses to correct for correlated defect recombination84. For scenarios (ii) and (iii), \({k}_{{SFT}}^{2}\) was replaced with the sink strength of loops and dislocations. \({J}_{v}^{{em}}\) is the vacancy emission rate22,68, and \(N\) is the number of vacancies contained in a void. \({J}_{v}^{{em}}/N\) is the total volume loss rate of a void.

Dual-ion irradiation of polycrystalline copper

High-purity (99.99%) polycrystalline copper disks obtained from Johnson–Mathey were cold rolled to achieve a 50% reduction in thickness. Prior irradiation, the disks were mechanically and electropolished at −40°C using a 33% HNO3/67% CH3OH solution with a 5 V applied potential. A thermocouple wire was spot welded to the back of the TEM disks to monitor the irradiation temperature. Oak Ridge dual-beam Van de Graaff irradiation facilities85 were used for irradiation. Simultaneous dual-ion beam (4 MeV Fe & 0.2–0.4 MeV He ions) irradiation was conducted at 440 °C, reaching a peak damage level of 13 dpa and helium concentration of 48 appm at a depth of approximately 1.2 μm, as calculated with Ed = 30 eV for Cu according to SRIM 2013 Q-C vacancy.txt method86. The total fluence of the Fe ion beam was 9.5\(\times\)1019, and the dose rate was 3.3\(\times\)10-3 dpa/s. The helium beam energy was ramped from 0.2 to 0.4 MeV at 0.025 Hz, resulting in uniform helium implantation at depths of 0.5–1.1 μm.

STEM Experiments

The STEM imaging and 4D-STEM exit-wave power-cepstrum (EWPC) measurements were conducted using a JEOL NEOARM aberration-corrected STEM operating at an acceleration voltage of 200 kV. Atomic-resolution STEM imaging was performed with a semi-convergence angle of ~13.6 mrad and 4D-STEM data for EWPC measurements was acquired using a ~3.4 mrad semi-convergence angle, resulting in a probe diameter of <1 nm. CBED patterns were captured with the beam direction aligned along a Cu \(\left\langle 1\bar{1}0\right\rangle\) zone axis on a pnCCD pixelated detector with a frame size of 264 × 264 pixels, an acquisition rate of 1000 frames per second, and at 512 × 512 real space probe positions. The data was binned by four in each real space dimension to increase the signal-to-noise ratio of the CBED patterns and by an additional two in each reciprocal space dimension to enhance the data processing rate. Details of the EWPC method used to make lattice strain measurements are available in a separate report55 and related codes are available at https://github.com/ElliotPadgett/PC-STEM.