Ballistic effects on the copper precipitation and re-dissolution kinetics in an ion irradiated and thermally annealed Fe–Cu alloy Free

Precipitation often occurs in materials that contain one or more solute elements, and is of particular significance in the field of nuclear structural materials. Due to the creation and accumulation of atomic defects under irradiation, structural materials in nuclear reactors are subject to a host of irradiation effects, such as irradiation hardening and embrittlement, irradiation creep, and dimensional instability, which severely deteriorate the designed properties of these materials.^1–3 One strategy developed in the past decades to mitigate the detrimental irradiation effects is to introduce a high number density of fine nanoscale particles through thermal precipitation prior to the deployment of the materials in nuclear reactors.^4,5 Such precipitates (PPTs) are believed to enhance recombination of vacancies and interstitials during irradiation, reduce the net accumulation of irradiation defects, and hence alleviate irradiation damage to the microstructure and properties of the matrix material. On the other hand, undesirable precipitation can take place in nuclear structural materials during the course of irradiation. Copper rich PPTs and nickel-manganese rich PPTs (“late blooming phase”) have been found to form in reactor pressure vessel (RPV) steels during service and been considered leading factors in the observed embrittlement of the materials.^6–8 

Precipitation under concurrent irradiation and heat is a complex phenomenon. Irradiation is known to enhance the diffusion of solute atoms and accelerate formation of PPT phases that are favored by equilibrium thermodynamics, but PPTs that are not normally formed under pure thermal conditions have also been observed after irradiation. Pre-existing PPTs, such as those in the ODS (Oxide Dispersion Strengthened) steels, have been reported to remain stable under irradiation in some studies, but shrink or coarsen in others.^9–15 

In the area of nuclear fuels, it has been known for a long time that fission gas bubbles in UO₂ are subject to ballistic re-dissolution induced by fission fragments.¹ Various efforts have been made to quantify the ballistic re-dissolution rate of the fission gas bubbles. A recent example is the combined Monte Carlo and molecular dynamics (MD) simulation work performed by Schwen et al. which, as the authors stated, provided a far more accurate evaluation of the re-dissolution parameter for Xe bubbles in UO₂ than previous analytical results.¹⁶ The same group of authors subsequently incorporated the ballistic re-dissolution mechanism in their Monte Carlo simulations of Xe bubble evolution¹⁷ in UO₂ and compositional patterning in computer model binary alloys^18,19 under irradiation on a longer time scale. These works have demonstrated the need for an accurate account of the ballistic re-dissolution in the modeling of irradiation behavior of nuclear fuels and potentially nuclear structural materials.

Dilute Fe–Cu alloys have been of interest to the nuclear structural materials community due to their close relevance to the embrittlement issue of the RPV steels. Cu precipitation from the Fe matrix, under irradiation and, in particular, thermal anneal conditions, has been studied using various experimental and computational methods (e.g., Refs. 20–24). Nevertheless, the effect of ballistic re-dissolution on Cu precipitation kinetics has not received much attention, possibly due to the lack of an accurate assessment of the ballistic re-dissolution rate.

In a recent study, Certain et al. performed molecular dynamics (MD) simulations of cascade induced re-dissolution of copper PPTs in an iron matrix, and developed through statistical analysis of the MD results a cascade re-dissolution parameter defined as the number of re-dissolved Cu atoms per cluster (PPT) atom per collision event.²⁵ The cascade re-dissolution parameter provides a new opportunity to more thoroughly and quantitatively assess the ballistic effects on the Cu precipitation/re-dissolution kinetics under irradiation, and to demonstrate how important (or not) the incorporation of the ballistic re-dissolution is for the understanding of the complex phenomena of precipitation and PPT stability under concurrent irradiation and heat.

Here we present a comprehensive atomistically based continuum model that incorporates both thermal and ballistic effects, including the newly established cascade re-dissolution parameter, for the Cu precipitation/re-dissolution kinetics in an Fe-0.78 at. % Cu alloy under varying thermal and/or ion irradiation conditions. We show that the material may exhibit different overall behavior, i.e., PPT formation, re-dissolution, or coarsening, depending on the dominating mechanism under each specific condition. We also compare our model predictions with previous experiments and limited atom probe tomography (APT) data acquired in this study.

The basic atomistic mechanisms behind the precipitation phenomenon are the diffusion and interactions of solute atoms. As illustrated in Fig. 1, a monomer solute atom (here substitutional Cu) can combine with another monomer, when they encounter each other along their diffusion paths, to form a dimer; a dimer, when approached by a diffusing monomer, can become a trimer, and so on. This clustering process leads to a continuous growth in size space and to the formation of PPTs that are big enough to be detected by various experimental characterization techniques. In the inverse direction, a dimer, a trimer, or a bigger PPT can emit monomers as a result of thermal dissociation or ballistic collisions, which leads to a reduction in cluster sizes and re-dissolution of solute atoms from PPTs into the matrix. Similar to the rate theory in the field of nuclear materials commonly used to treat kinetics of irradiation defects (vacancies, interstitials, and their various forms of clusters, etc.),^1,2,26–28 here we treat the capturing interactions of solute atoms as a second order chemical reaction, and the emission of monomers from a cluster/PPT as a first order chemical reaction. The overall kinetic evolution is then described by the following system of differential equations:

where C stands for the concentration, $kn+$ and $kn−$ are the rate constants for an n-Cu cluster’s capturing and emitting Cu monomers, respectively. The capturing rate constant is determined by the sizes and diffusivities of the two reactants as

where r is the radius, and D is the diffusivity, as Waite derived for homogeneous diffusion-limited reactions based on probability analysis.²⁹ According to Ref. 29, the capturing rate constant represents the probable rate at which the reacting species approach each other by diffusion. Indeed, if only one species is diffusing, Eq. (2) can also be derived by solving the Fickian diffusion equation for the steady state flux of the diffusing species into the static species.² In general, r in Eq. (2) should be the capturing/interaction radius which may be affected by the local strain around a cluster. Here the cluster size is used as a good approximation for r, considering the substitutional character of Cu atoms in Fe matrix and the similar sizes of Cu and Fe atoms. For the diffusivity, we only consider the monomer Cu to be mobile, i.e., D_n = D₁ (for n = 1) or 0 (for n > 1), consistent with the majority of the previous experimental and computational studies on Fe–Cu alloys (e.g., Refs. 22 and 23). It is noted that setting Cu clusters to be mobile while using a much lower monomer diffusivity than in the literature was recently shown to be also able to reproduce certain experiments at 500 °C.²¹ However, considering the relatively limited validation of the Cu cluster mobility, here we choose the conventional description with only Cu monomer being mobile. The emission rate constant (only defined for n > 1) under concurrent irradiation and heat is expressed as

where r and D have the same definitions as above, C₀ is the atomic number density of the matrix, $EnB$ is the binding energy of a Cu monomer to an n-Cu (n > 1) cluster, SICRD is a size-independent cascade re-dissolution parameter equal to approximately 1 per PKA (Primary Knock-on Atom) that is derivable from the recent MD simulations²⁵ as detailed in the section Case III of Results and Discussion, a₀ is the lattice constant of the iron matrix, ϕ is the ion flux, and $∂2NPKA≥1keV∂l∂Nion$ is the number of PKAs (≥1 keV) created per ion per unit depth. The first term on the right hand side (RHS) of Eq. (3) represents thermally activated emission and is formulated based on micro-balance with the inverse capturing process, i.e., $kn−=k1,n−1+C0exp−EnB/kBT$⁠, which is derivable by setting, in the reaction Cu₁ + Cu_n−1⇋Cu_n, the forward and backward rates to be equal and the concentrations to be at the thermodynamic equilibrium values. The second term on the RHS of Eq. (3) represents the emission resulting from collision cascades initiated by PKAs inside the PPT or within a distance of one lattice parameter outside the PPT which, according to Ref. 25, are the ones primarily responsible for the cascade re-dissolution mechanism. The term $∂2NPKA≥1keV∂l∂Nion$ can be obtained for any given ion species and energy and target material by using SRIM (Stopping and Range of Ions in Matter)³⁰ in the full cascade mode and performing statistical analysis of the SRIM output file “COLLISON.txt.” SRIM is a binary collision code that calculates the position of atomic collisions subsequent to the entrance of an ion into a material and associated atomic displacements and energy dissipation using statistical algorithms (for more details of SRIM, see Ref. 30). SRIM has been known to have the problem of reporting incorrect number of displacements³¹ when running in the full cascade mode, but here we will use the SRIM PKA data instead of the SRIM reported displacements. The binding energy $EnB$ is related to n by the commonly used capillary approximation,²⁶ i.e., $EnB=E1F+E2B−E1Fn2/3−n−12/3/22/3−1$⁠, where $E1F$ is the Cu-monomer formation energy in the iron matrix, and $E2B$ is the binding energy of a di-cluster of Cu atoms in the iron matrix. The capillary description of the binding energy accounts for both the interfacial energy between the precipitate and the matrix and the chemical potential difference (neglecting the entropic contribution which is discussed in the section Case I of Results and Discussion) between the solid solution and the PPT, and in general captures the correct thermodynamic trend as n increases. It is possible that at very small sizes, e.g., n = 3–10, the actual binding energy somewhat deviates from the description, but this is generally not expected to have a strong influence on the overall kinetic evolution.

Eq. (1) is the general equation system used to model all the different cases in this work. The initial condition for Eq. (1) varies with the case and will be articulated separately. Overall, the initial condition is of only one type throughout this work—known non-zero concentrations for certain species and zero concentrations for all the others. The initial concentration is non-zero for only monomer in one case, and for a range of cluster sizes in the other cases.

The PARASPACE (PARAllel SPAtially-dependent Cluster Evolution) code introduced in Ref. 26 is used to perform the modeling in this work. The spatial dependence option in PARASPACE is turned off by setting the number of spatial grids to 1 and removing the cross-grid diffusion flux along with the boundary conditions. In PARASPACE, differential equations are solved by an implicit method with an adaptive order and time step following the algorithms of Shampine et al.³² The linear algebraic equations incurred are solved by the ILUPACK developed by Bollhoefer et al.³³ The design, the structure, and the advanced features of the PARASPACE code can be found in Ref. 26.

In this study, we model three cases representing distinct thermal and/or irradiation conditions for the Fe-0.78 at. % Cu alloy: Case I. thermal anneal at 450 °C for 24 h; Case II. thermal anneal at 450 °C for 24 h followed by additional thermal anneal at −20 and 300 °C for 7 h; Case III. thermal anneal at 450 °C for 24 h followed by 5 MeV Ni ion irradiation with a flux of 9.8 × 10¹² cm⁻² s⁻¹ at −20 and 300 °C for 7 h. The choice of these cases is based on considerations of possible variations in the precipitation/re-dissolution behavior as well as practicability of performing corresponding experiments. Due to the very low solubility of Cu in Fe matrix, Case I is anticipated to induce PPT formation in the subject alloy. Cases II and III are chosen to explore and contrast the Cu precipitation/re-dissolution kinetics at presence of pre-existing PPTs under pure thermal conditions and mixed irradiation and thermal conditions. For each case, we perform a corresponding experiment and use atom probe tomography to determine the presence/absence of potential PPTs and, if they are present, measure their average radius and number density. The experimental details are provided in the supplementary material. The experiment for Case I is intended to calibrate key parameters in the model. The experiments for Cases II and III are to serve as initial validation of the corresponding model predictions. While more extensive model validation employing a myriad of experimental techniques and/or covering broader conditions is yet to be pursued in the future, the primary focus of this paper is to communicate the important mechanistic insights from the model results.

In Case I, the alloy starts from a supersaturated solution state with $Cn|t=0=0.0078C0forn=1or0forn>1$⁠, where 0.0078 is the total Cu content in the alloy composition. The second term on the RHS of Eq. (3) vanishes since this is a purely thermally driven process. Copper monomers migrate with the thermal diffusivity $D1=Dthermal=D0exp−Em/kBT$⁠, where D₀ is a pre-factor, and E_m is the migration energy. The key parameters determining the kinetics of precipitation include D₀, E_m, $E1F$ and $E2B$⁠. Previous studies reported values of 0.4-0.6 cm² s⁻¹ for D₀, 2.3-2.5 eV for E_m (e.g., Refs. 22 and 23), ∼0.5 eV for $E1F$ and ∼0.15 eV for $E2B$ (e.g., Ref. 23). For this case, our corresponding experiment reveals formation of Cu PPTs with an average radius of 1.3 nm and a number density of 5.1 × 10⁻⁴ nm⁻³ (listed in Table I) as measured by APT, after the 24 h anneal of this alloy at 450 °C. It is noted that all the specimens in this study have been solution treated at a high temperature (775 °C) and then quenched to obtain a supersaturated state of copper, prior to the anneal and/or irradiation processes in Cases I-III.

Using the APT data for Case I as a reference and fixing D₀ to be 0.5 cm² s⁻¹, we adjust/calibrate E_m, $E1F$ and $E2B$ near the literature values. The calibration involves multiple iterations of (re-)setting the parameters, solving the differential equation system up to the 24 h anneal time, finding the predicted average radius and number density of PPTs of APT resolvable sizes (radius ≥ 0.5 nm in this study), and comparing the predicted values with the experimental measurements. The calibrated values hence obtained for E_m, $E1F$ and $E2B$ are 2.32, 0.435, and 0.18 eV, respectively, corresponding to a model predicted average radius of 1.2 nm and a number density of 6 × 10 ⁻⁴ nm⁻³, for the APT resolvable PPTs, very close to the APT measurements.

The solubility limit (SL) of copper in BCC (body-centered-cubic) iron at different temperatures can be calculated from the calibrated formation energy (⁠$E1F$⁠) of Cu-monomer using the relationship $SL=exp−E1FkBT$⁠. The result is plotted as a solid line (red) in Fig. 2, where data reported from thermoelectric power (TEP) experiments²⁴ are also included (blue crosses). The TEP data in weight percent can be found in Table 2 of Ref. 24 and are here presented in atomic percent. The good agreement shown in Fig. 2 between the present work and the TEP study on the temperature dependent Cu solubility limit provides a validation for the calibrated value of $E1F$ listed above. Note that in the above formulation of the solubility limit, we have neglected the entropy (s) contribution to the formation energy which would otherwise give an extra coefficient in front of the exponential, i.e., $SL=expskBexp−E1FkBT$⁠. The fact that the TEP data agree quite well with a single temperature-dependent exponential without a constant coefficient indicates the minor role of entropy.

It is clear in Fig. 2 that the total Cu content in the subject alloy Fe-0.78 at. % Cu exceeds the Cu solubility limit at −20, 300, 450 °C, the anneal temperatures for Cases I-III, meaning that equilibrium thermodynamics would require formation/existence of Cu precipitates at all these temperatures. In addition, Fig. 2 also shows that the total Cu content, 0.78 at. %, is below the solubility limit at the solution treatment temperature, 775 °C, indicating that Cu is fully dissolved during the solution treatment, prior to the 450 °C anneal in Cases I-III. This re-affirms the validity of the initial condition used for Case I.

In Case II, the alloy is subjected to a 7 h secondary anneal at −20 or 300 °C following a pre-anneal at 450 °C for 24 h. The precipitation/re-dissolution kinetics in this case is purely driven by thermal activation, same as in Case I. The results, i.e., the concentrations of Cu monomer and all different sized Cu clusters, from Case I become the initial condition for the secondary anneal in Case II, while the rate equations (Eq. (1)), and expressions of monomer capturing (Eq. (2)) and monomer emitting (first term on the RHS of Eq. (3)) rate constants remain the same for the secondary anneal (although quantitatively changed due to the change of temperature).

With the physical parameters as calibrated earlier in Case I, the model is used to predict the fate of the pre-anneal introduced PPTs after the −20 or 300 °C secondary anneal. Fig. 3 shows the PPT size distribution curves (dC/dr = C_n × dn/dr = C_n × 3n/r) at the ends of the pre-anneal and the secondary anneal. It can be seen from Fig. 3 that the model predicts no change in the PPT size distribution after the secondary anneal at either −20 or 300 °C, indicating that the secondary anneal at these two temperatures has no impact on the pre-anneal introduced PPTs. As shown in Table II, the model predicted average radius and number density of APT resolvable (radius ≥0.5 nm in this study) PPTs remain the same as those at the end of the pre-anneal. This is in good agreement with the APT experiments corresponding to Case II which also reveal no variation in the average resolvable-PPT radius and number density with respect to Case I, as shown in Table I.

The remaining Cu monomer concentration in the matrix at the end of the 24 h 450 °C pre-anneal is 0.25 nm⁻³, still above the equilibrium concentration as determined by the product of the matrix atomic number density and the Cu solubility limit, i.e., C₀ × SL, which is 0.079 nm⁻³ at 450 °C. When the 450 °C pre-annealed alloy is subsequently exposed to a secondary anneal at −20 or 300 °C, this remaining Cu monomer concentration becomes even farther above the equilibrium concentration at the new temperature, which is 1.8 × 10⁻⁷ nm⁻³ at −20 °C and 0.013 nm⁻³ at 300 °C. Equilibrium thermodynamics would lead to further reduction of Cu monomer concentration in the matrix through growth of existing PPTs and/or formation of new PPTs, but this does not occur, at least within the 7 h secondary anneal timeframe, due to the slow kinetics, i.e., low thermal diffusivity of Cu monomers at −20 and 300 °C as determined by the relatively large migration energy of 2.32 eV, which corresponds to a theoretical diffusion length of 2.1 × 10⁻¹⁴ and 0.17 nm at these two temperatures, respectively, for a period of 7 h.

In Case III, the alloy is subjected to a 24 h 450 °C pre-anneal and then 5 MeV Ni ion irradiation at −20 or 300 °C for 7 h. The ion irradiation at the controlled temperatures represents a mixed condition that includes not only irradiation but also the same secondary anneal as in Case II.

Irradiation has two major effects on the precipitation/re-dissolution kinetics, namely, cascade re-dissolution of precipitates and irradiation enhanced diffusivity of Cu monomers (through generation of excessive vacancies that assist Cu diffusion). Based on MD simulations of PKA and precipitate interactions, Certain et al. recently developed a re-dissolution parameter defined as the number of re-dissolved Cu atoms per cluster (PPT) atom per collision event²⁵ which they plotted as a function of initial PKA energy (spanning 1–100 keV) for 1, 3, and 5 nm diameter precipitates in Fig. 7 of Ref. 25. Ignoring the weak dependence of this parameter on the initial PKA energy, and multiplying this parameter with the number of Cu-monomers in the 1, 3, and 5 nm diameter precipitates, we find that for all the studied precipitate sizes, approximately one Cu atom is re-dissolved from a precipitate per PKA (collision event). The rate (in the unit of one per second) of monomers being re-dissolved by cascades from a PPT is then equal to this size-independent cascade re-dissolution parameter (SICRD) of one per PKA multiplying the rate of creation of qualified PKAs (energy ≥1 keV, located inside the PPT or within a distance of one lattice parameter outside the PPT²⁵), as represented by the second term on the RHS of Eq. (3). Within this term, $∂2NPKA≥1keV∂l∂Nion$⁠, i.e., the number of PKAs (≥1 keV) created per ion per unit depth, is generally depth dependent for ion irradiation. In this work, we focus on the depth of 0.5 μm based on the following considerations: (1) it is reasonably far from the surface so that surface effects can be ignored; (2) the (depth) gradient of PKA and defect generation around this depth is relatively small, according to SRIM,³⁰ so that a spatially dependent model is not necessary; (3) it is far from the depth (∼1.8 μm) where most Ni ions are stopped so that potential effects of the stopped ions can be ignored. By running SRIM³⁰ for 5 MeV Ni ions in iron (displacement energy 40 eV), and performing statistical analysis of the PKA position and energy data in the resulting “COLLISON.txt” file, we find $∂2NPKA≥1keV∂l∂Nion≈0.032PKAnm−1$ ion⁻¹ around the depth of 0.5 μm. Note that we are neglecting the influence of the Cu atoms or PPTs on the generation of PKAs, which is reasonable considering the low total concentration (0.78 at. %) of Cu. It is also noted that although the incident ion energy is 5 MeV, the fraction of PKAs with energy greater than 100 keV is very small, being less than 0.5%, according to SRIM. Hence, it is believed safe to use the result from the MD simulations of the cascade re-dissolution performed with PKA energy up to ∼100 keV.

Irradiation enhanced diffusivity of Cu monomers can be treated as

where D_thermal is the thermal diffusivity as used for Cases I and II, and C_v and $Cceq$ are the vacancy concentration under irradiation and thermal equilibrium, respectively. The equilibrium vacancy concentration is determined by the vacancy formation energy $EvF$ as $Cveq=C0exp−EvFkBT$⁠. A range of $EvF$ values, ∼1.6-2.2 eV, can be found in the literature, obtained from experiments, ab initio, or MD studies, and here we choose to use 1.7 eV around which good agreement between MD calculation and more than one experiment was reported.³⁴ To calculate the vacancy concentration (averaged over the 7 h irradiation period) under irradiation, separate defect (vacancy, interstitial, and their respective clusters) cluster dynamics modeling is carried out that takes into account intra-cascade defect-cluster production and cluster mobility, in a similar way to that described in Ref. 26 with the following changes: (1) material and material/irradiation related quantities (listed in Table III), (2) negligence of the spatial dependence (weak around the depth of 0.5 μm) of defect evolution in this study (bulk ion irradiation), and (3) negligence of mobility of interstitial clusters bigger than I₂₀ for simplicity. Note that the production rates (listed in Table III) of small defects/defect-clusters are obtained by combining SRIM PKA (around the 0.5 μm depth) energy data with PKA energy-dependent intra-cascade production probabilities from previous MD studies, as detailed in Ref. 26.

The average vacancy concentration obtained in this way is 10 and 1 × 10⁻⁵ nm⁻³, for the irradiation at −20 and 300 °C, respectively. Apparently the 10 nm⁻³ for the −20 °C is too large to be physical, considering the iron atomic number density of 84.6 nm⁻³. Before reaching this large concentration, spatial correlation among vacancies and interstitials/interstitial-clusters should have already intervened, but this cannot be treated by cluster dynamics models which are based on random mean field theory. Hence, we perform additional defect dynamics calculation at 600 °C (resulting in a vacancy concentration of 1 × 10⁻⁶ nm⁻³) and estimate the vacancy concentration for the −20 °C irradiation by extrapolating the concentrations for 300 and 600 °C according to an Arrhenius relationship, and we find 0.0346 nm⁻³.

Taking into account the cascade re-dissolution and the irradiation enhanced diffusivity in Eqs. (1)-(3) and using the concentrations from Case I (at the end of the 24 h pre-anneal at 450 °C) as the initial condition, the model is used to predict the fate of the pre-anneal introduced PPTs in the subsequent ion irradiation at −20 and 300 °C. Shown in Fig. 4 is the predicted size distribution of PPTs after the ion irradiation at −20 °C for 0, 3.5, and 7 h. As can be seen in Fig. 4, the irradiation at −20 °C significantly reduces the number density of the bigger precipitates, which is because of the cascade re-dissolution mechanism. It is interesting to note that Fig. 4 also shows a decrease in the Cu monomer number density. This is because the remaining monomers in the matrix at the end of the pre-anneal acquire an enhanced diffusivity under irradiation and hence are captured by small to intermediate sized precipitates. This loss of monomers from the matrix cannot be compensated by the cascade re-dissolved monomers from the bigger precipitates which are also subject to capturing by small to intermediate sized precipitates. It is important to point out that the cascade re-dissolution rate has a pronounced dependence (∼r³) on the size of a precipitate, as expressed in Eq. (3). This implies that, for the irradiation at −20 °C where thermal effects are weak, growth due to irradiation enhanced monomer diffusion dominates the fate of small PPTs, while shrinkage due to cascade re-dissolution dominates the fate of the bigger PPTs. It can be expected that as irradiation continues indefinitely (beyond 7 h), a static size distribution profile of PPTs will be established due to the balance of these two factors. This is indeed confirmed by additional model calculations that are extended to 12, 16, 20, and 24 h of irradiation at the same temperature (−20 °C), as shown in Fig. 5.

Compared with the state at the end of the 450 °C pre-anneal, the model predicts an overall re-dissolution behavior of the pre-anneal formed PPTs during the −20 °C irradiation. As listed in Table II, the model predicts an average radius of 0.7 nm and a number density of 5.6 × 10⁻⁴ nm⁻³ for the APT resolvable (radius ≥ 0.5 nm in this study) PPTs after 7 h irradiation at −20 °C, as opposed to the 1.2 nm and 6 × 10⁻⁴ nm⁻³ (model predicted) at the start of the irradiation. The corresponding experiments including the 450 °C pre-anneal, the −20 °C Ni ion irradiation, and the post-irradiation APT analysis (with caution in sample extraction from the 0.5 μm depth; for details, see the supplementary material) reveal an average PPT radius and PPT number density of 0.6 nm and 3.8 × 10⁻⁴ nm⁻³, respectively, as listed in Table I. The experimental results not only substantiate the model predicted overall re-dissolution behavior of the existing PPTs under this condition but also are in fairly good quantitative agreement with the model.

Because the irradiation vacancy concentration of 0.0346 nm⁻³ for the −20 °C irradiation is not directly from a dedicated defect calculation, we have performed additional modeling for this case using a vacancy concentration one order of magnitude greater or smaller than the 0.0346 nm⁻³, to check the sensitivity of the model predictions to the vacancy concentration. It is found that the sensitivity is relatively weak within the two orders of magnitude span of the vacancy concentration. The model predicted average PPT radius at the end of the 7 h irradiation is 0.66 and 0.76 nm, and the PPT number density is 5.1 × 10⁻⁴ and 6.0 × 10⁻⁴ nm⁻³, when using a vacancy concentration of 0.003 46 and 0.346 nm⁻³, respectively. These are quite close to the 0.7 nm and 5.6 × 10⁻⁴ nm⁻³ obtained above using a vacancy concentration of 0.0346 nm⁻³.

In contrast to the −20 °C irradiation, the pre-annealed alloy displays quite different behavior under 300 °C irradiation, as predicted by the model and shown in Fig. 6. The small to intermediate-sized pre-existing PPTs experience a significant decrease in the number density during the 7 h irradiation at 300 °C. The size distribution curve is shifted to the lower right as new bigger PPTs are formed, at the cost of the smaller precipitates. This is a typical characteristic of Ostwald ripening. In Case II we have already shown that at 300 °C thermal effects alone are not able to modify the size distribution of the pre-existing PPTs. We have also shown that, in Case III, irradiation at −20 °C, where thermal effects are even weaker, leads to re-dissolution of the pre-existing PPTs. The differences from these earlier cases suggest that under irradiation at 300 °C, thermal effects are operating together with irradiation effects, pushing the balance between cascade induced shrinkage and diffusivity-enhanced growth towards bigger cluster sizes. Due to the rapid increase in the cascade re-dissolution rate constant with PPT size as expressed by Eq. (3), it is still expected that the evolution of the size distribution profile of the PPTs will come to a stop after a sufficiently long time of irradiation at 300 °C. Indeed, such a trend of approaching a static size distribution is clearly exhibited in Fig. 7 by model calculations for extended irradiation at this temperature beyond 7 h.

The average radius and the number density of APT resolvable (radius ≥0.5 nm in this study) PPTs predicted by the model for the pre-annealed alloy after the 300 °C irradiation for 7 h are 3.2 nm and 4.9 × 10⁻⁵ nm⁻³, respectively, as listed in Table II. The measured values in the corresponding experiments are 2.5 nm and 3.5 × 10⁻⁵ nm⁻³, respectively, as listed in Table I. Comparing with the values for Case I in the respective tables, both the model and the experiments suggest an overall coarsening behavior of the pre-existing PPTs during the 300 °C irradiation. The quantitative agreement between the model and the experiments after 7 h irradiation at 300 °C is reasonably good.

As already described, we have compared our model predictions with the solubility data from thermoelectric power experiments in the literature²⁴ and the average radius and number density of resolvable PPTs measured by atom probe tomography in this work for the ending states of the different cases. The comparison has revealed reasonably good agreement between the model and the experiments in general. This serves as initial but certainly not full validation of the model. Future experiments covering a broader range of temperatures, irradiation doses, and anneal/irradiation durations are needed to validate the model more thoroughly.

Nevertheless, as stated earlier, the primary purpose/focus of this paper is to communicate the mechanistic insights disclosed by the model, particularly the ballistic effects on solute precipitation and re-dissolution kinetics, which we believe is important for the understanding of the complex phenomena of precipitation and precipitate stability under concurrent irradiation and heat.

We have presented a physically based kinetic model for the Cu precipitation/re-dissolution kinetics in ion irradiated and/or thermally annealed Fe-0.78 at. % Cu. Our model incorporates thermal and irradiation enhanced diffusion of atomic Cu, clustering of Cu into different sized precipitates, thermal dissociation of the precipitates and, in particular, a cascade re-dissolution parameter that has been made available by recent molecular dynamics simulations. Three cases of processing conditions have been examined using the model: Case I — anneal at 450 °C for 24 h only; Case II — (pre-)anneal at 450 °C for 24 h followed by a secondary anneal at −20 or 300 °C; Case III — (pre-)anneal at 450 °C for 24 h followed by 5 MeV Ni ion irradiation at −20 or 300 °C. Case I results in the formation of precipitates with a certain size distribution which sets the initial condition for the subsequent secondary anneal and/or ion irradiation in Cases II and III. The secondary anneal at either −20 or 300 °C in Case II has no impact on the pre-existing precipitates. The ion irradiation at −20 °C in Case III results in re-dissolution of the pre-existing precipitates while the irradiation at 300 °C leads to their coarsening. The differences among the cases are interpreted based on the equilibrium thermodynamics, thermally driven kinetics, and irradiation/ballistic effects. The model and the cases presented here may be useful in improving the understanding of the complex phenomena of precipitation and precipitate stability in nuclear structural materials under varying irradiation and thermal conditions.

See supplementary material for the experimental details.

The authors would like to thank and acknowledge D. Klingensmith and Professor G. R. Odette of University of California – Santa Barbara for supplying the alloy samples used in this study, and the Environmental Molecular Sciences Laboratory (EMSL), a Department of Energy collaborative scientific user facility located at Pacific Northwest National Laboratory, for providing access to its ion beam and atom probe. D.X. and B.D.W. acknowledge financial support from the U.S. Department of Energy, Office of Fusion Energy Sciences under Grant No. DE-FG02-04GR54750 and the U.S. Department of Energy, Office of Nuclear Energy’s Nuclear Energy University Programs (NEUP).