⇦Chinese Physics Letters, 2020, Vol. 37, No. 1, Article code 016103 An Incremental Model for Defect Production upon Cascade Overlapping * Yi Wang (王忆)**, Wensheng Lai (赖文生), Jiahao Li (李家好) Affiliations The Key Laboratory of Advanced Materials (MOE), School of Materials Science and Engineering, Tsinghua University, Beijing 100084 Received 8 October 2019, online 23 December 2019 *Supported by the National Key Research and Development Program of China (No. 2017YFB0702201) and the National Natural Science Foundation of China (No. 51571129).
**Corresponding author. Email: yiwang_cn@126.com Citation Text: Wang Y, Lai W S and Li J H 2020 Chin. Phys. Lett. 37 016103    Abstract An analytic incremental model is proposed to predict the defect production upon cascade overlapping. By resolving the coupled annealing events during cascade overlapping, this model handles cascade overlapping with multiple pre-existing defects of different sizes and number densities. The model is first parameterized and then applied to bcc-Fe. The proposed model satisfyingly reproduces the defect production obtained by molecular dynamics simulations with various radiation damage levels and defect cluster size distributions. The present model provides an essential description of the primary source of radiation damage, especially for high dose irradiation, and could be used in conjunction with reactive diffusion models for better understanding of radiation damage. DOI:10.1088/0256-307X/37/1/016103 PACS:61.82.Bg, 61.80.-x, 61.72.-y, 61.80.Az © 2020 Chinese Physics Society Article Text Collision cascades are the primary source of radiation defects and hence are the root reason of material property degradation upon irradiation. Defect production by a cascade in originally undamaged metals is determined mainly by the energy of the primary knocked-on atom (PKA).[1] Presuming non-overlapping, the radiation damage is usually evaluated as a linear combination of the defect production by individual PKAs, in equivalence with the displacements per atom (DPA).[2,3] However, new cascades will eventually overlap with radiation-damaged zone, disconnecting the actual defect productions from the DPA dose.[4] Indeed, during ion irradiation, the defect productions in metals become saturated with DPA doses.[5] Thus, consideration of cascade overlapping effects should be essential for accurate prediction of radiation damage and its effects at high DPA doses. Towards that end, previous studies have mainly used molecular dynamics (MD) to investigate the picosecond cascade process. Gao et al.[6] studied cascade overlapping with a single cascade debris generated by low energy PKAs ($\leqslant $5 keV) in bcc-Fe. Stoller et al.[7] studied similar configurations but with debris generated by relatively high energy PKAs ($\geqslant$10 keV). They both reported significant reduced defect production. Nordlund et al.[8] discovered that the reduction is induced by the thermal spike annealing. Recently, the statistical behaviors of cascade overlapping with a single defect cluster in bcc metals have been revealed by Byggmästar et al.[9] on self-interstitial atoms (SIA) cluster, also by Granberg et al.[10] and Fellman et al.[11] on vacancy (Vac) clusters. Defect production by multiple sequential overlapping cascades has also been investigated by MD, which gave direction observations of the saturation of defect production.[12–14] Despite its own powers, MD is costly and difficult to illustrate the statistical rules. Analytic modelling is desirable to describe statistical behavior of defect production upon cascade overlapping. Gao et al.[6] first found a Gaussian rule on the reduction of defect production with respect to the distance between overlapped cascades. Byggmästar et al.[9] established a correlation between the Gaussian parameters and the size of cascade liquid region. However, so far, the analytic modelling approach is only capable to model cascade overlapping with a simple damaged zone containing either a single cascade debris or defect cluster, but unable to handle general damaged zones containing multiple pre-existing defects of different sizes and number densities. Such cascade overlapping conditions are expected to be inevitable for the bcc-Fe based alloys in nuclear energy applications, which will endure high DPA doses at least on the order of 10$^{-2}$ for a life cycle of decades.[15] Also, ion radiation experiments may further enhance the overlapping probability due to the contracted exposure with high dose rates and restraint of diffusion. Thus, this work is motivated to extend the analytic approach to model the incremental change of defect production upon cascade overlapping with general damaged zones containing multiple pre-existing defects, choosing the bcc-Fe as a case studied. The present model focuses on the statistical behavior of the defect production upon overlapping by a new cascade in bcc-Fe. Herein, it features an incremental description of defect production, and the effects of pre-existing defects are treated using the mean-field approximation. The defect production is quantified as the change of Frenkel pairs. Following Gao et al.[6] on the defect production, instead of directly study the newly produced defects, it is more convenient to investigate the relative reduction compared with the ideal non-overlapping state, $$ {\rm \delta }n_{\rm F} =n_{{\rm F}-\infty } -n_{{\rm F-reduc}},~~ \tag {1} $$ where ${\rm \delta }n_{\rm F}$ is the interested incremental defect production, $n_{{\rm F}-\infty}$ is the defect production of ideal non-overlapping state (i.e., cascades separated by infinite distance), $n_{\rm F-reduc}$ is a to-be-examined quantity on the reduction of defect production. The reduction of defect production is induced by the thermal spike annealing, of which two mechanisms exist:[8] (i) recombination between newly produced defects and pre-existing defects due to liquid region capture, (ii) recombination between pre-existing Vac and SIA defects due to thermal diffusion upon cascade thermal spike affection. We assume that the evolution of radiation defects could be divide into two independent processes: the locally annealing process as controlled by the two above-mentioned mechanisms and the equilibrium diffusion-controlled recovery and/or clustering. Hence, to predict the defect production under overlapping, the analytic solution of the locally annealing process has to be resolved. Here, we divide liquid region capture mechanism (i) into two sub-scenarios, thus we re-group the annealing events into three scenarios. In scenario I, paired Vac sites and SIAs from pre-existing defects are captured and each recombines with newly produced defects, leading to a net reduction of pre-existing defects. In scenario ${\rm {I\!I}}$, unpaired Vac sites or SIAs are captured and recombine with newly produced defects, leading to suppressed production of new defects. In scenario ${\rm{I\!I\!I}}$, corresponding to mechanism (ii), disassociated SIAs from pre-existing SIA defects recombine with Vac sites from pre-existing Vac defects via thermal diffusion upon cascade thermal spike affection, leading to another net reduction of pre-existing defects. Now, it becomes convenient to establish a formula to decompose $n_{\rm F-reduc}$ as $$\begin{align} n_{\rm F-reduc} =\,&\min\left(n_{\rm F-reduc, I}+n_{\rm F-reduc, {\rm {I\!I}}},n_{{\rm F}-\infty} \right)\\ &+n_{\rm F-reduc, {\rm {I\!I}}},~~ \tag {2} \end{align} $$ % where $n_{{\rm F-reduc, I}}$ ($n_{{\rm F-reduc, {\rm {I\!I}}}}$) corresponds to the number of those paired (unpaired) Vac sites/SIAs from pre-existing defects which recombine with newly produced defects through liquid region capture, their joint contribution should be limited by $n_{{\rm F}-\infty}$ in a statistical sense, $n_{{\rm F-reduc, {\rm {I\!I}}}}$ corresponds to the number of those Vac sites/SIAs from pre-existing defects which recombined during thermal diffusion. Assume that newly produced defects recombine with $m_{{\rm Vac}}$ Vac sites ($m_{\rm SIA}$ SIAs) from pre-existing Vac (SIA) defects, $q_{\rm SIA}$ SIAs disassociate from pre-existing SIA defects and diffuse away for recombination, and $Q_{{\rm Vac}}$ pre-existing Vac sites are available for recombination, we have $$\begin{align} &n_{{\rm F-reduc, I}} =\min\left({m_{{\rm Vac}},m_{\rm SIA} } \right),~~ \tag {3} \end{align} $$ $$\begin{align} &n_{{\rm F-reduc, {\rm {I\!I}}}} =\left| {m_{{\rm Vac}} -m_{\rm SIA} } \right|,~~ \tag {4} \end{align} $$ $$\begin{align} &n_{{\rm F-reduc, {\rm {I\!I}}}} =\min\left({q_{\rm SIA},Q_{{\rm Vac}} } \right).~~ \tag {5} \end{align} $$ Thus, Eqs. (2)-(5) turn the problem of solving ${\rm \delta }n_{\rm F}$ into a problem accounting the recombined numbers of atoms from various pre-existing defects. Among the quantities $m_{{\rm Vac}}$, $m_{\rm SIA}$, $q_{\rm SIA}$ and $Q_{{\rm Vac}}$, only $Q_{{\rm Vac}}$ is directly observable from the configuration of pre-existing defects. However, when a new cascade overlapping with a single defect cluster, only the scenario ${\rm {I\!I}}$ annealing process exists, so that $m_{{\rm Vac}}$ or $m_{\rm SIA}$ becomes observable as $$ m_{\rm def} \left\{ {{\rm single clus}} \right\}=n_{\rm F-reduc} \left\{ {{\rm single clus}} \right\},~~ \tag {6} $$ where subscript ${\rm def}$ refers to Vac or SIA type, and clus short-aliases for 'cluster', and the same hereinafter. Meanwhile, when an SIA cluster is overlapped, it suffers a loss of size statistically.[9] The lost SIAs (numbered by $g_{\rm SIA})$ can only recombine with newly produced Vac defects in the absence of pre-existing defects. However, in the presence of pre-existing Vac defects, the lost SIAs from one pre-existing SIA cluster may recombine with either the newly produced Vac defects or those pre-existing Vac. Considering about the conservation of SIAs, $q_{\rm SIA}$ becomes observable as $$\begin{alignat}{1} &q_{\rm SIA} \left\{ {{\rm single clus}} \right\}\\ =\,&g_{\rm SIA} \left\{ {{\rm single clus}} \right\}-m_{\rm SIA} \left\{ {{\rm single clus}} \right\},~~ \tag {7} \end{alignat} $$ where $q_{\rm SIA} \left\{ {{\rm single clus}} \right\}$ describes a potential contribution from this specific SIA cluster to the scenario ${\rm {I\!I\!I}}$ recombination, and the actual contribution depends also on the amount of pre-existing Vac defects, as shown in Eq. (5). The evaluation of $m_{\rm def} \left\{ {{\rm single clus}} \right\}$ and $g_{\rm SIA} \left\{ {{\rm single clus}} \right\}$ is achieved based on Gaussian rules, $$\begin{alignat}{1} &m_{\rm def} \left\{ {{\rm single clus}} \right\}=\mu \left({r,R^{\rm def} } \right)\\ &=\begin{cases} N_{0} \left({R^{\rm def} } \right),& r < R^{\rm def}, \hfill \\ N_{0} \left({R^{\rm def} } \right)\exp\Big[ {-\Big({\frac{r-R^{\rm def} }{\alpha r_{\rm c} }} \Big)^{2}} \Big], & r\geqslant R^{\rm def}, \hfill \\ \end{cases}~~~~ \tag {8} \end{alignat} $$ $$\begin{align} &g_{\rm SIA} \left\{ {{\rm single clus}} \right\}=\min\big({K^{{\rm SIA}},{\rm \gamma }({r,K^{{\rm SIA}} })} \big)\\ &=\min\Big({K^{{\rm SIA}},\lambda n_{{\rm F}-\infty } \exp\Big[ {-\Big({\frac{r}{\eta r_{\rm c} }} \Big)^{2}} \Big]} \Big),~~ \tag {9} \end{align} $$ where $r$ is the distance between the defect cluster and the center of the cascade, $r_{\rm c}$ is a characteristic radius for the size of the cascade liquid region, $K^{\rm def}$ is the number of Vac sites/SIAs belonging to the defect cluster and is equivalently related with the effective defect cluster radius $R^{\rm def}$ by $R^{\rm def} =a({\frac{K^{\rm def} }{2}\frac{3}{4\pi }})^{1/3}$ ($a$ is the lattice constant), $N_{0} \left({R^{\rm def} } \right)$ is a function independent of $r$. In the equations, $\alpha$, $\lambda$ and $\eta$ are fitting parameters. Hereafter, $\mu \left({r,R^{\rm def} } \right)$ will be referred as the recombination function since it corresponds to the number of Vac sites/SIAs from a defect cluster recombined with newly produced defects, and ${\rm \gamma }\left({r,K^{{\rm SIA}} } \right)$ be referred as the ideal loss function corresponding to the number of lost SIAs from a defect cluster. Equation (8) is proposed by Byggmästar et al.,[9] and based on their data we propose Eq. (9). To collect the combined effects from multiple pre-existing defects, we use a mean-field approximation, which presents an average in spatial space for a stochastically introduced new cascade. In the following, treatment of the recombination function $\mu \left({r,R^{\rm def} } \right)$ is presented as an example. First, the contribution of a defect cluster with radius of $R^{\rm def}$ to the quantity $m_{\rm def}$ is given by an integration of Eq. (8), $$ M_{\rm def} \left({R^{\rm def} } \right)=\mathop \int\limits_{{\rm \mathit{\Omega} }} \rho \left({R^{\rm def} } \right)\mu \left({r,R^{\rm def} } \right){d\mathit{\Omega} },~~ \tag {10} $$ where $\mathit{\Omega}$ is an integration domain representing the affected region by the new cascade, $\rho \left({R^{\rm def} } \right)$ is the number density of the defect cluster with radius $R^{\rm def}$. Then, by summing up $M_{\rm def} \left({R^{\rm def} } \right)$ from defect clusters of different sizes, one obtains $$\begin{align} m_{\rm def} &=\mathop \sum\limits_{i} M_{\rm def} \left({R_{i}^{\rm def} } \right)\\ &=\mathop \sum\limits_{i} \mathop \int\limits_{{\rm \mathit{\Omega} }} \rho \left({R_{i}^{\rm def} } \right)\mu \left({r,R_{i}^{\rm def} } \right){d\mathit{\Omega} },~~ \tag {11} \end{align} $$ where $M_{\rm def} \left({R_{i}^{\rm def} } \right)$ is the contribution by defect cluster with the $i$th size $R_{i}^{\rm def}$. Using a similar procedure, contributions for the ideal loss function is given by $$\begin{alignat}{1} q_{\rm SIA} =\,&\mathop \sum\limits_{i} \mathop \int\limits_{\rm \mathit{\Omega}} \rho \Big({K_{i}^{\rm SIA}} \Big)\Big\{\max\Big[0,\min\Big(K_{i}^{{\rm SIA}},\\ &{\rm \gamma}\Big(r,K_{i}^{{\rm SIA}}\Big)\Big)-\mu \Big(r,R_{i}^{\rm SIA}\Big)\Big]\Big\}{d\mathit{\Omega}}.~~ \tag {12} \end{alignat} $$ Equations (1)-(5) are the controlling equations of the current incremental defect production model, while the parameters $m_{{\rm Vac}}$, $m_{\rm SIA}$ and $q_{\rm SIA}$ are determined by Eqs. (11) and (12), correspondingly. As is seen, the current model correlates the incremental defect production ${\rm \delta }n_{\rm F}$ with the number density $\rho \left({R^{\rm def} } \right)$ of pre-existing defects, and this implies a dependence of ${\rm \delta }n_{\rm F}$ on the size distribution of pre-existing defects. Hence, the model can be further coupled with object kinetic Monte Carlo (OKMC)[16,17] or cluster dynamics (CD),[18] which describes the evolution of number density and size distribution of defects due to reactive diffusion process, to study the evolution of radiation defects upon continuous irradiation. To apply the current model, Eqs. (8) and (9) need to be properly parameterized, which means the function $N_{0} \left({R^{\rm def} } \right)$, parameters $\alpha$, $\lambda$ and $\eta$ have to be fitted. With a normalization, $N_{0} \left({R^{\rm def} } \right)/n_{{\rm F}-\infty}$ is independent of the morphology of the defect cluster and temperature, but only dependent on the relative size of the defect cluster.[9] We propose a simple empirical function to analytically describe $N_{0} \left({R^{\rm def} } \right)/n_{{\rm F}-\infty}$, $$ \frac{N_{0} \left({R^{\rm def} } \right)}{n_{{\rm F}-\infty } }=\beta \frac{K^{\rm def} /n_{{\rm F}-\infty } }{1+K^{\rm def} /n_{{\rm F}-\infty } },~~ \tag {13} $$ where $\beta$ is a parameter depending on defect cluster of Vac or SIA type, and provided $\beta \leqslant 1$ it is ensured that $N_{0} \leqslant n_{{\rm F}-\infty}$ and $N_{0} \leqslant K^{\rm def}$.
cpl-37-1-016103-fig1.png
Fig. 1. Fittings of the analytic model based on MD data. Here (a) is a schematic of the fitting procedure; (b), (c) and (d) correspond to Eqs. (13), (8) and (9), respectively. The scatters are extracted from MD data from Ref. [9] for SIA clusters and from Ref. [10] for Vac clusters. Solid (dashed) lines are fitted analytic functions for SIA (Vac) clusters.
Table 1. Fitted parameters for current analytic model of incremental defect production.
$\alpha$ $\beta$ $\lambda$ $\eta$
Vac clusters 1.50 0.26
SIA clusters 1.00 1.00 0.80 1.04
A schematic of the fitting procedure is given in Fig. 1(a). Specifically, in Fig. 1(b), $\beta$ is fitted for $N_{0} /n_{{\rm F}-\infty}$. Then, in Fig. 1(c), $\alpha$ is further fitted for the recombination function $\mu \left({r,R^{\rm def} } \right)$, with the corresponding $N_{0} /n_{{\rm F}-\infty}$ values provided by the fitted $\beta$ and Eq. (13). Separately, in Fig. 1(d), $\lambda$ and $\eta$ are fitted for the ideal loss function ${\rm \gamma }\left({r,K^{{\rm SIA}} } \right)$. The prerequisite MD data for fitting is extracted from the simulation results by Byggmästar et al.[9] on SIA clusters and by Granberg et al.[10] on Vac clusters for bcc-Fe using the M07B potential.[19] The fitted values of the parameters are summarized in Table 1. As is seen, all the values of $\alpha$ and $\eta$ are larger than 1.0. This implies that the affected region by the annealing process goes beyond the liquid region. Besides these parameters, variables $n_{{\rm F}-\infty}$ and $r_{\rm c}$ are the case dependent inputs and determined by the energy of PKA. It is worthy to note that the parameterization implicitly depends on the definition of $r_{\rm c}$. Following the literature,[8,9] with a spherical approximation, $r_{\rm c}$ should be calculated according to a maximum cascade damaged volume within which atoms have average kinetic energies higher than $\frac{3}{2}k_{{\rm B}} T_{m}$ (i.e., local temperatures higher than the melting point $T_{m}$). Moreover, it is appropriate to truncate the integration domain ${\rm \Omega}$ (see Eqs. (11) and (12)) as a finite spherical zone with a radius of $4r_{\rm c}$, where the Gaussian rules sufficiently converge toward zero, as shown in Fig. 1. Additionally, split cascades could be decomposed into spherical-like sub-regions based on morphology analysis,[20] potentially extending the fitness of spherical approximation on liquid region size. However, this work focuses on spherical-like cascades, while coupling with the morphology decomposition will be studied in the future. To validate the current model, MD simulations were carried out using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS).[21] The atomistic interactions in bcc-Fe are described by the M07 potential,[19] with DFT-calibrated short-range interactions[12] (i.e., the M07B potential). Adaptive time steps were used to keep the time integration sufficiently accurate. The simulation blocks were first equilibrated using the NPT ensemble, and then each collision cascade was simulated using the NVE ensemble as initialized by imparting the velocity of a randomly selected atom according to a given PKA energy. Three PKA energies were considered, $T_{\rm d} = 2$, 4 and 8 keV, respectively. All simulations were conducted at an initial temperature of 50 K, which is lower than the temperature of stage $I_{\rm D2}$ recovery (107 K) in bcc-Fe,[22,23] so that equilibrium diffusion is prohibited. The produced defects are analyzed using Voronoi method, and the defect clusters are identified using OVITO[24] with a cutoff distance of 3.0 Å (between the second and third nearest neighbor distances of bcc-Fe). To obtain the case-dependent variables $n_{{\rm F}-\infty}$ and $r_{\rm c}$, single cascades in undamaged bcc-Fe are simulated, with a sampling space of 15 PKA directions and 5 PKA positions for statistics. Their values are listed in Table 2.
Table 2. The values of input variables $n_{{\rm F}-\infty}$ and $r_{\rm c}$ determined for bcc-Fe at 50 K. The MD simulations were conducted within cubic boxes, which contain $N\times N\times N$ conventional cells of bcc. The data in brackets are predicted by the arc-DPA model[25]. The error bar is given as the standard error, and the errors of $r_{\rm c}$ are below 10$^{-2}$ Å, hence not shown.
$n_{{\rm F}-\infty}$ $r_{\rm c}$(Å) $N$
PKA: 2 keV 10.5$\pm$0.2 (8.3) 10.9 64
PKA: 4 keV 16.7$\pm$0.4 (15.0) 13.1 64
PKA: 8 keV 27.8$\pm$0.8 (27.6) 15.9 80
cpl-37-1-016103-fig2.png
Fig. 2. Cluster size distribution of the LD and HD configurations. Scatter data are from MD and the dashed line is a fitted inversed power law.
Damage level is a primary factor affecting the probability of cascade overlapping. To illustrate the effect of damage level on defect production, two radiation damaged configurations with different damage levels, LD and HD, were prepared. As measured by the ratio between the number of Frankel pairs and the total number of atomic sites (221, 184 sites), the damage levels are $\zeta_{\rm F} \left({{\rm LD}} \right)=0.0027$ and $\zeta_{\rm F} \left({{\rm HD}} \right)=0.0036$, respectively. The size distributions of defect clusters in LD and HD configurations are shown in Fig. 2. As is seen, they closely follow a common inversed power law of $F\left({K^{\rm def}} \right)=0.754\left({K^{\rm def}} \right)^{-2.5}$ (by definition, the number density of defects is $\rho \left({K_{i}^{\rm def} } \right)=\frac{\zeta_{\rm F} F\left({K_{i}^{\rm def} } \right)}{\left({a^{3}/2} \right)\mathop \sum\nolimits_{K^{\rm def}=1}^{K^{\rm def}=K_{\max}^{\rm def} } K^{\rm def}F\left({K^{\rm def}} \right)}$ ($a =2.86$ Å)). Thus, the differences of the two configurations on defect production should be dominated by the damage level. The defect production by a new cascade in LD and HD were investigated by MD and the incremental model. MD simulations were performed by introducing a random PKA, and were sampled for 64 times for statistics. The incremental model used the fitted inversed power law for numerical simplicity. The calculated defect productions are given in Fig. 3(a). As is seen, the incremental model accurately reproduced the results of MD. The ${\rm \delta }n_{\rm F}$ is significantly smaller in HD configurations than that in LD configurations, as is expected due to higher probability of overlapping. Interestingly, ${\rm \delta }n_{\rm F}$ gradually becomes negative with PKA energy ($T_{\rm d}$), which means the pre-existing defects are partially erased by high energy PKAs. To understand the annealing mechanism behind the erasure, contributions from different annealing scenarios are plotted in Fig. 3(b). As is seen, the reduction mainly comes from the scenario $\rm {I\!I\!I}$ annealing process (i.e., the thermal diffusion-controlled recombination) and its contribution increases significantly with PKA energy. Deduced from our modelling, during irradiations with broad spectra of PKA energies (e.g., ion radiation), defects produced by low energy PKAs may be partially erased by high energy PKAs, and hence the primary radiation damage does not necessarily always increase upon cascades.
cpl-37-1-016103-fig3.png
Fig. 3. Defect production in LD and HD configurations: (a) comparison of the value of ${\rm \delta }n_{\rm F}$ predicted by the incremental model and calculated by MD simulations, (b) decomposition of the contributions on ${\rm \delta }n_{\rm F}$ from different annealing event scenarios.
The size distribution of the defect clusters is a secondary factor affecting the cascade overlapping, which is readily seen from Eqs. (8) and (9). During continuous irradiation, the defect clusters usually grow through clustering, changing the size distribution. The mechanism of clustering varies due to the dose rate effects. Under low dose rates, clustering is driven by diffusion,[26] while under the ultra-high dose rates (e.g., in MD), clustering is supposedly a direct result of collision cascades.[8,27] Nevertheless, sequential cascades in MD provide a chance to test our model on the coupled effects of the change of damage level and defect size distribution. The detailed mechanism of clustering, however, is out the scope of this work. We performed two MD simulations of sequential cascades, by PKAs with energies of 4 keV and 8 keV, respectively. Each cascade was initialized by a random PKA, followed by NVE ensemble simulation of cascade decay for at least 15 ps, then by NVT ensemble simulation of cooling down. The distributions of defect clusters still follow an inversed power law of $F\left({K^{\rm def}} \right)\sim \left({K^{\rm def}} \right)^{-\tau \left({{\rm DPA}} \right)}$ during the accumulation of radiation damage. However, the exponent of the inversed power law of SIAs gradually decreases as $\tau^{{\rm SIA}}\left({{\rm DPA}} \right)/\tau^{{\rm SIA}}\left(0 \right)=1-8.89{\rm DPA}$ ($\tau^{{\rm SIA}}\left(0 \right)=3)$ due to clustering, while exponent of the inversed power law for Vac clusters $\tau^{{\rm Vac}}\left({{\rm DPA}} \right)$ ($\tau^{{\rm Vac}}\left(0 \right)=2)$ hardly changes, as shown in Fig. 4(a). Informing the change of defect cluster size distribution to the incremental model, the evolution of damage level is predicted, and is shown in Fig. 4(b). As is seen, the MD results are satisfactorily reproduced by the model with an explicit consideration of defect clustering. In fictitious configurations with constant cluster size distribution, the damage level saturates too early, and is lower in values compared with the actual condition. This result suggests that clustering of defects may postpone the saturation of radiation damages. Conversely, the damage level should be easy to saturate provided with restrained clustering of defects, which is in agreement with the observations in high-entropy alloys.[13]
cpl-37-1-016103-fig4.png
Fig. 4. Evolution of radiation damage by sequential cascades. (a) Change of the inversed power exponent of the defect cluster distribution, dashed and dotted lines are fitted from the scatters of MD data. (b) Change of the radiation damage level $\zeta_{\rm F}$. The dash-dotted lines are predicted by the analytic incremental model with consideration of the change of defect cluster size distribution, thin solid lines are predicted damaged level with a constant cluster size distribution using the $\tau \left(0 \right)$ exponents, scatters are MD data. DPA dose is evaluated according to the NRT model.[2,3]
In conclusion, an analytic incremental defect production model is proposed based on decomposition of the annealing events during cascade overlapping. Applying the model, the defect production in damaged bcc-Fe is accurately predicted. It is observed that high energy PKAs are capable to partially erase previously introduced damages via diffusion-controlled recombination, and clustering of defects postpones radiation damage saturation. The present model has a minimal parameter set and can easily be applied to different damage conditions with varying damage levels and defect cluster size distributions. Hence the model could be coupled with OKMC or CD simulations for a better understanding of the radiation damage by continuous irradiation. The main limitation of the current model is that it is only applicable to sphere-like cascades. To extend the model into split-cascades due to high energy PKA, coupling with the sub-cascade decomposition analysis[20] appears to be necessary, and will be studied in the future. References The Displacement of Atoms in Solids by RadiationComputer simulation of atomic-displacement cascades in solids in the binary-collision approximationA proposed method of calculating displacement dose ratesPrimary radiation damage: A review of current understanding and modelsRadiation-annealing effects in energetic displacement cascadesComputer simulation study of cascade overlap effects in α-ironSecondary factors influencing cascade damage formationPoint defect movement and annealing in collision cascadesCollision cascades overlapping with self-interstitial defect clusters in Fe and WCascade overlap with vacancy-type defects in FeRadiation damage in tungsten from cascade overlap with voids and vacancy clustersEffects of the short-range repulsive potential on cascade damage in ironMechanism of Radiation Damage Reduction in Equiatomic Multicomponent Single Phase AlloysDamage accumulation in ion-irradiated Ni-based concentrated solid-solution alloysEmbrittlement of nuclear reactor pressure vesselsSimulation of radiation damage in Fe alloys: an object kinetic Monte Carlo approachSimulation of the nanostructure evolution under irradiation in Fe–C alloysOn the transfer of cascades from primary damage codes to rate equation cluster dynamics and its relation to experimentsComparison of empirical interatomic potentials for iron applied to radiation damage studiesA model of defect cluster creation in fragmented cascades in metals based on morphological analysisFast Parallel Algorithms for Short-Range Molecular DynamicsMultiscale modelling of defect kinetics in irradiated ironThe resistivity recovery of high purity and carbon doped iron following low temperature electron irradiationVisualization and analysis of atomistic simulation data with OVITO–the Open Visualization ToolImproving atomic displacement and replacement calculations with physically realistic damage modelsThe nanostructure evolution in Fe–C systems under irradiation at 560KDefect structures and statistics in overlapping cascade damage in fusion-relevant bcc metals
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