Governing equations of point defects and defect clusters

The general concept and approach of CD have been described in detail and validated in the literature [4-6, 12, 13]. The main considerations and important assumptions made in the current modeling framework are as follows:

(1) Mobile species include point defects such as self-interstitial atoms (SIA) and vacancies; small self-interstitial clusters contain two, three, and four SIAs. Only point defects can be emitted from clusters.

(2) Formation of small defect clusters (vacancy and interstitial clusters) directly from collision cascades.

(3) All mobile species exhibit 3D diffusion.

(4) For mobile interstitial clusters, the migration energy is constant and independent of their size.

(5) The pre-exponential factor of the diffusion coefficient of the interstitial clusters is the fitting parameter, which varies with the irradiation temperature and size of the interstitial clusters.

(6) Grain boundaries, dislocation lines, and surfaces are intrinsic sinks for mobile species.

The first four hypotheses in this model are based on the CD model described in detail in [12] and [13]. As stated in reference [13], the diffusion coefficient of interstitial clusters is calculated by the method in [18], the diffusion pre-exponential factor D_n0 decreases monotonically with cluster size n according to the power law D_n0=D₀n^-S. Ab initio research in [20] shows that as the temperature gradually increases from 200 K to 1600 K, the pre-exponential factor of the diffusion coefficient of Fe atoms in the fcc structure Ni-Fe alloy also increases. Therefore, we assume that the parameter S decreases with increasing temperature, such that the pre-exponential factor of the diffusion coefficient of small self-interstitial clusters also increases with increasing temperature; this is qualitatively consistent with the conclusion in [20]. The irradiation temperature reported in Ref. [1] are 300 ℃ and 400 ℃, while the temperature in Ref. [2] is 350 ℃. When the temperature is 300 ℃, S = 4.5, which is consistent with the value used in [13]. When the temperature is increased to 350 ℃ and 400 ℃, S is taken to be 4.3 and 4.0, respectively, through the conclusion in [20] and by fitting the calculated results with the experimental results.Grain boundary and dislocation lines are intrinsic sinks that capture mobile species [21]. A surface sink is added to the model to account for the surface effects of ion irradiation. For the sake of description, the vacancies and SIA are denoted by V and I, respectively, and small self-interstitial clusters containing two, three, and four SIAs are labeled as I₂, I₃, and I₄, respectively. In addition, n denotes the number of point defects in a defect cluster.

Based on the above assumptions, a series of governing equations are constructed to describe the evolution of dislocation loops. The governing equations in this model for SIA and vacancy point defects are structured as follows: $\begin{array}{c}\frac{\text{d}{C}_I}{\text{d}t}=G_{\text{irra}}^I+2g_{I_2}^I{C}_{I_2}+{\displaystyle \sum _{n=\mathrm{1,2,3}}{k}_{V_n}^{I_{n+1}}}{C}_{V_n}{C}_{I_{n+1}}\\ +{\displaystyle \sum _{n=3}^N{g}_{I_n}^I}{C}_{I_n}-k_I^V{C}_I{C}_V-2k_I^I{C}_I{C}_I-{\displaystyle \sum _{n=2}^N{k}_{I_n}^I}{C}_{I_n}{C}_I\\ -{\displaystyle \sum _{n=2}^N{k}_{V_n}^I}{C}_{V_n}{C}_I-K_d^I{D}_I\left(C_I-C_I^{\text{eq}}\right)\\ -K_{\text{gb}}^I{D}_I\left(C_I-C_I^{\text{eq}}\right)-K_{\text{sf}}^I{D}_I\left(C_I-C_I^{\text{eq}}\right)\end{array}$ (1) $\begin{array}{c}\frac{\text{d}{C}_V}{\text{d}t}=G_{\text{irra}}^V+2g_{V_2}^V{C}_{V_2}-k_I^V{C}_I{C}_V+{\displaystyle \sum _{n=3}^N{g}_{V_n}^V}{C}_{V_n}\\ -{\displaystyle \sum _{n=2}^N{k}_{I_n}^V}{C}_{I_n}{C}_V-2k_V^V{C}_V{C}_V-{\displaystyle \sum _{n=2}^N{k}_{V_n}^V}{C}_{V_n}{C}_V\\ +{\displaystyle \sum _{n=1,2,3,4}{k}_{V_{\left(n+1\right)}}^{I_n}}{C}_{I_n}{C}_{V_{\left(n+1\right)}}-K_d^V{D}_V\left(C_V-C_V^{eq}\right)\\ -K_{\text{gb}}^V{D}_V\left(C_V-C_V^{\text{eq}}\right)-K_{\text{sf}}^V{D}_V\left(C_V-C_V^{\text{eq}}\right)\end{array}$ (2) The form of the governing equations describing the change in the concentrations of interstitial clusters In (n=2,3,4) and vacancy clusters Vn (n=2,3,4) is $\begin{array}{c}\frac{\text{d}{C}_{I_n}}{\text{d}t}=G_{\text{irra}}^{I_n}-g_{I_n}^I{C}_{I_n}+g_{I_{n+1}}^I{C}_{I_{n+1}}-K_{\text{sf}}^{I_n}{D}_{I_n}{C}_{I_n}\\ +{\displaystyle \sum _{m=\mathrm{1,}m+n\le 4}{k}_{V_m}^{I_{m+n}}}{C}_{I_{m+n}}{C}_{V_m}-{\displaystyle \sum _{m=1}^N{k}_{V_m}^{I_n}}{C}_{V_m}{C}_{I_n}\\ +{\displaystyle \sum _{m=\mathrm{1,2}m\le n}{k}_{I_m}^{I_{n-m}}}{C}_{I_m}{C}_{I_{n-m}}-K_d^{I_n}{D}_{I_n}{C}_{I_n}\\ -K_{\text{gb}}^{I_n}{D}_{I_n}{C}_{I_n}-\left({\displaystyle \sum _{m=1}^N{k}_{I_m}^{I_n}}{C}_{I_m}{C}_{I_n}+k_{I_n}^{I_n}{C}_{I_n}{C}_{I_n}\right)\end{array}$ (3) $\begin{array}{c}\frac{\text{d}{C}_{V_n}}{\text{d}t}=G_{\text{irra}}^{V_n}-g_{V_n}^V{C}_{V_n}+g_{V_{n+1}}^V{C}_{V_{n+1}}-k_{V_n}^V{C}_{V_n}{C}_V\\ +k_{V_{n-1}}^V{C}_{V_{n-1}}{C}_V-{\displaystyle \sum _{m=1}^4{k}_{V_n}^{I_m}}{C}_{V_n}{C}_{I_m}\\ +{\displaystyle \sum _{m=1}^4{k}_{V_{m+n}}^{I_m}}{C}_{V_{m+n}}{C}_{I_m}\end{array}$ (4) Disregarding the irradiation cascade generation term $G_{\text{irra}}^{V_n}$ in Eq. (4), a general form of the governing equations for vacancy clusters with sizes greater than four can be obtained. The governing equation for immobile dislocation loops with a size greater than four is as follows: $\begin{array}{c}\frac{\text{d}{C}_{I_n}}{\text{d}t}={\displaystyle \sum _{m=\mathrm{1,2}m\le n}{k}_{I_m}^{I_{n-m}}}{C}_{I_m}{C}_{I_{n-m}}-{\displaystyle \sum _{m=1}^4{k}_{I_n}^{I_m}}{C}_{I_n}{C}_{I_m}\\ +k_{I_{n+1}}^V{C}_{I_{n+1}}{C}_V-k_{I_n}^V{C}_{I_n}{C}_V+g_{I_{n+1}}^I{C}_{I_{n+1}}-g_{I_n}^I{C}_{I_n}.\end{array}$ (5) Specifically, CI and CV are the concentrations of SIA and vacancy point defects, respectively, whereas $C_I^{\text{eq}}$ and $C_V^{\text{eq}}$ are the thermal equilibrium concentrations of SIA and vacancies, respectively. The density of interstitial clusters/loops comprising n SIAs is denoted by $C_{I_n}$, and that of vacancy clusters is denoted by $C_{V_n}$. The thermal equilibrium concentrations of the dislocation loops $C_{I_n}^{\text{eq}}\left(n\ge 2\right)$ and vacancy clusters $C_{V_n}^{\text{eq}}\left(n\ge 2\right)$ are zero, $G_{\text{irra}}^{I_n}$ and $G_{\text{irra}}^{V_n}$ are generation rates of small self-interstitial clusters (n≤4) and small vacancy clusters (n≤4), respectively, directly from intra-cascade clustering [2, 6, 22]. $k_{I_n}^{\theta }$ and $k_{V_n}^{\theta }$ are the rate coefficients of the dislocation loops and vacancy clusters of size n, respectively, that absorb mobile defect θ. Dislocation loops and vacancy clusters can emit only SIAs and vacancy point defects, respectively. $g_{I_n}^I$ is the rate at which dislocation loops emit SIA, and $g_{V_n}^V$ is the rate at which vacancy clusters emit vacancies. DI and DV are the diffusion coefficients of the SIAs and vacancies, respectively, with $D_{I_n}$ being the diffusion coefficient of the mobility interstitial clusters. $K_{\text{d}}^{\theta }$, $K_{\text{gb}}^{\theta }$ and $K_{\text{sf}}^{\theta }$ are the sink strengths of dislocation lines, grain boundary, and surfaces, respectively. The formalisms of the rate coefficient and sink strength etc. is discussed in detail in Sect. 2.2.

The model described in Sect. 2.1 corresponds to Model-4 in this study, which represents an mobile interstitial cluster with a maximum size of four. In this study, to analyze the influence of mobile interstitial clusters on the density and size of dislocation loops predicted by the CD model, we construct three types of cluster dynamics models: those in which only a single SIA can move, those in which at most two interstitial clusters can move, and those in which interstitial clusters of size three can move; these cluster dynamics models correspond to Model-1, Model-2, and Model-3, respectively. For vacancy clusters, Model-1 to Model-4 assume that only vacancies can move. The input parameters, such as the irradiation and material parameters, are the same for Model-1 to Model-4, as shown in Table 1 and Table 2 in Sect. 2.4. The capture of mobile defects by dislocation lines, grain boundaries, and surface sinks has been considered in the four models. The main difference between the models is the size of the maximum mobile interstitial cluster. As the maximum mobile interstitial cluster gradually increases, the generation and disappearance terms required in the governing equations describing the evolution of defects of a certain size in different models change accordingly. All models assume that mobile defects undergo 3D diffusion; that is, the rates of mutual reactions between defects are calculated according to the 3D-3D expression. For brevity, Section 2 only introduces the construction of Model-4 in detail; the other three models are similar to Model-4.

Rate coefficients

The generation rates of defects from the in-cascade are taken from Ref. [6] which considered the formation of clusters with sizes greater than four unlikely. The defect-generation terms are as follows: $\begin{array}{l}{G}_{\text{irra}}^I=\eta G_{\text{dpa}}\left(1-f_{I_2}-f_{I_3}-f_{I_4}\right)\\ G_{\text{irra}}^{I_n}=\frac{\eta G_{\text{dpa}}{f}_{I_n}}{n}\left(n\le 4\right)\\ G_{\text{irra}}^V=\eta G_{\text{dpa}}\left(1-f_{V_2}-f_{V_3}-f_{V_4}\right)\\ G_{\text{irra}}^{V_n}=\frac{\eta G_{\text{dpa}}{f}_{V_n}}{n}\left(n\le 4\right)\end{array}$ (6) G_dpa denotes the damage rate under irradiation and η is the cascade efficiency. $f_{I_n}$ and $f_{V_n}$ are the generation fractions of interstitial clusters and vacancy clusters of size n that survive the reorganization events following the cascade, respectively.

$k_I^V$ is the characteristic annihilation rate of the SIA and vacancy point defects, which can be expressed as: $\begin{array}{l}{k}_I^V=4\pi r_{IV}\left(D_I+D_V\right)\hfill \end{array}$ (7) where rIV is the recombination radius.

By adopting the formalism in Ref. [23], the rate coefficients of dislocation loops and vacancy clusters that absorb mobile defects are: $\begin{array}{ll}{k}_{I_n}^{\theta }=2\pi \left(D_{I_n}+D_{\theta }\right)\left(r_{I_n}+r_{\theta }\right)Z_{I_n}^{\theta }\hfill & \left(\theta =I\mathrm{,}V\right),\hfill \end{array}$ (8) $\begin{array}{cc}{k}_{V_n}^{\theta }=4\pi \left(D_V+D_{\theta }\right)\left(r_{V_n}+r_{\theta }\right)& \left(\theta =I\mathrm{,}V\right).\end{array}$ (9) In Eq. (8), $Z_{I_n}^{\theta }$ is the bias factor of the dislocation loops, which can be calculated using the method described in [5]. In particular, the diffusion coefficient $D_{I_n}$ is calculated using the method described in Ref. [18]: $\begin{array}{l}{D}_{I_n}=D_0{n}^{-s}\exp \left(-\frac{E_m^I}{k_{\text{B}}T}\right)\left(n\le 4\right)\hfill \end{array}$ (10) The rate coefficients for the emission of point defect SIAs and vacancies by dislocation loops and vacancy clusters are [24] $g_{I_n}^I=\frac{k_{I_{n-1}}^I}{V_{\text{at}}}\exp \left(-\frac{E_{I_n}^{\text{b}}}{k_{\text{B}}T}\right)$ (11) $g_{V_n}^V=\frac{k_{V_{n-1}}^V}{V_{\text{at}}}\exp \left(-\frac{E_{V_n}^{\text{b}}}{k_{\text{B}}T}\right)$ (12) The binding energies $E_{I_n}^{\text{b}}$ and $E_{V_n}^{\text{b}}$ in Eqs. (11) and (12) can be determined using the following extrapolation law [25] $E_{I_n}^{\text{b}}=E_I^{\text{f}}+\frac{E_{I_2}^{\text{b}}-E_I^{\text{f}}}{2^{\frac{2}{3}}-1}\left(n^{2/3}-{\left(n-1\right)}^{2/3}\right)$ (13) $E_{V_n}^{\text{b}}=E_V^{\text{f}}+\frac{E_{V_2}^{\text{b}}-E_V^{\text{f}}}{2^{\frac{2}{3}}-1}\left(n^{2/3}-{\left(n-1\right)}^{2/3}\right)$ (14) where $E_I^{\text{f}}$ and $E_V^{\text{f}}$ are the formation energies of the SIAs and vacancies, respectively. $E_{I_2}^{\text{b}}$ and $E_{V_2}^{\text{b}}$ are the binding energies for the interstitial and vacancy clusters of size two, respectively.

The sink strength of dislocation lines is $K_{\text{d}}^{\theta }=Z_{\text{D}}^{\theta }\rho {}_{\text{dis}}\left(\theta =I\mathrm{,}V\right)$ (15) where $Z_{\text{D}}^{\theta }$ is the bias for the dislocation lines absorbing SIA or vacancies. $Z_{\text{D}}^I=1.1$ and $Z_{\text{D}}^V=1.0$ are adopted from [6]. ρ_dis is the density of the dislocation lines, assuming that the density of dislocation lines ρ_dis remains unchanged during irradiation.

The expressions for the sink strengths of the grain boundary and surfaces were adopted in [26]. They can be given as: $K_{gb}^{\theta }=\frac{6\sqrt{S_m^{\theta }}}{d_{gb}}\left(\theta =I\mathrm{,}V\right)$ (16) $K_s^{\theta }=\frac{\sqrt{s_m^{\theta }}}{d}\frac{1}{\text{coth}\left(\sqrt{s_m^{\theta }}d\right)-\frac{1}{\sqrt{s_m^{\theta }}d}}\left(\theta =I\mathrm{,}V\right)$ (17) where d_gb is the grain size and d is the half thickness of the thin foil. $S_m^{\theta }$ represents the total sink strength of the medium without surfaces. Similar to [4], the bias of the dislocation loops to the interstitial clusters I₂, I₃ and I₄ is $Z_{I_n}^{I_2}=Z_{I_n}^{I_3}=Z_{I_n}^{I_4}=Z_{I_n}^I$, the bias of the dislocation lines to the interstitial clusters I₂, I₃ and I₄ is $Z_{\text{D}}^{I_2}=Z_{\text{D}}^{I_3}=Z_{\text{D}}^{I_4}=Z_{\text{D}}^I$, the sink strength of the grain boundary and surface to the interstitial clusters I₂, I₃ and I₄ are $k_{\text{gb}}^{I_2}=k_{\text{gb}}^{I_3}=k_{\text{gb}}^{I_4}=k_{\text{gb}}^I$ and $k_{\text{sf}}^{I_2}=k_{\text{sf}}^{I_3}=k_{\text{sf}}^{I_4}=k_{\text{sf}}^I$.

The coupled equation, Eq. (1)- (5), along with the input parameters detailed in Sect. 2.4, are considered in the evolution system of dislocation loops involving the largest interstitial clusters I_(n=117000). Thousands of equations must be solved by using this system, which requires a significant amount of simulation time. The Grouping method [30] and the Fokker-Planck method [4-6] have been used to reduce the number of equations to be solved in the system to reduce the simulation time. Parallel-solution methods reduce the simulation time of the system [31]. In this study, the discrete rate equations are transformed into Fokker-Planck equations to improve simulation efficiency. Its specific form is described in Sect. 2.3.

Fokker-Planck method

The dislocation loop evolution system described by the CD model contains a large number of ordinary differential equations(ODEs), which need to be solved. One of the main reasons for the increase in the simulation time of the cluster dynamics is the number of ODEs in the model; the more equations there are, the longer the CD simulation requires. In addition, the stiffness of the equations is also a major factor that increases the time and complexity of solving CD models. The simulation efficiency can be improved by reducing the number of equations in the system and selecting an appropriate ODE solver. The Fokker-Planck(F-P) method has been applied to reduce the number of ordinary differential equations in cluster dynamics models [4-6, 27, 32]. When the F-P method is applied, the part of the dislocation loop evolution system described by the discrete rate equations is called the discrete part, and the part that transforms the discrete rate equation into the F-P equation through Taylor expansion is the continuous part [32]. In this study, we assume that the interstitial and vacancy clusters in the discrete part range from I_n=1 to I_n=100 and $V_{n=1}$ to $V_{n=100}$, when n > 100; that is, the equations within the continuous part, Eqs. (4) and (5) can be transformed into F-P equations as follows: $\frac{\partial C\left(x\mathrm{,}t\right)}{\partial t}=-\frac{\partial }{\partial x}\left\{F\left(x\mathrm{,}t\right)C\left(x\mathrm{,}t\right)-\frac{\partial }{\partial x}\left[D\left(x\mathrm{,}t\right)C\left(x\mathrm{,}t\right)\right]\right\}$ (18) where F(x, t) is the drift term related to the growth in cluster size x.D(x, t) is the diffusion term related to the nucleation of clusters [32].

For interstitial clusters, the drift term $F_I\left(x\mathrm{,}t\right)$ and the diffusion term DI(x, t). $F_I\left(x\mathrm{,}t\right)={\displaystyle \sum _{m=\mathrm{1,2,3,4}}m{k}_{I_x}^{I_m}}{C}_{I_m}-g_{I_x}^I-k_{I_x}^V{C}_V$ (19) $D_I\left(x\mathrm{,}t\right)=\frac{1}{2}\left[{\displaystyle \sum _{m=\mathrm{1,2,3,4}}{m}^2{k}_{I_x}^{I_m}}{C}_{I_m}+g_{I_x}^I+k_{I_x}^V{C}_V\right]$ (20) For vacancy clusters, $F_V\left(x\mathrm{,}t\right)=k_{V_x}^V{C}_{V_x}{C}_V-{\displaystyle \sum _{m=\mathrm{1,2,3,4}}m{k}_{V_x}^{I_m}}{C}_{V_x}{C}_{I_m}-g_{V_n}^V$ (21) $D_V\left(x\mathrm{,}t\right)=\frac{1}{2}\left[{\displaystyle \sum _{m=\mathrm{1,2,3,4}}{m}^2{k}_{V_x}^{I_m}}{C}_{I_m}+g_{V_n}^V+k_{V_x}^V{C}_V\right]$ (22) Eq. (18) can be discretized into the following form by using the central difference method [33] $\begin{array}{c}\frac{\partial C\left(x_i\mathrm{,}t\right)}{\partial t}=\frac{2}{\Delta x_{i+1}+\Delta x_i}[\frac{1}{2}\left(F\left(x_{i-1}\mathrm{,}t\right)C\left(x_{i-1}\mathrm{,}t\right)-F\left(x_{i+1}\mathrm{,}t\right)C\left(x_{i+1}\mathrm{,}t\right)\right)\\ (\frac{D\left(x_{i+1}\mathrm{,}t\right)C\left(x_{i+1}\mathrm{,}t\right)-D\left(x_i\mathrm{,}t\right)C\left(x_i\mathrm{,}t\right)}{\Delta x_{i+1}}\\ -\frac{D\left(x_i\mathrm{,}t\right)C\left(x_i\mathrm{,}t\right)-D\left(x_{i-1}\mathrm{,}t\right)C\left(x_{i-1}\mathrm{,}t\right)}{\Delta x_i})]\end{array}$ (23) In Eq. (23), subscript i is the index of the divided grid point, the dislocation loop size at index i is denoted xi, and the relationship between the dislocation loop size at index i and the dislocation loop size at index i+1 is $x_{\left(i+1\right)}=x_i+\Delta x_i$, where Δxi is the step size of the grid at index i. To reduce the number of equations, it is necessary to divide the nonuniform grid, that is, Δxi=1.01Δx_(i-1). Using the F-P method, the number of F-P equations describing the evolution of the dislocation loop density in the continuous part is 750, and the number of F-P equations describing the evolution of the vacancy cluster density in the continuous part is 900. Combined with the number of discrete rate equations, only 1850 equations must be solved in the system to describe the evolution of dislocation loops with a maximum radius of 42 nm. The accurate discrete rate equations require hundreds of thousands of equations to describe the evolution of dislocation loops; therefore, the F-P method significantly reduces the number of equations in the system and improves the efficiency of the solution. The numerical solution of this dislocation loops evolution system is calculated using the Fortran wrappers of ODEPACK algorithms, the ODE solver such as CVODE in which is suitable for solving stiffness problems [34].

Input parameters

The HR3 austenitic stainless steel was irradiated with Fe⁺ in Ref. [1] The SA304 and CW316 stainless steels were irradiated with Fe⁺ in Ref. [2]. The irradiation and material parameters from the two studies can be used as input parameters for the CD model in this study, as shown in Tables 1 and 2.

Effect of mobile interstitial clusters on dislocation loops evolution

Suppose that the size of the largest mobile interstitial cluster in the CD model gradually increases from one to four, corresponding to Model-1, Model-2, Model-3 and Model-4 respectively. Figure 1 shows the dislocation loop size distribution simulated by the four models at a dose of 0.145 dpa under the same irradiation conditions.

Fig. 1Full-screen View

(Color online) Dislocation loops size distribution simulated by Model-1 to Model-4, and modeling at G_dpa=2.9×10^-4 dpa/s, t = 500 s, T = 350 ℃

The ordinate in Fig. 1 is the logarithm of the dislocation loop density; therefore, the lowest density of dislocation loops cannot be considered as zero. In this study, we use $C_{\text{loop}}=10\times {10}^{-11}{\text{ cm}}^{-3}\approx 0{\text{ cm}}^{-3}$ for the plot. However, in the CD model, similar to [4], we set the number of evolution equations describing the dislocation loops and vacancy clusters sufficiently large to ensure that the integral value describing the maximum defect cluster in the system is zero. The blue, red dotted, orange, and purple curves shown in Fig. 1 show the dislocation loop size distribution curves predicted by Model-1, Model-2, Model-3 and Model-4, respectively. The blue vertical dashed line indicates the dislocation loops radius R_loop=0.478 nm corresponding to the peak dislocation loops density predicted by Model-1; Model-2 and Model-3 predicted dislocation loops peak densities corresponding to the same radius R_loop=0.524 nm. The dislocation loops radius corresponding to the dislocation loops peak densities predicted by Model-2 and Model-3 is marked with an orange vertical dashed line. The purple vertical dashed line marks the dislocation loop size R_loop=2.5 nm corresponding to the peak dislocation loop density predicted by Model-4. Figure 1 shows that under the same irradiation conditions, when the dislocation loops average radius R_loop<0.478 nm, Model-1 predicts that the dislocation loops density gradually increases, and when R_loop=0.478 nm, the dislocation loops density reaches the peak $C_{\text{loop}}^{\text{peak}}=5.9\times 17{\text{ cm}}^{-3}$, when the dislocation loops density predicted by Model-1 is zero, R_loop=1.8 nm, the total number density of dislocation loops predicted by Model-1 is $C_{\text{loop}}=1.3\times 19{\text{ cm}}^{-3}$. When the dislocation loop radius R_loop<0.524 nm, the dislocation loop density predicted by Model-2 and Model-3 increases slowly, and the dislocation loop density reaches the peak values of $C_{\text{loop}}^{\text{peak}}=2.09\times 17{\text{ cm}}^{-3}$ and $C_{\text{loop}}^{\text{peak}}=1.57\times 17{\text{ cm}}^{-3}$, respectively, at R_loop=0.524 nm. When the predicted value of the dislocation loops density of Model-2 is zero, R_loop=2.1 nm, and when the predicted value of the dislocation loops density of Model-3 is zero, R_loop=2.25 nm. The total densities of dislocation loops predicted by Model-2 and Model-3 are $C_{\text{loop}}=7.22\times 18{\text{ cm}}^{-3}$ and $C_{\text{loop}}=6.43\times 18{\text{ cm}}^{-3}$. In the prediction of Model-4, when R_loop<2.5 nm, the dislocation loop density continues to increase. At R_loop=2.5 nm, the dislocation loop density reaches the peak value $C_{\text{loop}}^{\text{peak}}=4.28\times 13{\text{cm}}^{-3}$. At R_loop=3.9 nm, the dislocation loop density is zero. Model-4 predicts that the number density of dislocation loops is $C_{\text{loop}}=3.22\times 15{\text{ cm}}^{-3}$. A comparison of the simulation results of the four models is shown in Fig. 1; this comparison shows that, under the same irradiation conditions, as the mobile interstitial cluster gradually increases from SIA to interstitial cluster I₄, the peak $C_{\text{loop}}^{\text{peak}}$ of the dislocation loop density and the number of dislocation loop densities predicted by Models-1 to Models-4 start to decrease, and the radius corresponding to the peak dislocation loop density increases. By comparing the radius corresponding to a dislocation loop density of zero, we can see that the distribution of the dislocation loop size predicted by Model-1 to Model-4 becomes wider and trending towards a larger size.

The curves in Fig. 2 and Fig. 3 show a comparison of the number density and the average diameter of the dislocation loops when the four models are simulated to 10 dpa, respectively. The scatter values are the TEM observations of the number density and average diameter of the dislocation loops, respectively, in Ref. [2].

Fig. 2Full-screen View

(Color online) Comparison of the average loop number density at 10 dpa simulated by Model-1 to Model-4 with the experimental data in [2]. Modeling at G_dpa=2.9×10^-4 dpa/s, T = 350 ℃

Fig. 3Full-screen View

(Color online) Comparison of the average loop diameter at 10 dpa simulated by Model-1 to Model-4 with the experimental data in [2]. Modeling at G_dpa=2.9×10^-4 dpa/s, T = 350 ℃

Model-1 in Fig. 2 predicts a blue curve corresponding to the highest dislocation loop density. The dislocation loop densities predicted by Model-2 and Model-3 are indicated by red dashed and orange curves, respectively. Figure 2 shows that the dislocation loop densities predicted by Model-2 and Model-3 are almost identical, with a very small difference. In fact, the dislocation loop density predicted by Model-2 is higher than that predicted by Model-3. Compared with the other three models, Model-4 predicts the lowest dislocation loop density. In general, with an increase in the largest mobile interstitial cluster size, the number density of the dislocation loops simulated from Model-1 to Model-4 decreases. With an increase in the radiation dose, the dislocation loop density predicted by the four models gradually increased, and the growth rate of the dislocation loop density decreased; that is, the dislocation loop density tended to saturate with the increase in the dose.

In contrast to Fig. 2, compared with the other three models, the average diameter of dislocation loops predicted by Model-4 in Fig. 3 is the highest, whereas that predicted by Model-1 is the lowest. Similar to Fig. 2, the average diameters of the dislocation loops predicted by Model-3 and Model-2 were almost the same, but the average diameter of the dislocation loops predicted by Model-3 was larger than that predicted by Model-2. Therefore, in Fig. 3, the average diameter of the dislocation loops simulated from Model-1 to Model-4 increased with the increasing cluster size of the largest mobile interstitial cluster. As the irradiation dose increased, the average diameter of the dislocation loops simulated by Model-1 to Model-3 tended to stabilize after 1 dpa, with almost no growth. However, the average diameter of the dislocation loops predicted by Model-4 still has a high growth rate after 1 dpa, and does not tend to saturate even at 10 dpa; the predicted value of Model-4 is also much larger than the predicted values of the other three models.

The simulation results shown in Figs. 1, 2 and 3 can be attributed to three reasons: on the one hand, the dislocation loop number density obtained by Model-4 in Fig. 2 is the smallest; however, the average diameter of the dislocation loops obtained using Model-4 in Fig. 3 is the largest. In addition, the dislocation loop size distribution simulated by Model-1 to Model-4 in Fig. 1 evolves to a larger size at 0.145 dpa. These results indicate that increasing the size of mobile interstitial clusters can promote dislocation loop growth. In fact, the molecular dynamics simulation in [8] demonstrates that interstitial clusters can move, and the experiment in [1] demonstrates that interstitial clusters can migrate and merge to promote dislocation loop growth. Therefore, the increase in the size of the mobile interstitial clusters may be the main reason for the increase in the size of the dislocation loops simulated in Model-4. Second, with the increase in the size of mobile interstitial clusters, they are captured by dislocation loops, vacancy clusters, surfaces, grain boundaries, and dislocation line sinks, which can explain the density of dislocation loops obtained from Model-1 to Model-4 in Fig. 2 decreases with an increase in the size of the mobile interstitial clusters. Third, because the largest in-cascade interstitial cluster is assumed to be four in this study, when the mobile interstitial cluster in the CD model is less than three, an in-cascade interstitial cluster greater than three is still generated; however, an in-cascade interstitial cluster greater than three can’t be diffused and migrated. These non-migratory clusters themselves produce an accumulation, resulting in an increase in the density of dislocation loops, and also serve as a sink for mobile interstitial clusters, thus inhibiting the growth of dislocation loops to a large size distribution. This explains the reasons that lead to the almost less than 2.25 nm radius distribution of dislocation loops predicted by Model-1 to Model-3 in Fig. 1 and the average diameter of the dislocation loops predicted by Model-1 to Model-3 in Fig. 3, which consistently approximates 2 nm.

The density and average diameter of the dislocation loops in SA304 and CW316 steels irradiated by Fe⁺ ions observed by TEM in [2] are compared with the simulation data of Model-1 to Model-4. The experimental results agree well with the simulation results of Model-4. The reasons for the better simulation results of Model-4 may be summarized in two points. First, the fraction of in-cascade interstitial clusters obtained in [22] shows that there are few in-cascade interstitial clusters with a size greater than four; therefore, it is reasonable to assume that the maximum mobile interstitial cluster size is four. In addition, as mentioned previously, the mobility of the interstitial clusters in the model may be the main reason why the simulation results of Model-4 are in good agreement with the experimental results. The experimental results agree well with the simulation results of Model-4, indicating that it is reasonable to consider the mobility of the interstitial clusters in the CD model.

Effect of temperature on mobile interstitial clusters

In this section, the CD model(Model-4) constructed in this study is used to simulate the evolution of dislocation loops in HR3 steel, as observed during in-situ electron microscopy in [1]. Figures 4 and 5 show comparisons between the CD predictions and the experimental data for the number density and average diameter of dislocation loops at 300 ℃ and 400 ℃, respectively.

Fig. 4Full-screen View

(Color online) Comparison of the loop number density at 6 dpa simulated by CD with the experimental data in [1] at different temperatures

Fig. 5Full-screen View

(Color online) Comparison of the average loop diameter at 6 dpa simulated by CD with the experimental data in [1] at different temperatures

The experimental values of the number density and average diameter of the dislocation loops shown in Figs. 4 and 5 are affected by both the irradiation temperature and irradiation dose. The experimental values in Fig. 4 demonstrate that, as the temperature increases, the number density of dislocation loops decreases. This is because high temperatures promote the growth and aggregation of dislocation loops, while limiting the nucleation of new dislocation loops [1]. As mentioned in the model description, the pre-exponential factor of the diffusion coefficient of mobile interstitial clusters in this paper varies with the size of mobile interstitial clusters n and temperature T. Based on this hypothesis, T = 300 ℃, S = 4.5, T = 400 ℃, S = 4.0. With the increase in temperature, the value of S decreases and the pre-exponential factor of the diffusion coefficient of interstitial clusters increases; thus, the diffusivity of interstitial clusters increases. Enhancing the diffusion ability of interstitial clusters can promote the migration and accumulation of dislocation loops.

The CD model in Fig. 4 predicts the number density of dislocation loops under irradiation conditions of 300 ℃ and 400 ℃, respectively. With increasing temperature, the predictions of the CD model remain consistent with the experimental data. Similarly, the average diameters of the dislocation loops simulated using the CD model, as shown in Fig. 5, are in good agreement with the experimental data, and the average diameter of the dislocation loops increases with increasing temperature.

As shown in Fig. 4, the experimental value of the dislocation loop density reaches saturation at 0.5 dpa and begin to decrease with increasing radiation doses. Ref. [1] assumed that the increase of irradiation dose promotes the merging of dislocation loops and formation of network dislocations, which limits the increase in the number density of dislocation loops during irradiation. Stoller constructed an evolution model of network dislocation under irradiation, which means that the variation in network dislocation density due to the possible growth and unfaulting of dislocation loops, activation of Bardeen-Herring sources, and irradiation-enhanced climb of dislocations have been considered in [35]. The CD model in this study does not consider the network dislocation evolution, i.e., the density of network dislocations is constant. Therefore, as the irradiation dose increases, the CD model predicts that the number density of dislocation loops slowly increases and tends to saturate. This trend differs slightly from that of the experimental data for the number density of dislocation loops. In addition, the CD predictions of the average diameters of the dislocation loops still tend to grow toward larger sizes. The dislocation loop number density is slightly larger than the experimental values shown in Fig. 4 for CD simulations up to 5 dpa and 6 dpa; this may have contributed to the larger average diameter of the dislocation loops in Fig. 5 for CD predictions with irradiation doses of up to 5 dpa. Adding a network dislocation evolution mechanism to the CD model should be considered in future studies.