Point defect dynamics in bcc metals

We present an analysis of the time evolution of self-interstitial atom and vacancy (point defect) populations in pure bcc metals under constant irradiation flux conditions. Mean-field rate equations are developed in parallel to a kinetic Monte Carlo (kMC) model. When only considering the elementary processes of defect production, defect migration, recombination and absorption at sinks, the kMC model and rate equations are shown to be equivalent and the time evolution of the point defect populations is analyzed using simple scaling arguments. We show that the typically large mismatch of the rates of interstitial and vacancy migration in bcc metals can lead to a vacancy population that grows as the square root of time. The vacancy cluster size distribution under both irreversible and reversible attachment can be described by a simple exponential function. We also consider the effect of highly mobile interstitial clusters and apply the model with parameters appropriate for vanadium and $\alpha$-iron.

Figure 1

Self-interstitial (∎), $n_i$, and vacancy (◇), $n_v$, densities as a function of time for $\Gamma ={10}^2,{10}^3,{10}^4,{10}^5$, and ${10}^6$ ($\Gamma$ increases from the top to bottom of the figure). Time is measured in units of the inverse interstitial hopping rate $a^2∕D_i$. The solid lines show the result of direct numerical integration of Eqs. (1) with ${\kappa }_v={\kappa }_i={\kappa }_{is}={\kappa }_{vs}=21$, and the symbols correspond to the results of the kMC simulations in a periodic simulation box. The straight solid lines have slope 1 and $1∕2$.

Figure 2

Interstitial densities $n_i$ (∎), free vacancy density $n_v$ (◇) as well as the total vacancy density $n_{v\mathrm{tot}}$ (*) and vacany cluster density $n_c$ (▴) as a function of time for (a) $\Gamma ={10}^5$, (b) $\Gamma ={10}^7$, and (c) $\Gamma ={10}^9$. The thin solid lines show the result of direct numerical integration of Eqs. (13) with ${\kappa }_i^m={\kappa }_v^m=25$ and a $m_{\max }=20$ (see text).

Figure 3

Plot of the cluster size distribution $n_c\left(m\right)$ found from the kMC simulations with $\Gamma ={10}^5$ of Fig. 2a at four different times $t{D}_i∕a^2=3\times {10}^4$ (∎), $3\times {10}^5$ (◆), $3\times {10}^6$ (*), and $3\times {10}^7$ (▴) in regimes III and IV. The straight lines correspond to $n_v{\left(t\right)}^m∕{\left[\alpha \sqrt{\beta }{n}_i\left(t\right)\right]}^{m-1}$.

Figure 4

(a) Interstitial density $n_i$ (∎), free vacancy density $n_v$ (◇) as well as the total vacancy density (▴) and vacancy cluster density $n_c$ (*) as a function of time for $\Gamma ={10}^5$, and ${10}^7$ for reversible aggregation, where $\delta =0.01$. (b) Cluster size distribution $n_c\left(m\right)$ for the case of $\Gamma ={10}^5$ at four different times in regimes III and IV. The straight lines correspond to $n_v{\left(t\right)}^m∕{[\alpha \sqrt{\beta }{n}_i\left(t\right)+{\kappa }_d^1∕{\kappa }_i^1{\alpha }^{\prime }]}^{m-1}$ with ${\kappa }_d^1∕{\kappa }_i^1=0.25$.

Figure 5

Interstitial densities $n_i$ (∎), free vacancy density $n_v$ (◇), total vacancy density $n_{v\mathrm{tot}}$ (▴), vacany cluster density $n_c$ (*), and density of immobile interstitial cluster (◻) as a function of time for $\Gamma ={10}^6$.

Figure 6

Interstitial, vacancy, interstitial cluster, and vacancy cluster densities for $\Gamma ={10}^5$ and $\Gamma ={10}^7$ (symbols as in Fig. 5). Interstitial clusters (◻) diffuse and rotate with the same rate as single interstitials. The interstitial cluster density first rises due to nucleation events, but then rapidly drops as mobile clusters collide with sinks.

Figure 7

Interstitial, vacancy, interstitial cluster, and vacancy cluster densities (symbols as in Fig. 5) with parameters ${\alpha }^{\prime }={10}^2$, $n_s={10}^{-5}$, for (a) $\Gamma ={10}^9$, (b) $\Gamma ={10}^{11}$, and (c) $\Gamma ={10}^{13}$ representative for V or $\alpha \text{−}\mathrm{Fe}$ at 600 K. Thin solid lines show the result of numerically integrating Eqs. (13) again for ${\kappa }_m=25$ and $m_{\max }=20$. The solid lines have slope 1 and $1∕2$, respectively.