## 2 Materials and Methods
### Model of Dislocation-Precipitate Interaction  
We use the model of dislocation motion in pure aluminum and dislocation-precipitate interaction offered in [34; 57; 58]. This model assumes that the dislocation overcomes precipitate forming the Orowan loop with energy losses for formation of additional dislocation segments. Description of free motion of dislocation between the interactions with precipitates and motion of segments during the Orowan loop formation takes into account the phonon friction. This model does not consider the cross-slip or climb of segments into an adjacent plane, since these processes do not significantly affect the stress to overcome the precipitate at single interaction. Figure 3 shows the main stages of dislocation motion and interaction with \(\theta^{\prime}\) precipitate. For simplicity, we consider a rectangle inclusion with the length \(l\) along the direction of the perfect Burgers vector and width \(d\) along the vector tangent to the dislocation. The distance between \(\theta^{\prime}\) inclusions in the direction of the \(z\)-axis is denoted as \(D\), and the distance in the direction of the \(x\)-axis is \(L\). Following an MD statement, the periodic boundary conditions are applied along the \(z\)-axis; in this case, the dislocation interacts with the precipitate chain. The dislocation segments are considered as straight ones. Edge and screw segments are directed along the \(z\)- and \(x\)-axis, correspondingly, as given in Figure 3. We use two generalized coordinates \(x\) and \(a\) to describe the positions of the moving edge and screw segments; other segments are immobilized at \(\theta^{\prime}\) inclusion. We suppose that the energy of the segments adjacent to the inclusion consists only of the core energy \(\varepsilon_{p}\), since their elastic field is screened by the inclusion. This value, \(\varepsilon_{p}\), is the model parameter that depends on  
Figure 2: Dislocation and \(\theta^{\prime}\) precipitate of 5 nm diameter in MD system after energy minimization. (**a**)—view along \([11\overline{2}]\) direction, (**b**)—view along \([111]\). Atoms with a centrosymmetry parameter more than 5 square angstroms are depicted. Aluminum atoms are colored in gray, copper atoms are red. Shockley partial dislocations are colored in green. the temperature and type of precipitate. Our previous work [57] shows that this parameter has the same value for edge and screw segments. In this work, we fit it to MD results for \(\theta^{\prime}\) precipitate. For segments not in contact with the inclusion, the total energy is the sum of the core energy \(\varepsilon_{1}=0.2\) eV/\(b\) and the energy of the elastic field. In the case of screw segments, it is \(\varepsilon_{s}=\varepsilon_{1}+\varepsilon_{2s}\ln(R/R_{0})\), and \(\varepsilon_{e}=\varepsilon_{1}+\varepsilon_{2e}\ln(R/R_{0})\) in the case of edge segments. \(R_{0}=4b\) is the radius of the dislocation core; \(R\) represents the distance to the nearest obstacle or half the distance to the opposite dislocation segment; \(\varepsilon_{2s}=Gb^{2}/[4\pi]\) and \(\varepsilon_{2e}=Gb^{2}/[4\pi(1-v)]\) are obtained from the elastic theory of dislocations; \(G\) and \(v\) are the shear modulus and Poisson's ratio of aluminum, respectively; and \(b\) is the magnitude of the Burgers vector. We consider four stages of dislocation-precipitate interaction (Figure 3): (A) movement towards the inclusion; (B) wrapping around the inclusion; (c) the Orowan loop closing; and (D) movement from the inclusion. Motion of dislocation segments is caused by the forces, which can be obtained by differentiation of the total energy of the dislocation line along the generalized coordinates \(x\) and \(a\)[57; 58; 34]. At stages (B) and (C), the increase in the length of the dislocation line during the Orowan loop formation is controlled by these forces. This increase in length provides the dislocation deceleration.