4.3.1. Results for Ni Sample

Based on the hypotheses described in Section 4.2 and considering the B4 slip system in the crystal II of the studied Ni sample, the out-of-surface component of the Burgers vector is about 0.15 nm. The maximum slip step height is about 9.1 nm at *d* ≈ 3.28 μm and the slip step height at GB due to slip transmission and/or dislocation absorption is about 0.86 nm. Thus, based on Equation (2), the number of dislocations in the pile-up is 61 and the number of transmitted and/or absorbed dislocations is 6. Therefore, the number of dislocations in the pile-up is equal to 55. The applied stress is 289.4 MPa as measured in experiment, and it corresponds to a resolved shear stress *τ* = 129.0 MPa. The thicknesses of Grains I and II are *H*<sup>I</sup> = 4.30 μm and *H*II = 3.73 μm as shown in Figure 1b. With all these parameters, the slip step height distribution due to dislocations in the pile-up was simulated for different critical forces while considering or not the effect of free surfaces as shown in Figure 8a.

Comparing the result of Simulation 1 without free surfaces to Simulation 2 with two free surfaces, it is found that the free surfaces have an influence on dislocations behavior. From the theory, it is known that free surfaces have always an attractive effect on dislocations. When the dislocations are closer to the first free surface Λ<sup>1</sup> than the second one Λ<sup>2</sup> (see Figure 7a), the total force without considering the stress field of the last dislocation is always towards Λ1, and so dislocations move towards the last dislocation to reach equilibrium. However, the dislocations around GB are nearly in the middle of both free surfaces, thus the effects of the two free surfaces are balanced out by each other. Therefore, the free surfaces have much more effects on the dislocations which are near the free surface rather than the ones which are located around GB. These two simulation results are close to the experimental measurement, but there are still some discrepancies. Compared to the experimental measurement, the dislocations are closer to GB without free surfaces, while they are closer to the first free surface when considering the effect of free surfaces.

As discussed in Section 4.1, it is actually necessary to consider the effect of critical force *Fc* (see Equation (1)). After the analysis of the results with different value of *Fc*, it is found that *Fc* moves the dislocations towards the GB. When considering the effect of free surfaces with a material's critical force *Fc* = 0.003 N/m in Equation (1) (equivalent to a resolved shear stress *τ<sup>c</sup>* = 12 MPa), the simulation result is closer to the experimental measurement as shown in Figure 8a with Simulation 3. Here, the value is higher than the theoretical value of lattice friction for pure FCC crystals which is around 1 ∼ 2 MPa [45]. The reason might be due to the sample preparation, such as defects coming from FIB polishing. Furthermore, as discussed in Section 4.2 (3), the P–K force is not zero on the last fixed dislocation. The total resolved shear stresses on this fixed dislocation are 6.6 MPa for Simulation 1, 175.0 MPa for Simulation 2 and 71.2 MPa for Simulation 3. The value found in Simulation 2 is quite high for a FCC crystal and appears unrealistic.

**Figure 8.** (**a**) simulation of slip step height profile for Ni with different discrete dislocation pile-up simulation conditions (*b*GB is always the same as *b*In). Simulation 1: without free surfaces. Simulation 2: with two free surfaces. Simulation 3: with two free surfaces and a non zero material dependent critical force *Fc* = 0.003 N/m; (**b**) simulation of slip step height profile for *α*-Brass with different discrete dislocation pile-up simulation conditions (always with two free surfaces). Simulation 1: *<sup>b</sup>*GB = *<sup>b</sup>*In. Simulation 2: *<sup>b</sup>*GB = *<sup>b</sup>*In − *<sup>b</sup>*Out (A6). Simulation 3: *<sup>b</sup>*GB = *<sup>b</sup>*In − *<sup>b</sup>*Out (A6) with a non zero material dependent critical force *Fc* = 0.011 N/m.
