*3.4. Grain Boundary E*ff*ects on Mechanical Behavior*

Grain boundary (GB) strengthening mechanism is based on the observation that GBs impede dislocation movement. Since the adjacent grains differ in orientation, it requires more energy for a dislocation to change slip direction and slip plane and transfer into the adjacent grain [59,60]. The transferability or continuity of slip systems in the two grains across a GB is described by a geometrical compatibility factor (GCF), m'=cos(ϕ)×cos(k) [54–58]. The bigger the GCF is, the easier slip transmission happens. ϕ is the angle between the slip plane normal directions and *k* is the angle between the slip directions. We examined this in two bi-crystal micropillars.

The bi-crystal 1 has the orientation relation: [403]A//[1]B, (010)A//(010)B between Grains A and B. The compression direction is parallel to [403]A and [1]B. The maximum SF in Grain A is 0.478 for (121)A and the second highest SF is 0.455 for (121)A. The maximum SF in Grain B is 0.483 for (112)B and the second highest SF is 0.479 for (112)B. Video 8 records the in situ compression testing of the pillar, showing slip transmission across the GB. The morphology of the micropillar after compression is shown in Figure 7a. The slip traces on the surface of the deformed pillar enable us to identify the activated slip planes, (121)A with a SF of 0.455 in grain A and (011)B with a SF of 0.355 in grain B, as depicted in Figure 7b. We computed geometrical compatibility factors associated with any pair of slip systems (121)A/{011}B and (121)A/{112} across the GB, the factor associated with the pair of slip systems (121)A/(011)B is the largest, m' = 0.84, which accounts for the observed slip transmission across the GB in Figure 7a and no apparent strain hardening during the compression as shown in Figure 7e.

**Figure 7.** (**a**,**c**) Two bi-crystal micropillars after compression at a strain of 0.3. (**b**) A schematic of bi-crystal 1 showing two preferred slip systems. (**d**) Two schematics of bi-crystal 2 showing two pairs of two preferred slip systems. Yellow dashed lines indicate grain boundaries. The arrows show the slip vectors. (**e**) True stress-strain curves and strain hardening rates.

The bi-crystal 2 has the orientation relation: [11 5 9]C // [135]D and (9 4 2)//(2 1 10) between Grains C and D. The compression direction is parallel to [11 5 9]C and [135]D. The highest SF in Grain C is 0.466 for (121)C and the second highest SF is 0.428 for (110)C. The highest SF in Grain D is 0.483 for (211)D and the second highest SF is 0.482 for (101)D. Video 9 records the in situ compression testing of the pillar, showing slip transmission across the GB. The morphology of the micropillar after compression in Figure 7c shows two sets of slip transmission across the GB of the bi-crystal. According to the slip traces on the surface of the deformed pillar, the activated slip systems are identified to be (101)D with a SF of 0.482 and (101)C with a SF of 0.25, as illustrated in Figure 7d. The geometrical compatibility factor associated with the two slip systems (101)D/(101)C is 0.44. We computed the geometrical compatibility factors associated with all pairs of slip systems (101)D/{110}C and (101)D/{112}C, the largest factor m' = 0.74 is associated with the pair of slip systems (101)D/(121)C. However, this pair of slips did not apparently activate. One possible reason is ascribed to the low SF of 0.15 associated with the slip system (121)C<111>. The another pair of slips are identified to be (121)C with the highest SF of 0.466 in grain C and (011)D slip with a SF of 0.24 in grain D, as illustrated in Figure 7d. The geometrical compatibility factor associated with the two slip systems is 0.75. We computed the geometrical compatibility factors

associated with all pairs of slip systems (121)C/{110}D and (121)C/{112}D, the largest factor m' = 0.79 is associated with the pair of slip systems (121)C/(112)D, but the slip system (112)D<111> has a low SF of 0.20. Compared to the case of bi-crystal 1, there are two sets of slip transmissions associated with slip systems (101)D/(101)C and (121)C/(011)D, the pair of (101)D/(101)C has small geometrical compatibility factor but large SFs and the pair of (121)C/(011)D has big geometrical compatibility factor but small SFs. Thus, the bi-crystal 2 exhibits a high flow strength, as shown in Figure 7e.
