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## Energy Minimization 

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The optimization of grain-boundary structures at zero temperature made use of a simu- lation block with two grains separated by a flat grain boundary as shown in Fig 1. The cell was periodic in the plane of the boundary and non-periodic perpendicular to it. The grains were sandwiched between two slabs parallel to the grain-boundary plane. The atoms in each of these slabs were fixed relative to each other and could only undergo a rigid body motion. The slabs could move normal to the boundary, allowing volume expansion to maintain zero normal stress, and in the plane of the boundary, avoiding any resistance from the surfaces to translational movement of the grains relative to each other. Multiple trial configurations were built in order to minimize the boundary energy with respect to relative translations of the grains and with respect to the number of atoms in the grain boundary, as described in Ref. [41]. The energy of each trial configuration was minimized using the conjugate gradient method. After minimization, the energy of the grain boundary was computed as the total energy of the unconstrained atoms, less the bulk crystal energy for the same number of 

