text: Incoherent grain boundaries are those in which there is a significant mismatch in crystallographic orientation between adjacent grains. This results in a discontinuity in the crystal lattice across the boundary, and the formation of a variety of defects such as dislocations, stacking faults, and grain boundary ledges.The presence of these defects creates a barrier to the motion of dislocations and leads to a strengthening effect. This effect is more pronounced in materials with smaller grain sizes, as there are more grain boundaries to impede dislocation motion. In addition to the barrier effect, incoherent grain boundaries can also act as sources and sinks for dislocations. This can lead to localized plastic deformation and affect the overall mechanical response of a material.
  When small particles are formed through precipitation from supersaturated solid solutions, their interphase boundaries may not be coherent with the matrix. In such cases, the atomic bonds do not match up across the interface and there is a misfit between the particle and the matrix. This misfit gives rise to a non-coherency strain, which can cause the formation of dislocations at the grain boundary. As a result, the properties of the small particle can be different from those of the matrix. The size at which non-coherent grain boundaries form depends on the lattice misfit and the interfacial energy.