  Grain boundaries (GBs), an important constituent of polycrystalline
materials, have a wide range of manifestion and significantly affect the
properties of materials. Fully understanding the effects of GBs is stalemated
due to lack of complete knowledge of their structures and energetics. Here, for
the first time, by taking graphene as an example, we propose an analytical
energy functional of GBs in angle space. We find that an arbitrary GB can be
characterized by a geometric combination of symmetric GBs that follow the
principle of uniform distribution of their dislocation cores in straight lines.
Furthermore, we determine the elusive kinetic effects on GBs from the
difference between experimental statistics and energy-dependent thermodynamic
effects. This study not only presents an analytical energy functional of GBs
which could also be extended to other two-dimensional materials, but also sheds
light on understanding the kinetic effects of GBs in material synthesizing
processes.
