  We study the atomic structure of twisted bilayer graphene, with very small
mismatch angles ($\theta \sim 0.28^0$), a topic of intense recent interest. We
use simulations, in which we combine a recently presented semi-empirical
potential for single-layer graphene, with a new term for out-of-plane
deformations, [Jain et al., J. Phys. Chem. C, 119, 2015] and an often-used
interlayer potential [Kolmogorov et al., Phys. Rev. B, 71, 2005]. This
combination of potentials is computationally cheap but accurate and precise at
the same time, allowing us to study very large samples, which is necessary to
reach very small mismatch angles in periodic samples. By performing large scale
atomistic simulations, we show that the vortices appearing in the Moir\'e
pattern in the twisted bilayer graphene samples converge to a constant size in
the thermodynamic limit. Furthermore, the well known sinusoidal behavior of
energy no longer persists once the misorientation angle becomes very small
($\theta<1^0$). We also show that there is a significant buckling after the
relaxation in the samples, with the buckling height proportional to the system
size. These structural properties have direct consequences on the electronic
and optical properties of bilayer graphene.
