  This paper is concerned with the sharp interface limit for the Allen-Cahn
equation with a nonlinear Robin boundary condition in a bounded smooth domain
$\Omega\subset\mathbb{R}^2$. We assume that a diffuse interface already has
developed and that it is in contact with the boundary $\partial\Omega$. The
boundary condition is designed in such a way that the limit problem is given by
the mean curvature flow with constant $\alpha$-contact angle. For $\alpha$
close to $90${\deg} we prove a local in time convergence result for
well-prepared initial data for times when a smooth solution to the limit
problem exists. Based on the latter we construct a suitable curvilinear
coordinate system and carry out a rigorous asymptotic expansion for the
Allen-Cahn equation with the nonlinear Robin boundary condition. Moreover, we
show a spectral estimate for the corresponding linearized Allen-Cahn operator
and with its aid we derive strong norm estimates for the difference of the
exact and approximate solutions using a Gronwall-type argument.
