  In a previous work, two of the authors proposed a new proof of a well known
convergence result for the scaled elementary connected vacant component in the
high intensity Boolean model towards the Crofton cell of the Poisson hyperplane
process. In this paper, we consider the particular case of the two-dimensional
Boolean model where the grains are discs with random radii. We investigate the
second-order term in this convergence when the Boolean model and the Poisson
line process are coupled on the same probability space. A precise coupling
between the Boolean model and the Poisson line process is first established, a
result of directional convergence in distribution for the difference of the two
sets involved is derived as well.
