  We study the effective electrostatic interactions between a pair of charged
colloidal particles without salt ions while the system is confined in two
dimensions. In particular we use a simplified model to elucidate the effects of
rotational fluctuations in counterion distribution. The results exhibit
effective colloidal attractions under appropriate conditions. Meanwhile,
long-range repulsions persist over most of our studied cases. The repulsive
forces arise from the fact that in two dimensions the charged colloids cannot
be perfectly screened by counterions, as the residual quadrupole moments
contribute to the repulsions at longer range. And by applying multiple
expansions we find that the attractive forces observed at short range are
mainly contributed from electrostatic interactions among higher-order electric
moments. We argue that the scenario for attractive interactions discussed in
this work is applicable to systems of charged nanoparticles or colloidal
solutions with macroions.
