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. By applying multiple expansions, we find that the attractive forces observed at short range are mainly contributed by 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.
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