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The configurational entropy of colloidal particles in a confined space

We calculate the configurational entropy of colloidal particles in a confined geometry interacting as hard disks using Monte Carlo integration method. In particular, we consider systems with three kinds of boundary conditions: hard, periodic and spherical. For small to moderate packing fraction $ϕ$ values, we find the entropies per particle for systems with the periodic and spherical boundary conditions tend to reach a same value with the increase of the particle number $N$, while that for the system with the hard boundary conditions still has obvious differences compared to them within the studied $N$ range. Surprisingly, despite the small system sizes, the estimated entropies per particle at infinite system size from extrapolations in the periodic and spherical systems are in reasonable agreement with that calculated using thermodynamic integration method. Besides, as $N$ increases we find the pair correlation function begins to exhibit similar features as that of a large self-assembled system at the same packing fraction. Our findings may contribute to a better understanding of how the configurational entropy changes with the system size and the influence of boundary conditions, and provide insights relevant to engineering particles in confined spaces.