In this experimental work, the thermodynamics and self-organization of classical two-dimensional Coulomb clusters are studied as a function of the cluster size. The experiments are carried out in a DC glow discharge Argon plasma in the Dusty Plasma Experimental (DPEx) device for clusters with different number of particles. Hexagonal symmetry around each individual particle is quantified using the local orientational order parameter ($|{psi_6}|$) for all the configurations. The screened Coulomb coupling parameter, which plays a key role in determining the thermodynamic nature of a Coulomb cluster, is estimated using Langevin dynamics and found to be sensitive to the number of particles present in the cluster. In addition, the process of self-organization and the dynamics of individual particles of the cluster as it changes from a metastable state to the ground state are examined through the estimation of dynamic entropy. Our findings suggest an intimate link between the configurational ordering and the thermodynamics of a strongly coupled Coulomb cluster system - an insight that might be of practical value in analysing and controlling the micro dynamics of a wider class of finite systems.
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