Is there an intrinsic limit to the size of 2D supracrystals built from weakly interacting nanoparticles?

This work establishes that an extremely stringent, but dismissed thermodynamic condition governs the size of perfect 2D supracrystals that consist of weakly interacting nanoparticles. This severe condition is ultimately imposed by the large modulus of the negative entropy, which accumulates in any large defect-free (or defectless) crystal, owing to the high demand for the precise ordering of its arrayed elements. This thermodynamic toll is especially difficult to compensate for when interaction enthalpies within the lattice and its underlying substrate are weak, as is required for these arrays, which are valuable for innovative applications such as optics, electronics and magnetism. This constraint is formulated here by a simple scale law which predicts the maximum size achievable for 2D supracrystals as a function of their surface and edge enthalpies. This scale law analysis is ultimately based on the recognition that the modulus of entropic contributions grows faster (as ln(N(0)!) in which N(0) is the number of perfectly arrayed particles) than that of enthalpic ones (proportional to N(0)) when the size of any perfectly ordered structure increases. Hence, disorder must be introduced into the network to relax the entropic demand to allow sufficient stability. This intrinsic characteristic is shown to be extremely prohibitory if the enthalpic interactions between the arranged objects are close to the thermal quantum, as is imposed for the applications mentioned above. To substantiate this concept, we have evaluated the thermodynamics of 2D supracrystals composed of nanoparticles arranged into the most common crystallographic arrays (square, c, square-centered, cc, or hexagonal, hex, lattices). The essentially identical results obtained for these there lattices establish that the scale law developed here is valid for most realistic lattices.