On the equilibrium of helical nanostructures with ordered mesopores

Helical conformation exists universally at different length scales. We present a new model to explain the energetics of a helical structure with ordered mesopores and successfully predict their equilibrium state. The formation of the helical structure, which is composed of twisted and hexagonally arrayed one-dimensional pore channels, should be understood at the macromorphology level through the competition between surface free energy reduction and torsion strain energy increase. Our model is established by first reverting a helical rod with experimentally defined parameters to a conjectured straight rod without intrinsic pore channel twisting, and then quantitatively calculating the variation of two competitive energies as a function of twist angle in the torsion process starting from the reverted straight rod. Through our model, a free energy curve is achieved, so that the equilibrium state and the helical structural parameters can be predicted, which are in good agreement with experimental results for helical rods synthesized by different surfactant templates. Moreover, our model can be successfully applied to explain the pitch-radius relationships in previous observations. Our achievement provides unique and fundamental understandings for the spontaneous mesoscopic helix formation, which are different from the microscopic helical structures such as DNA chains.