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Aqueous two-phase (ATP) extraction has been demonstrated as a fast, scalable, and effective separation technique to sort single-wall carbon nanotubes (SWCNTs) according to their diameter and chiral structure. The exact mechanism behind the chirality-dependent migration of SWCNTs between the two phases is however not completely understood, and depends on many parameters (e.g., choice of surfactants and their concentration, pH, temperature, ...), making it difficult to optimize the multivariable parameter space. In this work, we present a systematic study of the choice and concentration of specific surfactants on the ATP sorting, by performing a series of single-step ATP separations in which each time only one parameter is systematically varied, while monitoring the structure-specific migration of every SWCNT chirality between both phases with detailed wavelength-dependent spectroscopy. These systematic studies reveal that the diameter-dependent stacking of a discrete number of sodium deoxycholate molecules fitting around the SWCNT circumference determines the separation order in the form of a periodically modulated pattern as a function of SWCNT diameter. Addition of cosurfactants can be used to compete with the bile salt surfactant to enhance the separation yields, but does not affect the sorting order. The results are afterwards directly applied to predict the parameters required to separate specific chiral structures in just two ATP steps.