Quantitative theory of adsorptive separation for the electronic sorting of single-walled carbon nanotubes

Recently, several important advances in techniques for the separation of single-walled carbon nanotubes (SWNTs) by chiral index have been developed. These new methods allow for the separation of SWNTs through selective adsorption and desorption of different (n,m) chiral indices to and from a specific hydrogel. Our group has previously developed a kinetic model for the chiral elution order of separation; however, the underlying mechanism that allows for this separation remains unknown. In this work, we develop a quantitative theory that provides the first mechanistic insights for the separation order and binding kinetics of each SWNT chirality (n,m) based on the surfactant-induced, linear charge density, which we find ranges from 0.41 e(-)/nm for (7,3) SWNTs in 17 mM sodium dodecyl sulfate (SDS) to 3.32 e(-)/nm for (6,5) SWNTs in 105 mM SDS. Adsorption onto the hydrogel support is balanced by short-distance hard-surface and long-distance electrostatic repulsive SWNT/substrate forces, the latter of which we postulate is strongly dependent on surfactant concentration and ultimately leads to gel-based single-chirality semiconducting SWNT separation. These molecular-scale properties are derived using bulk-phase, forward adsorption rate constants for each SWNT chirality in accordance with our previously published model. The theory developed here quantitatively describes the experimental elution profiles of 15 unique SWNT chiralities as a function of anionic surfactant concentration between 17 and 105 mM, as well as phenomenological observations of the impact of varying preparatory conditions such as extent of ultrasonication and ultracentrifugation. We find that SWNT elution order and separation efficiency are primarily driven by the morphological change of SDS surfactant wrapping on the surface of the nanotube, mediated by SWNT chirality and the ionic strength of the surrounding medium. This work provides a foundational understanding for high-purity, preparative-scale separation of as-produced SWNT mixtures into isolated, single-chirality fractions.