Toward a Deterministic Nucleation Theory for Chirality-Controlled Nanotube Synthesis
The electronic properties of carbon nanotubes are governed by their chirality, specified by the integer indices (n,m). While chirality-controlled synthesis has achieved notable successes, theoretical understanding remains predominantly focused on post-nucleation growth. Two fundamental obstacles impede deeper insight: the absence of a clear description of nucleation cap topology and its connection to tube chirality, and an incomplete understanding of atomic-level mechanisms governing templated cap formation. Here we address these challenges directly. First, we develop a mathematically rigorous topological framework for carbon networks that provides both a concise definition of cap structures and a quantitative relationship between cap architecture and chirality-the vector sum rule. Second, contrary to conventional perspectives attributing chirality enrichment to edge matching during growth, we demonstrate that chirality is deterministically encoded during nucleation through selective formation of specific cap structures on catalyst surfaces. For the specific case of (12,6) nanotubes, we show that their enrichment arises from a six-fold symmetric cap with epitaxial matching to catalyst facets. Our deterministic nucleation theory not only provides a coherent explanation for chirality enrichment but also elucidates its pattern in chirality space. This work establishes a theoretical framework that redefines the field, shifting the paradigm from stochastic growth kinetics to deterministic nucleation programming and paving the way toward predictable synthesis.