Fundamental scaling laws for the direct-write chemical vapor deposition of nanoscale features: modeling mass transport around a translating nanonozzle

The nanometer placement of nanomaterials, such as nanoribbons and nanotubes, at a specific pitch and orientation on a surface, remains an unsolved fundamental problem in nanotechnology. In this work, we introduce and analyze the concept of a direct-write chemical vapor deposition (CVD) system that enables the in-place synthesis of such structures with control over orientation and characteristic features. A nanometer scale pore or conduit, called the nanonozzle, delivers precursor gases for CVD locally on a substrate, with spatial translation of either the nozzle or the substrate to enable a novel direct write (DW) tool. We analyze the nozzle under conditions where it delivers reactants to a substrate while translating at a constant velocity over the surface at a fixed reaction temperature. We formulate and solve a multi-phase three-dimensional reaction and diffusion model of the direct-write operation, and evaluate specific analytically-solvable limits to determine the allowable operating conditions, including pore dimensions, reactant flow rates, and nozzle translation speed. A Buckingham Π analysis identifies six dimensionless quantities crucial for the design and operation of the direct-write synthesis process. Importantly, we derive and validate what we call the ribbon extension inequality that brackets the allowable nozzle velocity relative to the CVD growth rate - a key constraint to enabling direct-write operation. Lastly, we include a practical analysis using attainable values towards the experimental design of such a system, building the nozzle around a commercially available near-field scanning optical microscopy (NSOM) tip as a feasible example.