The growth of nanowires via the vapor-liquid-solid (VLS) mechanism is a complex process governed by the interplay of mass transport, surface diffusion, and crystallization kinetics. A comprehensive kinetic model must account for the dynamic interactions between the vapor phase, the liquid catalyst droplet, and the solid nanowire, as well as the influence of diameter-dependent effects, tapering, and kinking. This article explores the fundamental principles of VLS nanowire growth kinetics, with a focus on Si, GaN, and III-V nanowires, and compares theoretical predictions with experimental observations.

At the core of VLS growth is the catalyst droplet, typically a liquid metal such as Au for Si nanowires or Ni for GaN nanowires. The droplet serves as a preferential site for precursor adsorption and decomposition, facilitating the supersaturation necessary for nanowire nucleation and elongation. The growth process begins with the diffusion of precursor species through the vapor phase to the droplet surface, followed by their incorporation into the liquid alloy. The solute atoms then diffuse through the droplet to the liquid-solid interface, where crystallization occurs. The rate-limiting step in VLS growth can vary depending on experimental conditions, but it is often determined by either precursor supply, surface diffusion, or crystallization kinetics.

Mass transport in the vapor phase is typically described by diffusion equations, accounting for the concentration gradient between the bulk vapor and the droplet surface. For a nanowire of diameter d, the precursor flux J_vapor can be expressed as proportional to the difference in precursor concentration between the far-field and the droplet surface, scaled by the diffusion length. In many cases, the vapor-phase diffusion is fast compared to other processes, but at low pressures or high growth rates, it can become limiting. Surface diffusion of adatoms on the nanowire sidewalls also plays a critical role, particularly for materials with high surface mobility such as Si. Adatoms adsorbed on the sidewalls can migrate to the catalyst droplet, contributing to axial growth or leading to radial growth (tapering) if they incorporate into the sidewalls instead.

The crystallization process at the liquid-solid interface is governed by the supersaturation of solute in the droplet, which drives the incorporation of atoms into the growing nanowire. The growth rate v can be modeled as a function of the solute concentration C_l and the equilibrium concentration C_eq at the interface. For a droplet of diameter d, the Gibbs-Thomson effect modifies C_eq due to curvature, leading to a diameter-dependent growth rate. Smaller diameters result in higher equilibrium concentrations, reducing the effective supersaturation and slowing growth. This effect is particularly pronounced for diameters below 20 nm, where growth rates can decrease significantly.

Tapering in nanowires arises when radial growth competes with axial growth. This can occur due to insufficient surface diffusion, where adatoms accumulate on the sidewalls instead of reaching the catalyst droplet, or due to changes in droplet composition that favor lateral overgrowth. Kinetic models incorporating tapering often include terms for sidewall adsorption and desorption, as well as diffusion barriers that influence adatom migration. Kinking, or sudden changes in growth direction, is another phenomenon observed in nanowires, particularly in III-V materials. Kinking can result from fluctuations in droplet composition, changes in supersaturation, or defects at the liquid-solid interface. Simulations of GaN nanowires, for example, have shown that local variations in the III/V ratio can lead to asymmetric growth and kinking.

In situ transmission electron microscopy (TEM) observations have provided critical insights into VLS growth dynamics. For Si nanowires, in situ TEM has revealed real-time changes in droplet shape and nanowire morphology during growth, confirming predictions of diameter-dependent growth rates. GaN nanowires have been observed to exhibit abrupt kinking under certain conditions, aligning with simulations that account for compositional instabilities in the catalyst droplet. III-V nanowires, such as those of InAs or GaAs, often show complex growth behaviors due to the interplay of group III and group V elements in the droplet, leading to variations in growth direction and tapering.

Catalyst alloying effects further complicate the growth kinetics. For Si nanowires, the Au-Si eutectic droplet changes composition as growth proceeds, affecting solute solubility and diffusion rates. In GaN nanowires, the Ni-Ga-N system exhibits complex phase behavior, with stoichiometric variations influencing nucleation and growth. III-V nanowires often require precise control of the group III to group V ratio in the droplet to maintain steady growth, as deviations can lead to kinking or termination. Kinetic models must therefore incorporate thermodynamic data on the catalyst alloy system to accurately predict growth behavior.

Comparisons between simulations and experiments highlight both successes and challenges in VLS growth modeling. For Si nanowires, models that include diameter-dependent supersaturation and surface diffusion accurately reproduce experimentally observed growth rates and tapering profiles. GaN nanowire simulations that account for nitrogen solubility limits in the catalyst droplet match in situ TEM observations of growth intermittency. However, discrepancies remain in predicting the exact conditions for kinking in III-V nanowires, suggesting that additional factors, such as interfacial strain or defect dynamics, may need incorporation into the models.

Future directions in kinetic modeling of VLS growth include refining descriptions of interfacial processes, such as the role of steps and defects at the liquid-solid boundary, and integrating more detailed thermodynamic data for multicomponent catalyst systems. Advances in computational power and algorithms will enable larger-scale simulations that capture the full complexity of nanowire growth, from nucleation to termination. Coupling these models with high-resolution in situ characterization techniques will further enhance predictive capabilities, enabling precise control over nanowire morphology and properties for applications in electronics, photonics, and energy conversion.