Wurtzite ZnO is polar along the c-axis: the (0001) Zn-terminated top facet has a much higher surface energy and far fewer stably-bound surface oxygens than the six nonpolar {10-10} side facets. Growth units (Zn(OH)₄²⁻ complexes releasing Zn²⁺ and OH⁻ as hexamethylenetetramine hydrolyzes in the bath) attach almost exclusively at the polar tip, so a seeded crystallite elongates far faster along c than it thickens — the classic hydrothermal route to vertically-aligned nanorod arrays.
Both facet growth rates follow Arrhenius kinetics driven by the bath's supersaturation ratio S = C/Ceq − 1:
dL/dt = A_axial · exp(−Ea_axial / RT) · S^1.5 (tip, polar facet)
dD/dt = A_radial · exp(−Ea_radial / RT) · S^1.5 · (1 − 0.9·θ_cap) (sides, nonpolar facets)
θ_cap is the fractional surface coverage by citrate ions, which adsorb selectively onto the nonpolar side facets (their carboxylate groups match the Zn²⁺ spacing there) and block growth-unit attachment — raising the capping-agent slider suppresses radial growth almost without touching the axial rate, which is exactly how real syntheses dial in aspect ratio.
- Temperature — raises both rates exponentially through the Arrhenius term; the radial facet has the higher apparent activation energy, so heat also modestly boosts the aspect ratio.
- Precursor concentration — sets the supersaturation S that drives both rate laws; near the equilibrium concentration (S≈0) growth stalls entirely.
- Citrate capping — selectively throttles the radial rate, turning stubby hexagonal crystallites into long thin nanorods.
Real-world relevance: this seeded hydrothermal route (zinc nitrate + HMTA, 60–200 °C, aqueous, no vacuum) is the standard low-cost way to grow the vertical ZnO nanorod arrays used in dye-sensitized solar cells, piezoelectric nanogenerators and UV photodetectors.