  Growth of hard--rod monolayers via deposition is studied in a lattice model
using rods with discrete orientations and in a continuum model with hard
spherocylinders. The lattice model is treated with kinetic Monte Carlo
simulations and dynamic density functional theory while the continuum model is
studied by dynamic Monte Carlo simulations equivalent to diffusive dynamics.
The evolution of nematic order (excess of upright particles, "standing--up"
transition) is an entropic effect and is mainly governed by the equilibrium
solution, {rendering a continuous transition} (paper I, J. Chem. Phys. 145,
074902 (2016)). Strong non--equilibrium effects (e.g. a noticeable dependence
on the ratio of rates for translational and rotational moves) are found for
attractive substrate potentials favoring lying rods. Results from the lattice
and the continuum models agree qualitatively if the relevant characteristic
times for diffusion, relaxation of nematic order and deposition are matched
properly. Applicability of these monolayer results to multilayer growth is
discussed for a continuum--model realization in three dimensions where
spherocylinders are deposited continuously onto a substrate via diffusion.
