  We investigate by molecular dynamics simulations [arXiv:1412.4537
[physics.plasm-ph], arXiv:1608.06789 [physics.plasm-ph]] a beam of electrons
released via field emission from a planar cathode surface of 1 ${\rm \mu m}^2$
with an inhomogeneous two-level work function, $\phi_{\rm low}$ and $\phi_{\rm
high}$. A rectangular grid, where each cell can have one out of two values of
the work function, is used as a model. The number of cells in the grid ranges
from 6x6 to 96x96. We compare a periodic checkerboard arrangement with
disordered distributions of patches. We perform multiple simulations and
randomize the pattern each time. We study the beam behavior by calculating the
position and velocity of each electron, r.m.s. emittance, and the brightness of
the electron beam. The emittance increases while brightness decreases, with the
mean distance between patches with $\phi_{\rm low}$ when they are in minority,
and they switch the behavior vs. the mean distance between patches with
$\phi_{\rm high}$ when these patches are in minority, respectively. The Coulomb
interaction between all particles is fully included in our simulations.
