  Understanding the source of sub-stellar polarimetric observations in the
optical and near-infrared is key to characterizing sub-stellar objects and
developing potential diagnostics for determining properties of their
atmospheres. Differential scattering from a population of aligned,
non-spherical dust grains is a potential source of polarization that could be
used to determine geometric properties of the dust clouds. This paper addresses
the problem of the spheroidal growth of dust grains in electrically activated
sub-stellar atmospheres. It presents the novel application of a mechanism
whereby non-spherical, elongated dust grains can be grown via plasma deposition
as a consequence of the surface electric field effects of charged dust grains.
We numerically solve the differential equations governing the spheroidal growth
of charged dust grains via plasma deposition as a result of surface electric
field effects in order to determine how the dust eccentricity and the dust
particle eccentricity distribution function evolve with time. From these
results, we determine the effect of spheroidal dust on the observed linear
polarization. Numerical solutions show that $e\approx 0.94$ defines a watershed
eccentricity, where the eccentricity of grains with an initial eccentricity
less than (greater than) this value decreases (increases) and spherical
(spheroidal) growth occurs. This produces a characteristic bimodal eccentricity
distribution function yielding a fractional change in the observed linear
polarization of up to $\approx0.1$ corresponding to dust grains of maximal
eccentricity at wavelengths of $\approx1 \mu$m, consistent with the near
infrared observational window. The results presented here are relevant to the
growth of non-spherical, irregularly shaped dust grains of general geometry
where non-uniform surface electric field effects of charged dust grains are
significant.
