  A highly-charged spherical colloid in a salt-free environment exerts such a
powerful attraction on its counterions that a certain fraction condenses onto
the surface of a particle. The degree of condensation depends on the curvature
of the surface. So, for instance, condensation is triggered on a highly-charged
sphere only if the radius exceeds a certain critical radius $\collrad^{*}$.
$\collrad^{*}$ is expected to be a simple function of the volume fraction of
particles. To test these predictions, we prepare spherical particles which
contain a covalently-bound ionic liquid, which is engineered to dissociate
efficiently in a low-dielectric medium. By varying the proportion of ionic
liquid to monomer we synthesise nonpolar dispersions of highly-charged spheres
which contain essentially no free co-ions. The only ions in the system are
counterions generated by the dissociation of surface-bound groups. We study the
electrophoretic mobility of this salt-free system as a function of the colloid
volume fraction, the particle radius, and the bare charge density and find
evidence for extensive counterion condensation. At low electric fields, we
observe excellent agreement with Poisson-Boltzmann predictions for counterion
condensation on spheres. At high electric fields however, where ion advection
is dominant, the electrophoretic mobility is enhanced significantly which we
attribute to hydrodynamic stripping of the condensed layer of counterions from
the surface of the particle.
