  Polyelectrolyte (PE) brushes have ubiquitous applications as surface
modifiers which regulate various structural and dynamic properties. Here, we
apply a continuous-space self-consistent field theory to study the structural
heterogeneity in PE brushes induced by competing electrostatic and hydrophobic
interactions. For brushes with high grafting densities, we find a series of
microphase-segregated morphologies with alternating polymer-rich and
polymer-poor layers in the direction normal to the substrate. The transitions
between multi-layer morphologies with consecutive numbers of condensed layers
as well as the melting transition to the fully swollen brush are all
discontinuous. We also elucidate that the segregated layers are formed by
different subpopulations of all chains, significantly different from the
scenario of the pearl-necklace structure formed by a single PE in poor
solvents. Furthermore, we bridge the microstructure of multi-layer brushes to
experimentally measurable reflectivity spectra, where the oscillation period
and amplitude in the spectra are shown to be very sensitive to the number of
layers and the sharpness of the polymer-solvent interface. The multi-layer
morphology predicted by our theory is in good agreement with the lamellae
structure experimentally observed at hydrated Nafion film interfaces.
