  The viscosity of AOT/water/decane water-in-oil microemulsions exhibits a
well-known maximum as a function of water/AOT molar ratio, which is usually
attributed to increased attractions among nearly spherical droplets. The
maximum can be removed by adding salt or by changing the oil to CCl$_4$.
Systematic small-angle X-ray scattering (SAXS) measurements have been used to
monitor the structure of the microemulsion droplets in the composition regime
where the maximum appears. On increasing the droplet concentration, the
scattering intensity is found to scale with the inverse of the wavevector, a
behavior which is consistent with cylindrical structures. The inverse
wavevector scaling is not observed when the molar ratio is changed, moving the
system away from the value corresponding to the viscosity maximum. It is also
not present in the scattering from systems containing enough added salt to
essentially eliminate the viscosity maximum. An asymptotic analysis of the SAXS
data, complemented by some quantitative modeling, is consistent with
cylindrical growth of droplets as their concentration is increased. Such
elongated structures are familiar from related AOT systems in which the sodium
counterion has been exchanged for a divalent one. However, the results of this
study suggest that the formation of non-spherical aggregates at low molar
ratios is an intrinsic property of AOT.
