  We use liquid state theory and computer simulations to gain insights into the
shape of the structure factor for fluids of particles interacting via a
combination of short-range attractions and long-range repulsions. Such systems
can reversibly morph between homogeneous phases and states comprising compact
self-limiting clusters. We first highlight trends with respect to the presence
and location of the intermediate-range order (IRO) pre-peak in the structure
factor, which is commonly associated with clustering, for wide ranges of the
tunable parameters that control interparticle interactions (e.g., Debye
screening length). Next, for approximately 100 different cluster phases at
various conditions (where aggregates range in size from six to sixty monomers),
we quantitatively relate the shape of the structure factor to physical
characteristics including intercluster distance and cluster size. We also test
two previously postulated criteria for identifying the emergence of clustered
phases that are based on IRO peak-height and -width, respectively. We find that
the criterion based on peak-width, which encodes the IRO thermal correlation
length, is more robust across a wide range of conditions and interaction
strengths but nonetheless approximate. Ultimately, we recommend a hybrid
heuristic drawing on both pre-peak height and width for positively identifying
the emergence of clustered states.
