  Star-to-star dispersion of r-process elements has been observed in a
significant number of old, metal-poor globular clusters. We investigate
early-time neutron-star mergers as the mechanism for this enrichment. Through
both numerical modeling and analytical arguments, we show that neutron-star
mergers cannot be induced through dynamical interactions early in the history
of the cluster, even when the most liberal assumptions about neutron-star
segregation are assumed. Therefore, if neutron-star mergers are the primary
mechanism for r-process dispersion in globular clusters, they likely result
from the evolution of isolated, primordial binaries in the clusters. Through
population modeling, we find that moderate fractions of GCs with enrichment are
only possible when a significant number of double neutron-star progenitors
proceed through Case BB mass transfer --- under various assumptions for the
initial properties of globular clusters, a neutron-star merger with the
potential for enrichment will occur in ~15-60% (~30-90%) of globular clusters
if this mass transfer proceeds stably (unstably). The strong anti-correlation
between the pre-supernova orbital separation and post-supernova systemic
velocity due to mass loss in the supernova leads to efficient ejection of most
enrichment candidates from their host clusters. Thus, most enrichment events
occur shortly after the double neutron stars are born. This requires
star-forming gas that can absorb the r-process ejecta to be present in the
globular cluster 30-50 Myr after the initial burst of star formation. If
scenarios for redistributing gas in globular clusters cannot act on these
timescales, the number of neutron-star merger enrichment candidates drops
severely, and it is likely that another mechanism, such as r-process enrichment
from collapsars, is at play.
