Mergers of binary black holes on eccentric orbits are among the targets for second generation ground-based gravitational-wave detectors. These sources may commonly form in galactic nuclei due to gravitational-wave emission during close flyby events of single objects. We determine the distributions of initial orbital parameters for a population of these gravitational-wave sources. Our results show that the initial dimensionless pericenter distance systematically decreases with the binary component masses and the mass of the central supermassive black hole, and its distribution depend sensitively on the highest possible black hole mass in the nuclear star cluster. For a multi-mass black hole population with masses between 5 Msun and 80 Msun, more than 40% of sources have an eccentricity greater than 0.1 when the gravitational-wave signal reaches 10 Hz or if it forms at higher frequencies, but only 7% of the sources with binary component masses less than 30 Msun remain eccentric at this level near the last stable orbit (LSO). The eccentricity at LSO is typically between 0.005-0.05 for the lower mass BHs, and 0.1-0.2 for the highest mass BHs. Thus, due to the limited low-frequency sensitivity, the currently known six quasicircular LIGO/Virgo sources are compatible with this originally highly eccentric source population. At the design sensitivity of these instruments, the measurement of the eccentricity and mass distribution of merger events may be a useful diagnostic to distinguish among different astrophysical binary formation channels.