  We assess the contribution of dynamical hardening by direct three-body
scattering interactions to the rate of stellar-mass black hole binary (BHB)
mergers in galactic nuclei. We derive an analytic model for the single-binary
encounter rate in a nucleus with spherical and disk components hosting a
super-massive black hole (SMBH). We determine the total number of encounters
$N_{\rm GW}$ needed to harden a BHB to the point that inspiral due to
gravitational wave emission occurs before the next three-body scattering event.
This is done independently for both the spherical and disk components. Using a
Monte Carlo approach, we refine our calculations for $N_{\rm GW}$ to include
gravitational wave emission between scattering events. For astrophysically
plausible models we find that typically $N_{\rm GW} \lesssim$ 10.
  We find two separate regimes for the efficient dynamical hardening of BHBs:
(1) spherical star clusters with high central densities, low velocity
dispersions and no significant Keplerian component; and (2) migration traps in
disks around SMBHs lacking any significant spherical stellar component in the
vicinity of the migration trap, which is expected due to effective orbital
inclination reduction of any spherical population by the disk. We also find a
weak correlation between the ratio of the second-order velocity moment to
velocity dispersion in galactic nuclei and the rate of BHB mergers, where this
ratio is a proxy for the ratio between the rotation- and dispersion-supported
components. Because disks enforce planar interactions that are efficient in
hardening BHBs, particularly in migration traps, they have high merger rates
that can contribute significantly to the rate of BHB mergers detected by the
advanced Laser Interferometer Gravitational-Wave Observatory.
