  We investigate the phase transition between an ergodic and a many-body
localized phase in infinite anisotropic spin-$1/2$ Heisenberg chains with
binary disorder. Starting from the N\'eel state, we analyze the decay of
antiferromagnetic order $m_s(t)$ and the growth of entanglement entropy
$S_{\textrm{ent}}(t)$ during unitary time evolution. Near the phase transition
we find that $m_s(t)$ decays exponentially to its asymptotic value
$m_s(\infty)\neq 0$ in the localized phase while the data are consistent with a
power-law decay at long times in the ergodic phase. In the localized phase,
$m_s(\infty)$ shows an exponential sensitivity on disorder with a critical
exponent $\nu\sim 0.9$. The entanglement entropy in the ergodic phase grows
subballistically, $S_{\textrm{ent}}(t)\sim t^\alpha$, $\alpha\leq 1$, with
$\alpha$ varying continuously as a function of disorder. Exact diagonalizations
for small systems, on the other hand, do not show a clear scaling with system
size and attempts to determine the phase boundary from these data seem to
overestimate the extent of the ergodic phase.
