  By means of time- and temperature-dependent magnetization measurements, we
demonstrate that the timescale of hydrogen diffusion in partially-hydrogenated
LaFe$_{13-x}$Si$_x$H$_y$ is of the order of hours, when the material is held at
temperatures close to its as-prepared Curie temperature, $T_{C0}$. The
diffusion constant is estimated to be $D \approx 10^{-15}$ - $10^{-16}$
m$^2$s$^{-1}$ at room temperature. We examine the evolution of a magnetically
phase-separated state upon annealing for 3 days at a range of temperatures
around $T_{C0}$, and show that the thermodynamic driving force behind hydrogen
diffusion and phase segregation may be attributed to the lower free energy of
hydrogen interstitials in the ferromagnetic state relative to the paramagnetic
state.
