The edge of an advancing or receding two-dimensional sessile drop is s
tudied using Monte Carlo dynamics and analytical arguments. It is foun
d that for disordered substrates an advancing angle theta(a) or a rece
ding angle theta(r), both different from the equilibrium angle, appear
and remain stable during a lifetime much larger than the time needed
to reach equilibrium with a pure substrate. Much larger means a ratio
tending to infinity in the thermodynamic limit.