"Blackmail" may not be the most accurate word. Prospero actually bullies and threatens Ariel. He reminds him of how Sycorax kept him imprisoned, forcing Ariel to thank Prospero for the act of releasing him, but then Prospero goes on to threaten Ariel with precisely the same kind of imprisonment if he is disobedient and refuses to do what Prospero orders. Note the threat from Act I scene 2 where Prospero does this:
If thou more murmur'st, I will rend an oak,
And peg thee in his knotty entrails till
Thou hast howled away twelve winters.
This threat is successful (and shocking) because it is precisely the way that Sycorax imprisoned Ariel. The relationship between Prospero and Ariel is one that is of particular interest in this masterful play, as Prospero can be viewed as being far from the benevolent ruler he sometimes tries to be taken for. He can be viewed as exploiting Ariel most cruelly for his own purposes.