Is it obligatory for one to supplicate silently or aloud for the angels to say “Ameen” to the supplication?

It is permissible to make supplication aloud or silently, but doing so silently is better, based on the Almighty's saying: (Call upon your Lord humbly and privately; indeed, He does not like aggressors) [7:55]. Also, because it is more conducive to sincerity. Praying silently does not negate the angels' 'Amin' (amen), for in Sahih Muslim: "There is no Muslim servant who supplicates for his brother in his absence except that the angel says: 'And for you the like of it.'" And the meaning of "in his absence" is in the absence of the one being supplicated for and in secret, because it is more effective in sincerity.