What if?

So I'm reading the Silmarillion for the first time and I am wondering, with Melkors end game being the destruction of Eä in its entirety, had he defeated the free peoples of Middle Earth, would Sauron eventually turn on him? Not only out of self preservation, but also his need for order and control? They have to very different end games so I am curious as to whether Tolkien ever wrote his thoughts on that. It seems to me, knowing Melkors end game and Saurons nature he would have to turn on him eventually right? Or am I missing something?