There you are, stuck in a cold war between teammembers in your team. They don’t want to help each other, maybe just don’t like each other or got pissed off with each other. And you, you are responsible for the project. What to do with two members in your team acting like children.

First what not to do.

Do not become the parent!
If you tell them them off and tell them they should stop quarreling or basically in any way stand above them and say: “stop it”. They will turn against you, like siblings, suddenly realizing who is the real enemy.

What to do.

First iterate the communal goal: “We are all in it together” (or something like it).
Then call them on the situation in an adult way, just state the situation or the obvious: “I am noticing there is a silence between you two”.
And then stay quiet! Let them talk. If they start to quarrel again, then raise them back to the adult level. You do that by asking them how the situation can be resolved between them to reach the goal of the project. Always iterate the communual goal again and again.

Most people will sputter several times. You have to apply the same to every sputter. Stay calm and above the situation. If you notice the sputters become less, you are doing it right.

Good luck. If you have any management questions I’m happy to answer them.

Background: I have 10+ years in different start-ups. My background is a PhD in Chemistry and have studied project management. I launched many products with international teams enjoyed working as a program manager under the CEO in a start-up company that was sold for $250M.
Currently I'm building a start-up company myself, that won many (subsidy) grants, and pitch awards. It launched its first product to market in December 2017 within two years