Not sure how should I be during that time and.... Should I initiate any conversation? If she talks about things, what should I do? How to deal with awkward silence which I am sure will be there for most part?
Would you have a chat about your relationship and what is going on if you went to the zoo with someone that isn't your partner? If not, then just chat about whatever you would chat about with a friend that you went to the zoo with.
If she starts a conversation, you can just listen and validate her feelings. If she starts attacking and blaming, tell her the conversation needs to be over. Be able to walk away if needed.
Fill the awkward silence with conversation with your daughter. Awkward silence being filled is a needy thing. Be pleasant and present, but you don't have to talk all the time just to fill the silence.
You are there to make the day about your daughter. Make it about her. If W joins you and is pleasant, then it is still about your daughter. If she is nasty and leaves, it is still about your daughter. If she wanders off and gets eaten by a tiger, it is still about your daughter. Be the dad you want to be for her.
Me40; W38; S12; D9 BD11/19/2018 D filed 12/20/18 D Final 7/2020 Being the best example I know how for my kids to see.