Separation of oil-water mixture and entropy

Imagine having the same volumes of olive oil and water in a bottle. When we shake the bottle long enough, we disperse the oil in the water and create an emulsion. However, if we wait, the oil will collect at the top and the water at the bottom. This process is spontaneous and therefore we can conclude that the entropy of the emulsion is lower than the entropy of the system with the oil and the water separated. The entropy is a measure of the available phase space for a given macro state. Thus, that available phase space grows during de-mixing.

But why?

My idea so far is the following. Entropy is an additive quantity. Thus, we should be able to calculate it for the water and oil phase separately and add them together to get the total entropy. Directly after mixing, both oil and water occupy the same spatial volume. After de-mixing they each only occupy half the volume. Thus, the volume of their phase spaces’ momentum parts must grow during de-mixing. But I cannot explain why that is. I suppose it has something to do with the van-der-Waals forces between the water molecules.