Gfilui

By the second law of thermodynamics, entropy tends to increase when the system is let on itself. And if entropy is a measure of disorder, how come mixing oil in water and letting the system reach equilibrium, ends up with the oil and water well separated? I see no disorder whatsoever, while in reality the entropy increased compared to the initial state (oil and water seemingly well mixed by e.g. shaking the container).

This is because entropy has almost nothing to do with the apparent order or disorder you can see with your naked eye. That's just a pop science simplification.

Compare the entropy of a dictionary and an identically sized book full of random gibberish. You might think the latter has a higher entropy, because the content is disordered. But the entropy of the words in the book is not even a million millionth of the total entropy, which overwhelmingly comes from thermal motion of the molecules in the paper. (Compare the number of characters in the book to the number of molecules, on the order of $10^23$.) If you hold the dictionary for even a second, the heat from your hand will make the entropy of the dictionary higher.

In the case of oil and water, it is energetically favorable for the oil to be separate from the water, because these molecules bind more strongly to themselves than to each other. The extra energy released is now available to thermal motion of the oil and water molecules, or to the surrounding air molecules, increasing the entropy of the universe. This overwhelms the decrease of entropy associated with clumping the oil together.