First things first, I’d like to say that the description sounds like [one thing](https://www.youtube.com/watch?v=NsQ4vv8ljQ4), but I think you [meant another](https://www.youtube.com/watch?v=75CtYoIMuOM). I’d like to strongly recommend not mixing ice and ***bubbling fats,*** unless you know what you’re doing and/or have sufficient safety precautions. The boiling point of oils is much higher than water, which causes flashboiling and potentially explosive steam creation; the second video I linked is at most bubbling ***soup*** which is still primarily water, and thus not bubbling fats.
To steal a turn of phrase from Hank Green, I think it stops becoming chemistry, and starts becoming physics. The reason why liquid fats are hydrophobic is because fats are nonpolar, and water is polar. Polar liquids and nonpolar liquids really don’t want to go into solution together.
But when you have an ice cube, which is far below the solidification temperature of those fats, it’s not so much a chemical “nope,” as it is a physical “yes.” The fats start to solidify, and any nucleation points will be on the ice. The fats stick to themselves, and it appears to me to be incredibly weakly bonded to the ice. It’s also likely melting a bit of the ice every time. Friction may be the only thing holding it on; perhaps van der Waals, but I think a combination of “The fats that stick are the cold ones and the hot ones don’t” and it being held still, with the fact that fats want to float, I could see this being entirely physical in nature, not chemical.
Solid fat won’t stick to solid ice, if they are both solid and cold at the time; they will not interact. Placing a cold substance in a less cool fluid will create a layer of frozen (recently fluid) stuff on the surface of the cold solid, and some of the solid at its surface will melt and mix in with the fluid converting to solid from the other side, so you end up with a mixed zone between the two where the one is intergrown with the other and that is what makes that layer stay. A little further away and the fat, even if solid, will easily separate from the contact zone.
The general point is that even though the oil isn’t all that happy being in contact with water, it doesn’t have a choice. The MUST BE some layer of contact between the two once the one is dropped into the other. That zone of contact is a zone of freezing of the liquid and slight melting of the solid, but things happen fairly fast so chemical segregation is imperfect and the zone is a mixture more than a solution.
Very few pure substances, like water or some specific oil, are totally immiscible, so not a single molecule of the one can dissolve into the other, and vice versa. They maximum amounts (concentrations) are just very low, so on a big scale, we can pretend that there is none (too little to make a big difference from our perspective, even if it is still there). Usually on the order of a few parts per million in fairly strongly incompatible substances. Trace contaminants do happen. Truly pure substances are extremely difficult to make.
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