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Oil and water normally don't mix, but with an emulsifier (egg yolk, mustard, or even starch) and the right technique, they create stable creamy sauces like mayonnaise, hollandaise, and vinaigrettes. The emulsifier coats the tiny oil droplets, keeping them suspended in water. If the emulsion breaks (too much heat or oil added too fast), the sauce separates and looks curdled.
Make mayonnaise by whisking together egg yolk and oil, adding oil slowly at first. Try adding oil too fast and watch it break. Repair a broken emulsion by starting fresh with a new egg yolk and slowly whisking in the broken sauce.
Oil and water are chemically incompatible: oil molecules are nonpolar (electrically neutral and symmetrical), while water molecules are polar (with slightly positive and negative ends). Polar molecules are attracted to other polar molecules, so water molecules cluster together and push out the nonpolar oil. If you shake oil and water in a jar, they appear to mix momentarily, but they quickly separate because there is no force holding the tiny oil droplets suspended in water. An emulsion is a stable mixture of oil and water — and the ingredient that makes it stable is an emulsifier.
An emulsifier is a molecule with a split personality: one end is hydrophilic (water-loving, polar) and the other is lipophilic (fat-loving, nonpolar). When dispersed in a mixture of oil and water, emulsifier molecules migrate to the interface between oil droplets and the surrounding water. They coat each droplet with their lipophilic ends pointing inward into the oil and their hydrophilic ends pointing outward into the water, forming a protective layer that prevents the droplets from coalescing back together. The emulsifier doesn't make oil and water "like" each other — it physically keeps them apart in a stable dispersed state. Lecithin in egg yolk is the emulsifier in mayonnaise and hollandaise. Mucilage compounds in mustard help stabilize vinaigrettes. Even finely ground mustard powder acts as a minor emulsifier.
Technique matters alongside chemistry. Adding oil too quickly means you're introducing more oil than the available emulsifier molecules can coat. The droplets merge before they can be stabilized, and the sauce breaks — separating into an oily layer and a watery layer. Adding oil in a thin, slow stream while whisking vigorously does two things: it breaks the oil into tiny droplets (giving each more surface area that needs coating) and it disperses the emulsifier to where it's needed. Once you've established a stable base emulsion, you can add oil more quickly — but the first tablespoon or two sets the foundation.
A broken emulsion is not a disaster. The ingredients have separated, but they are still all there and chemically unchanged. To rescue it, start fresh with a new egg yolk in a clean bowl, then slowly whisk in the broken sauce as if the broken sauce were the "oil" you're adding. The new yolk provides fresh lecithin to re-coat the now-combined oil droplets. This works because the emulsification process is purely physical — you're just reorganizing the arrangement of the same molecules. Heat breaks emulsions (by disrupting the lecithin coating), which is why hollandaise must be kept warm but not hot, and why adding cold butter to a beurre blanc is done off heat and in small increments.