An emulsion is a suspension of one liquid in another that would not normally mix. A great many sauces are emulsions, and they fail for the same small set of reasons.
The mechanism
Oil and water do not mix because their molecules interact differently.
An emulsion forces one to disperse through the other as tiny droplets, which requires two things.
Mechanical energy to break the droplets small enough, and an emulsifier to keep them from recombining.
Which is why whisking alone produces a temporary emulsion that separates, and why adding an emulsifier produces a stable one.
Emulsifiers
Molecules with a water-attracting end and a fat-attracting end, which position themselves at the boundary and prevent droplets merging.
Egg yolk contains lecithin, which is why it is the classic emulsifier for mayonnaise and hollandaise.
Mustard contains compounds that work similarly, which is why a small amount stabilises vinaigrette.
Starch and proteins from other sources also work, which is why pasta water emulsifies a sauce and why a small amount of cream stabilises certain sauces.
Why they break
Adding oil faster than it can be dispersed, which leaves droplets too large to stay suspended.
Insufficient emulsifier for the quantity of oil.
Temperature shock, particularly in warm emulsions where heat causes the proteins to coagulate.
And sitting, since emulsions are thermodynamically unstable and separate eventually regardless.
Fixing a broken emulsion
The standard rescue works on the same principle.
Start again with a fresh emulsifier — a yolk, a spoonful of water, a little mustard — and add the broken mixture into it very slowly, whisking.
Which re-establishes a stable base and incorporates the broken material into it.
Adding a small amount of water can also work when the emulsion has become too thick and tight to hold more oil.
Temperature
Ingredients at similar temperatures emulsify more readily.
Which is why mayonnaise made with a cold yolk and warm oil frequently fails, and why letting everything reach room temperature helps.
Warm emulsions like hollandaise have a narrow temperature window — too cool and the butter solidifies, too hot and the eggs scramble.
Gentle indirect heat and constant movement are what keep it in range.
Butter sauces
Butter is itself an emulsion, of water droplets in fat.
Which is why cold butter whisked into a reduced liquid produces a sauce — the milk solids act as emulsifier and the butter's own structure disperses.
Adding it cold and in pieces keeps the temperature low enough that it does not simply melt and separate.
Modern emulsifiers
Lecithin can be bought in isolation, and various hydrocolloids stabilise emulsions in ways traditional ingredients cannot.
Which is what allows commercial dressings to remain stable on a shelf for months, and it is why they behave differently from a made-fresh version.
The practical version
Add oil slowly at the start and faster once established.
Use an emulsifier appropriate to the sauce.
Keep temperatures matched and moderate.
And know that a broken sauce is generally recoverable, which removes the anxiety that causes people to rush.
Vinaigrette proportions
The conventional ratio of three parts oil to one part acid is a starting point rather than a rule.
Which varies with the acid — a mild vinegar tolerates less oil than a sharp one, and citrus behaves differently again.
Tasting on a leaf rather than from a spoon is the reliable test, since dressing tastes very different alone.
Blender emulsions
Mechanical energy from a blender produces smaller droplets than whisking, which makes the emulsion more stable.
Which is why blender mayonnaise rarely fails and hand-whisked mayonnaise frequently does.
It also produces a thicker, denser result, which some prefer and others do not.
Food safety with raw egg
Mayonnaise and similar preparations contain raw egg, which carries a small salmonella risk.
Pasteurised eggs eliminate it and are widely available.
Guidance in several countries identifies pregnant women, young children, elderly people and those with compromised immunity as groups who should avoid raw egg preparations unless pasteurised.
Refrigeration and prompt use address the remaining risk for everyone else.
Stable against temporary
Vinaigrette is a temporary emulsion — it separates within minutes and is shaken again before use.
Mayonnaise is stable, holding for days refrigerated.
The difference is the quantity and effectiveness of the emulsifier relative to the oil.
Which means a vinaigrette can be made stable by adding an egg yolk or more mustard, at the cost of it becoming a different sauce.
Reduction as a stabiliser
A reduced liquid contains concentrated proteins and gelatine, which act as emulsifiers.
Which is why butter whisked into a reduced stock produces a stable sauce and butter whisked into water does not.
It is also why pan sauces work — the reduction concentrates exactly what is needed.
Which means understanding one sauce in this family gives you most of the others.