Emulsifying
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This is the step where a lotion becomes a lotion, and it takes about a minute. Almost everything that determines whether it holds for two years happens in that minute.
Energy is the ingredient
An emulsion needs work put into it to break one phase into droplets small enough for the emulsifier to stabilize. Smaller droplets are more stable, because the rate at which they find each other and merge falls sharply with size.
A stick blender or homogenizer provides that energy. Stirring does not, and an emulsion made with a spoon is one that will separate on a schedule you will not like.
Which phase into which
For an oil-in-water emulsion, add the oil phase into the water phase while mixing. You are dispersing oil droplets into a continuous water phase, and pouring the water into the oil encourages the wrong arrangement and can cause the batch to invert.
Add steadily rather than all at once, with the mixer already running.
How long, and how you know
Until it is uniform and has visibly thickened, usually a minute or two at high shear rather than ten. Over-mixing is a real thing: it incorporates air, which is both a cosmetic defect and a route for oxidation, and excess energy can begin to break the structure you just built.
Then continue gentle mixing through the cool-down, because an emulsion that is left to stand while still hot and thin will cream before it sets.
What goes wrong here
Phases at different temperatures, which is the commonest cause of a batch that never looks right.
Insufficient shear, producing large droplets that merge over weeks.
Air incorporation from mixing too long or with the blade breaking the surface, which shows as a whipped appearance and shortens the product's life.
And an HLB mismatch, which no amount of technique repairs. If an emulsion will not form with proper equipment and matched temperatures, the problem is the formula and not the minute.