The two phases
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Almost every skincare product is two liquids that do not mix, persuaded to stay mixed. Once you read formulas as two phases rather than as a list, most of the craft becomes legible.
What an emulsion is
Droplets of one liquid suspended in another, held apart by an emulsifier that sits at the boundary. It is a metastable arrangement: energy was put in to create it and it is thermodynamically trying to come apart for the rest of its life.
Your job is to make the coming-apart slower than the shelf life.
Which way round it is
Oil in water is the common form for lotions and creams: oil droplets dispersed in a continuous water phase. It feels light, absorbs quickly, and rinses.
Water in oil is the reverse, used for richer barrier products. It feels occlusive and resists being washed off.
The difference is not cosmetic. The continuous phase decides how it feels, what it mixes with, and, crucially, where organisms can live.
Why the water phase is the dangerous one
Microorganisms need water. In an oil-in-water emulsion the water is continuous, which means anything introduced can move freely through the whole product.
That is why preservatives are chosen for water solubility and why pH, which is a property of the water phase, decides whether they work at all. An anhydrous product with no water phase is a different safety proposition entirely.
Reading a formula by phase
Group the ingredients: water phase, oil phase, emulsifier, cool-down additions, preservative. That grouping tells you the manufacturing sequence almost by itself, because the two phases are heated separately and combined hot, and anything heat-sensitive waits for the cool-down.
It also tells you where a problem probably lives. Separation is an emulsifier and process question. Contamination is a water phase and preservative question. Rancidity is an oil phase question. Almost nothing crosses over.