What is an egg emulsion? All good recipes for hollandaise sauce consist of beating egg yolks vigorously and slowly add melted butter over moderately low heat.  Care must be taken to not overheat the sauce or the eggs will scramble and it won't emulsify.  Likewise, the melted butter must not be added too quickly or it will not emulsify. When done properly like a good chef, what is happening here?  Why does the sauce thicken?  What is an egg emulsion and why must the butter be added so slowly rather than faster?
To know what's happening here you should know what "emulsifying" is.

First we need to know that fat is "non-polar" substance. what does that mean?

It means that all fats can't be dissolved in water which is "polar" substance.

So how can we make fat -which is non-polar- dissolve in water or any other polar substances?

We add "emulsifiers" like: egg yolk and many other chemical substances like Lecithin.

These emulsifiers are (polar and non-polar) so they can link fat with water together.

We add butter slowly so that the molecules of emulsifier can link the molecules of butter with molecules of sauce -which is polar-.

If there's no free molecules of emulsifier, every molecule of added butter will not be dissolved in the sauce.