All sorts of things happen. Yes, there will be some partly filled orbitals and free electrons. Conditions will change the situations at various points. As a rule, the further any particular spot deviates from neutrality, the stronger the forces for ejecting excess electrons, or capturing available electrons become. Examples of the resulting conditions often are particularly obvious when you peel apart layers of plastic and find them to have quite dramatic static charges that cause them to spark or attract each other in unexpected ways.

Of course, these things differ according to the material you work with. If it consists of large molecules, some of which break at the surface, you will get far messier results than if it is a crystal of say, salt or diamond, that splits neatly between planes of molecules, atoms, or ions.

The situation is not as tidy as that description makes it seem though. In fact it now is clear that most substances have a skin effect rather like the surface tension of water, where the molecules making up the surface are organised differently from the molecules in the mass. A few layers of them tend to form a usually comparatively undisciplined region which nowadays is the basis of some (if you'll excuse the expression) cutting-edge research.

This is no answer to the question, but I have been intrigued to read (I think) that ultimately fine probes have been made with one atom on the "point". And, I think, that the conduction of electricity to the surface of the object being scanned has been monitored. How does the electricity (electrons) get through the mainly empty atom ? ? ?  Can the single atom be regarded the same element as the thick bit ! !