Can Electrons Leave a Circuit Through an Open Branch?

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Hi. I'm not sure if this is the right place to pose this question as it is not homework question.
I'm studying RC circuits in second semester physics and I have a question about the electrons traveling through a circuit. Is it true that electrons won't travel down a branch of a circuit in parallel if it is not connected to anything on one end? If that is true, why do they not travel down those paths?
 

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Yeah it's true. If the electrons have nowhere to go, then there will be a pile up of charge. Obviously a pile up of charge can't last, since electrons repel. Another way to look at it is that the concentration of charge will result in an electric field that opposes the one that was driving the electrons in the first place. Equalization will happen quickly. You need a complete circuit to have a current.
 
Thank you for the reply.

Does that mean that electrons will continuously accumulate in the broken branch and then scatter?
 
The electrons can not jump out of a conductor at a free end unless it is at very high voltage to make a spark.
The electrons can move freely inside the bulk of the conductor, but kept there by the Coulomb forces of the positive ions. They say that the conduction electrons are confined in a "potential well" inside the metal, and they need a certain amount of energy (Work function, a few eV) ) to escape.

ehild