Will atoms evaporate given enough time? Thought experiment: someday in the distant future, space will be expanding faster than the speed of light.  Imagine that trillions of years in the future, you have a single hydrogen atom.  Will that atom evaporate into quarks and gluons?
Here's my thinking.  The location of the electron is a probability function.  Even though it's highly improbable, given enough time that electron will appear x millimeters away from the nucleus.  If the universe is expanding at an acceleration of A, there must be some x where the acceleration caused by the electromagnetic force isn't as strong as A.  If the electron appears outside of x, then it will be lost.
The same thinking then goes into all the other particles that are exchanged that make up the strong and weak forces.
Will atoms eventually evaporate?
 A: Hydrogen atoms should all end up ionized, but not for the specific reason you give. The reason you give only applies in a Big Rip scenario, which is not currently believed to be what will happen in our universe, although we can't yet rule it out.
For a classical, gravitationally bound system, the secular trend in the size is given by $\dot{r}/r\sim (d/dt)(\ddot{a}/a)$, where $a$ is the cosmological scale factor. See Can the Hubble constant be measured locally? . This is currently much too small to produce any measurable effects, and moreover in our current cosmological models it goes to zero at large times, because for a universe dominated by dark energy, $\ddot{a}/a$ is a constant. I would expect essentially the same null result for a hydrogen atom.
However, for much more ordinary thermodynamic reasons, we do expect all hydrogen atoms to be ionized in the distant future. See Dark age of universe when all fusion process ceases?
As far as we know, the electron is stable. The proton may or may not be stable. We don't know.
Some matter will end up in black holes and then presumably get recycled into Hawking radiation, but this is not expected to happen to most matter.
A: The answer to your question ultimately comes down to a showdown between dark energy and the fundamental forces.
Contrary to popular belief, each and every system (from the biggest scale to the smallest) is expanding (there are certain ideas that inside galaxies, where gravity is strong, matter is not expanding at all), the effects are just too little to measure.

There is no critical density, and all of the systems you mentioned are expanding, but the effect is much too small to measure.

How is expansion of universe seen locally?

The catch with dark energy is that it has a constant energy-density, despite the expansion of space1. To paint a simple picture, as space expands, more dark energy is "created" so that the energy-density of dark energy remains unchanged.

Is Dark Energy A Constant?
Now there are basically two cases:


*

*dark energy is going to eventually overcome the EM and the strong force, even at the smallest scales. In this case, what you mention will happen, and electrons will be separated from the nucleus, and even the nucleus itself will be separated into its constituents. This is where it is very important to understand what hadronization and confinement means. What is actually might happen is that as dark energy overcomes the confinement of the strong force, the color tube it creates will eventually break and create a new particle antiparticle pair.



The strong interaction has the property of confinement, which in the case of trying to separate a pair of quark qq¯ is adequately pictured by a colour tube stretching between the quarks until it breaks, producing new q′q¯′ pairs.

fate of a hadron in a big rip


*the fundamental forces will remain stronger then dark energy on the smallest scales. This might cause atoms to stay intact (though as you can see from the other answer they might fall apart due to thermodynamic reasons). Eventually most of the matter will fall to atoms and will be separated by space expansion and the only macro scale pieces of matter will be black holes. 


The fate of black holes again depends on the showdown between dark energy and the forces.

An answer that got deleted linked to this paper:Babichev, Dokuchaev, Eroshenko, "The Accretion of Dark Energy onto a Black Hole," arxiv.org/abs/astro-ph/0505618

Will the Big Rip tear black holes apart?
And for gravity vs dark energy, to our current knowledge, gravity on the small scale does win over dark energy.
Is dark energy around a black hole locally curved?
