The theory of Big Rip is gaining more credence now, due to the accurate observations of accelerated expansion of the universe. There doesn't seem to be enough evidence observed for a Big Crunch eventuality.
It is interesting to note at the time of big bang there didn't exist matter (protons, electrons, etc). After fraction of time the energy that took birth at big bang condensed into plasma-like material from which normal matter took came into existence.
At the fag-end of big rip, similarly all matter protons, electrons would seemingly rip apart and disintegrate and everything would dissipate into nothingness or dark energy (which is supposed to be the cause of this big rip we see today itself).
So in simpler words Big Bang would fizzle out into Big Rip, more like a firewood burns itself out into ashes (like they say in the bible -- "from ashes to ashes")