The difference between Gibbs and grand canonical potentials

Question

To my understanding, Helmholtz free energy is a measure of the amount of work a thermodynamical system can perform. Following the same reasoning Gibbs and Grand-canonical free energies measure the the part of the energy free to do work that is not by $PV$ or $mu N$ respectively.
Now, assuming $F=0$, you get: $$mu=P=0$$ so that $G=Phi=0$. But here seems to be the part that I am missing. Couldn't available work by chemical potential and expansion cancel each other so that $F=0$ without $G$ or $Phi$ zeroing as well?

L. Levrel · Answer

$V$ and $N$ are independent variables. If $F$ is identically zero, there can be no work, neither by expansion nor by chemical potential. So $Φ$ and $G$ must be zero, respectively.

The difference between Gibbs and grand canonical potentials

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