Let $\mathtt{DGA}$ be the category of non-negatively graded DG chain algebras, and $\mathtt{DGA}$* the category of non-negatively graded cochain algebras. Let $\mathtt{CommDGA}$ be the full subcategory of commutative algebras in $\mathtt{DGA}$, and $\mathtt{CommDGA}$* likewise for $\mathtt{DGA}$*.

There is a well-known (Sullivan) closed model structure on $\mathtt{CommDGA}$* (with fibrations being surjections), and there is an analogous structure on $\mathtt{DGA}$* (Jardine, “A closed model category structure for differential graded algebras,” http://ncatlab.org/nlab/files/JardineModelDG.pdf).

There is also a closed model category structure on $\mathtt{DGA}$ (see, e.g., Baues and Pirashvili, “Comparison of MacLane, Shukla, and Hochschild cohomologies”). In this case, though, the fibrations are maps which are surjections on positive (i.e. nonzero) degree terms.

The question is, is there a corresponding closed model structure on $\mathtt{CommDGA}$? I haven't been able to find anything in the literature, which seems surprising...