Is there a good description of the homotopy type of a free simplicial ring (or simplicial $R$-algebra) on a given simplicial set, in terms of the homotopy type of that simplicial set?

(This is mostly an idle question, but also motivated by the fact that it is a theorem of Milnor that a similar construction with the free group gives a model for the $\Omega \Sigma X$, perhaps believable in view of the fact that $\Omega \Sigma$ is supposed to be the left adjoint from spaces into grouplike $A_\infty $-(i.e., with a coherently associative multiplication law) spaces.)