Steiner Triple Systems Intersecting in Pairwise Disjoint Blocks

Two Steiner triple systems $(X,{\cal A})$ and $(X,{\cal B})$ are said to intersect in $m$ pairwise disjoint blocks if $|{\cal A}\cap{\cal B}|=m$ and all blocks in ${\cal A}\cap{\cal B}$ are pairwise disjoint. For each $v$, we completely determine the possible values of $m$ such that there exist two Steiner triple systems of order $v$ intersecting in $m$ pairwise disjoint blocks.