  We prove the existence of $(20-2K^2)$-dimensional families of
simply-connected surfaces with ample canonical class, $p_g=1$, and $1 \leq K^2
\leq 9$, and we study the relation with configurations of rational curves in K3
surfaces via $\mathbb Q$-Gorenstein smoothings. Our surfaces with $K^2=7$ and
$K^2=9$ are the first surfaces known in the literature, together with the
existence of a $4$-dimensional family for $K^2=8$.
