Here is a proof that $f(n)=30$ for all $n \geq 32$.  Let $I$ be the icosahedron.  Let $G$ be obtained from $I$ by adding a degree-$3$ vertex inside each face of $I$.  Then $G$ is a planar graph with $32$ vertices, $20$ of which have degree $10$, and $12$ of which have degree $3$.  Since no two degree-$3$ vertices are adjacent in $G$, every edge $e$ of $G$ satisfies $D(e) \geq 30$.  Adding isolated vertices to $G$ gives examples for all $n \geq 32$ (if you don't like this there are other ways to make larger connected examples with minimum degree $3$).