2 oops

If your question is simply whether the 4-D associahedron is dual to a simplicial polytope, the answer is yes, because all associahedra are simple polytopes. To see this, note that the vertices of $K_{d+2}$ correspond to strings of $d+2$ letters "saturated" by $d$ pairs of parentheses. The $d$ edges in the star of a vertex therefore correspond to removing any one of those $d$ pairs of parentheses.

However, deltahedra are simplicial polyhedra whose faces are all equilateral triangles, so maybe you are asking whether the simplicial polytopes dual to associahedra may be realized with faces that are regular simplices? Then the results of this paper of John Sullivan's which classifies "convex deltatopes" imply that the duals of higher dimensional associahedra cannot be convex deltatopes (I checked that the convex deltatopes he constructs do not have the right number of faces once the dimension is greater than 3), and I suspect that one may be able to show that the dual simplicial polytopes of associahedra can't be made into deltatopes at all.

On a side note I recommend changing the title of the question and making it more clear in the body precisely what you are asking. The reference to equilibrium positions of ions, while interesting, threw me off.

1

If your question is simply whether the 4-D associahedron is dual to a simplicial polytope, the answer is yes, because all associahedra are simple polytopes. To see this, note that the vertices of $K_{d+2}$ correspond to strings of $d+2$ letters "saturated" by $d$ pairs of parentheses. The $d$ edges in the star of a vertex therefore correspond to removing any one of those $d$ pairs of parentheses.

However, deltahedra are simplicial polyhedra whose faces are all equilateral triangles, so maybe you are asking whether the simplicial polytopes dual to associahedra may be realized with regular simplices? Then the results of this paper of John Sullivan's which classifies "convex deltatopes" imply that the duals of higher dimensional associahedra cannot be convex (I checked that the convex deltatopes he constructs do not have the right number of faces once the dimension is greater than 3), and I suspect that one may be able to show that the dual simplicial polytopes of associahedra can't be made into deltatopes at all.

On a side note I recommend changing the title of the question and making it more clear in the body precisely what you are asking. The reference to equilibrium positions of ions, while interesting, threw me off.