Spherical micelle geometry ($R=3v/a_0,n=36\pi v^2/a_0^3$)

In a sharp-core spherical micelle of aggregation number $n$, incompressible tail volume $v$ per molecule and effective head-group area $a_0$ give $nv=4\pi R^3/3$ and $n{a}_0=4\pi R^2$. Division gives the displayed radius and aggregation number. If the maximum tail extension is $l_c$, the model requires $R\le l_c$, equivalently the amphiphile packing parameter $v/(a_0{l}_c)\le 1/3$. Input dimensions violating that inequality cannot describe an exact spherical filled core without modifying the molecular geometry assumptions.