the homopolymer at no free energy cost. To distinguish between crystal structures within a particular topology, such as between hexagonal and square for the cylindrical topologies, we must examine the energy for packing the molecules into the particular geometry, which is performed below.

=3.0truein

Non-round unit cells (straight paths)
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We can produce an [*upper*]{} bound for different structures by assembling small pieces of the cylindrical or spherical micelles to fill the appropriate unit cell. Each wedge has a parabolic monomer chemical potential given by eq \[eq:mu\]. However, each wedge $\alpha$ has a slightly different shape and geometry, and thus has a distinct potential $\mu_{\alpha}$. Adjacent wedges are not in equilibrium with each other and will relax if allowed to do so. Hence the calculation yields an upper bound. To construct the unit cell of, [*e.g.*]{}, hexagonal cylinders, we assume a hexagonal dividing surface scaled down by $\phi^{1/2}$ and assemble the unit cell from tiny pie-shaped wedges extending from the center of the hexagon to the cell boundary. We make an analogous construction for square arrays of cylinders, or for FCC and BCC packings of spheres.

We calculate the volume-averaged stretching free energy per molecule using eq \[eq:avg\]. To calculate 