Not Even Metastable: Cubic Double-Diamond in Diblock Copolymer Melts

We study the thermodynamics of continuous transformations between two canonical, cubic network phases of block copolymer melts: double-gyroid, an equilibrium morphology for many systems, and double-diamond, often thought to be a close competitor. We use a strong-segregation approach to compute the free energy of double network morphologies as a function of two structural parameters that convert between two limiting cubic cases: a tetragonal stretch of the unit cell in combination with fusion of pairs of trihedal gyroid nodes into tetrahedral diamond nodes. For the simplest case of conformationally symmetric diblock melts, we find that cubic double-diamond sits at an unstable saddle point that is continuously deformable into the lower free energy gyroid, as well as a second metastable, tetragonal network composed of trihedral nodes. We confirm the broad instability of double-diamond at finite segregation using self-consistent field studies and further show that it derives directly from the entropic free energy cost of chain packing in the tubular domains of tetrahedral nodes. Correspondingly, we demonstrate two factors that quench the entropic cost of packing in the tubular domain─homopolymer blending and elastic asymmetry between the blocks─and promote double-diamond to a metastable state by way of a free energy barrier that separates it from double-gyroid.