Frank Elastic Constants of Semiflexible Polymer Solutions

We derive the Frank elastic constants for nematic solutions of semiflexible polymers. We plot these results as a function of the coarse-grained Maier-Saupe quadrupole aligning strength and polymer stiffness ranging from rigid to highly flexible. The derivation uses the random phase approximation and combines the exact results for the statistics of a worm-like-chain with polymer field theory using a spherical harmonic basis. The results are evaluated using a numerical inverse Laplace transform. We present the results in terms of microscopic features such as hairpins and polymer ends so the trends can be understood independently from the derivation. Key findings are that for rigid polymers $K_{bend}>K_{splay}>K_{twist}$ while for flexible polymers $K_{splay}>K_{bend}>K_{twist}$. For rigid polymers, the Frank elastic constants grow with the polymer length. For flexible polymers the elastic constants grow with the persistence length, which becomes the characteristic length scale, with the exception of $K_{splay}$ at high alignment strengths which grows with polymer lengths due to the elimination of hairpins.