Generalized Flory Theory for Rotational Symmetry Breaking of Complex Macromolecules

We report on spontaneous rotational symmetry breaking in a minimal model of complex macromolecules with branches and cycles. The transition takes place as the strength of the self-repulsion is increased. At the transition point, the density distribution transforms from isotropic to anisotropic. We analyze this transition using a variational mean-field theory that combines the Gibbs-Bogolyubov-Feynman inequality with the concept of the Laplacian matrix. The density distribution of the broken symmetry state is shown to be determined by the eigenvalues and eigenvectors of this Laplacian matrix. Physically, this reflects the increasing role of the underlying topological structure in determining the density of the macromolecule when repulsive interactions generate internal tension. Eventually, the variational free energy landscape develops a complex structure with multiple competing minima.

Figure 1

Two-dimensional density profiles obtained using the GBF variational method for a 36 node branched polymer with a maximum of $M=18$ nonzero mode expectation values. Top left: Graph of the molecule. Top right: $v/a^2=0.24$ in units of $k_{B}T$. The density profile is isotropic. Bottom right: $v/a^2=1.20$. A few low-lying modes have condensed. Bottom left: $v/a^2=2.05$. Most modes have condensed. White space bar: $5a$.

Figure 2

Density profiles obtained by Monte Carlo simulation for the same molecule and interaction strengths as Fig. 1. White space bar: 5a.