Irregularity of polymer domain boundaries in two-dimensional polymer solution

Polymer chains composing a polymer solution in strict two dimensions (2D) are characterized with irregular domain boundaries, whose fractal dimension ($\mathcal{D}^{\partial}$) varies with the area fraction of the solution and the solvent quality. {\color{black}Our analysis of numerical simulations of polymer solutions finds} that $\mathcal{D}^{\partial}$ in good solvents changes non-monotonically from $\mathcal{D}^{\partial}=4/3$ in dilute phase to $\mathcal{D}^{\partial}=5/4$ in dense phase, maximizing to $\mathcal{D}^{\partial}\approx 3/2$ at a crossover area fraction $\phi_{\rm cr}\approx 0.2$, whereas for polymers in $\Theta$ solvents $\mathcal{D}^{\partial}$ remains constant at $\mathcal{D}^{\partial}=4/3$ from dilute to semi-dilute phase. Using polymer physics arguments, we rationalize these values, and show that the maximum irregularity of $\mathcal{D}^\partial\approx 3/2$ is due to "fjord"-like corrugations formed along the domain boundaries which also maximize at the same crossover area fraction. Our finding of $\mathcal{D}^\partial\approx 3/2$ is, in fact, in perfect agreement with the upper bound for the fractal dimension of the external perimeter of 2D random curves at scaling limit, which is predicted by the Schramm-Loewner evolution (SLE).