Tunable fractal shapes in self-avoiding polygons and planar vesicles

The shapes of self-avoiding continuum and lattice polygons of N monomers in a plane are studied using Monte Carlo simulations and exact enumeration. To model vesicles, a pressure increment Δp=${\mathit{p}}_{\mathrm{in}}$-${\mathit{p}}_{\mathrm{out}}$, is included. For N≫1 and Δp=0, the usual universal fractal shapes appear; but for Δp≠0, continuously variable fractal shapes are found controlled by the variable x∝Δ${\mathit{pN}}^{2\nu }$ where ν=1/${\mathit{D}}_{\mathit{F}}$=3/4. Thus, the ratio of principal radii of gyration Σ(x)=〈${\mathit{R}}_{\mathit{G},\mathrm{m}\mathrm{i}\mathrm{n}}^2$〉/〈${\mathit{R}}_{\mathit{G},\mathrm{m}\mathrm{a}\mathrm{x}}^2$〉 changes smoothly from Σ(+∞)=1, for circles, through Σ(0)≃0.39, to Σ(-∞)≃0.23, which corresponds to branched polymers.