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 {Δ}p=pᵢₙ-pₒᵤₜ, is included. For N{}1 and {Δ}p=0, the usual universal fractal shapes appear; but for {Δ}p{≠}0, continuously variable fractal shapes are found controlled by the variable x{∝}{Δ}pN^2ν where {ν}=1/DF=3/4. Thus, the ratio of principal radii of gyration {Σ}(x)=〈RG,min²〉/〈RG,max²〉 changes smoothly from {Σ}(+{∞})=1, for circles, through {Σ}(0){}0.39, to {Σ}(-{∞}){}0.23, which corresponds to branched polymers.