Entropic surfaces represented by fluctuating 2D membranes are predicted to have desirable mechanical properties when unstressed, including a negative Poissonâ??s ratio (â??auxeticâ? behavior). Herein, we present calculations of the strain-dependent Poisson ratio of self-avoiding 2D membranes demonstrating desirable auxetic properties over a range of mechanical strain. Finite size membranes with unclamped boundary conditions have positive Poissonâ??s ratio due to spontaneous non-zero mean curvature, which can be suppressed with an explicit bending rigidity in agreement with prior findings. Applying longitudinal strain along a singular axis to this system suppresses this mean curvature and the entropic out-of-plane fluctuations, resulting in a new, molecular scale mechanism for realizing a negative Poissonâ??s ratio above a critical strain, with values significantly more negative than the previously observed zero-strain limit for infinite sheets. We find that auxetic behavior persists over surprisingly high strains of more than 20% for the smallest surfaces, with desirable finite size scaling producing surfaces with negative Poissonâ??s ratio over a wide range of strains. These results promise the design of surfaces and composite materials with tunable Poissonâ??s ratio by pre-stressing platelet inclusions or controlling the surface rigidity of a matrix of 2D materials.
