Self-avoiding tethered surfaces are always flat

Abstract

The scaling behavior of fully flexible elastic tethered surfaces has been debated for decades. Some theories predict that self-avoiding surfaces would crumple in the absence of bending rigidity, while most simulations suggested that they would remain flat. Recent simulations on ideal membranes with lattice perforations suggest that systematically removing surface area from a membrane may provide an alternative way to crumpling self-avoiding surfaces. We perform extensive numerical simulations of two models of fully flexible elastic tethered surfaces in which self-avoidance can be systematically and continuously tuned to the ideal limit. We show that in the thermodynamic limit, these surfaces remain flat with a size exponent $ν=1$ for any finite degree of self-avoidance, with or without membrane perforations.

Figures (9)

• Figure 1: Sketch of the surface model with polymers of length $n_p=8$ containing 85 hexagonal cells. The inset shows the detail of the connections in each node of the membrane. The membrane conformation is defined by two parameters: $n_p$, the number of monomer units per polymer segment between adjacent nodes, and $R$, the radius of a circle centered on the surface, which determines the extent of the network by excluding any hexagons whose centers lie outside this circular boundary.
• Figure 2: Scaling of the square of the radius of gyration, $R_g$, as a function of total number of nodes rescaled by the characteristic size of a self-avoiding polymer, $l_g=\sigma\, n_p^{0.588}$, for fish-net systems characterized by $n_p=8,16$ and $24$ monomers per polymer link. The solid lines are the fits to the data with the functional form $f(x)=a\,x^c+b$. The powers are $c=0.98\pm0.02$, $0.97\pm0.05$, and $1.01\pm0.03$ for $n_p=8, 16$ and 24 respectively.
• Figure 3: Top and side view of a thermalized configuration of the largest surface simulated in our study. Each link contains $n_p=24$ particles for a total of $N=37374$ particles.
• Figure 4: Radius of gyration, $R_g$, of the soft-sphere model as a function of the degree of self-avoidance $\beta\varepsilon$ for a $N=80\times80$ surface. The inset is the same plot for a $N=30\times30$ membrane the elements of which interact with a Gaussian and with an Exponential potential as discussed in the main text.
• Figure 5: Radius of gyration of the soft-sphere model as a function surface length $L$ for various $\varepsilon$. The solid lines are guides to the eye. The straight dashed lines serve to highlight the onset of the linear dependence of the curves. The left inset shows the asphericity $e$ of the surfaces as a function of $L$ for $\beta\varepsilon=0.001$. The right inset shows top and side views of a surface configuration for $\beta\varepsilon=10^{-5}$. Particles sizes have been greatly reduced to highlight their site multi-occupancy.