Universal behavior of crystalline membranes: Crumpling transition and Poisson ratio of the flat phase

We revisit the universal behavior of crystalline membranes at and below the crumpling transition, which pertains to the mechanical properties of important soft and hard matter materials, such as the cytoskeleton of red blood cells or graphene. Specifically, we perform large-scale Monte Carlo simulations of a triangulated two-dimensional phantom network which is freely fluctuating in three-dimensional space. We obtain a continuous crumpling transition characterized by critical exponents which we estimate accurately through the use of finite-size techniques. By controlling the scaling corrections, we additionally compute with high accuracy the asymptotic value of the Poisson ratio in the flat phase, thus characterizing the auxetic properties of this class of systems. We obtain agreement with the value which is universally expected for polymerized membranes with a fixed connectivity.

Figure 1

Lattice that has been employed in this work, for $L=6$, in which a four-color partition is illustrated. Points with the same color can be updated simultaneously. This partition has been used to speed up, in a highly parallel way, our numerical simulations using GPU processors.

Figure 2

Sample membrane configuration at $\kappa =2$ in the flat phase for $L=24$.

Figure 3

Behavior of the specific heat as a function of temperature for all simulated lattice sizes. The solid black curves provide the least-squares fit of the points near the maximum to a quadratic form as described in the text. We have used pentagons ($L=16$), rhombs ($L=24$), inverted triangles ($L=32$), triangles ($L=48$), circles ($L=64$), squares ($L=96$), and crosses ($L=128$) to mark the points in the figure.

Figure 4

Behavior of the $\kappa$ derivative of the squared gyration radius as a function of temperature for all simulated lattice sizes. The solid black curves provide the least-squares fit of the points near the maximum to a quadratic form as described in the text. We have used pentagons ($L=16$), rhombs ($L=24$), inverted triangles ($L=32$), triangles ($L=48$), circles ($L=64$), squares ($L=96$), and crosses ($L=128$) to mark the points in the figure.

Figure 5

Scaling of the maximum of the specific heat (pentagons) as a function of the size of the system. The solid line is a fit to Eq. (20).

Figure 6

Position of the maximum of the specific heat (lower data, see legend, pentagons) and of the maximum of $d{R}_g^2/d\kappa$ (inverted triangles) as a function of $L^{-1/\nu }$. In this plot we have employed $1/\nu =1/0.73$, as obtained from the fit of the peak of the specific heat. We have used the position of the specific heat maximum to obtain ${\kappa }_c$. Finally, we have fitted $d{R}_g^2/d\kappa$ by imposing the previous obtained values of $\nu$ and $\kappa$; hence, there is a single free parameter in the fit. The solid (specific heat) and dashed (gyration radius) lines are “linear” fits in this scale. Notice that the gyration radius presents stronger scaling corrections than the specific heat.

Figure 7

Scaling of the maximum of the $\kappa$ derivative (squares) of the squared gyration radius, $d{R}_g^2/d\kappa$, as a function of system size. The solid line is a fit to Eq. (25).

Figure 8

Comparison between the computations of the Poisson ratio using $g_{11}$ and $g_{22}$ as the denominator in Eq. (26); see legend. Notice the strongly anisotropic behavior of the Poisson ratio for $\kappa =1.1$ (bottom panel). For $\kappa =2$ (top panel), isotropy holds for $L>32$.

Figure 9

Comparison between the computations of the Poisson ratio using $g_{11}$ (squares) and $g_{22}$ (crosses) as the denominator in Eq. (26) for $\kappa =3$; see legend. Isotropy holds for $L\ge 24$.

Figure 10

Finite-size effects in the Poisson ratio for $\kappa =2$ and 3. We have fixed the scaling correction exponent to 1/2 and the asymptotic value to the analytical value, ${\sigma }_{\infty }=-1/3$. We have used crosses and a solid line fit ($\kappa =2$) and squares and a dashed line fit ($\kappa =2$).