Statistical mechanics of asymmetric tethered membranes: Spiral and crumpled phases

We develop the elastic theory for inversion-asymmetric tethered membranes and use it to identify and study their possible phases. Asymmetry in a tethered membrane causes spontaneous curvature, which in general depends on the local in-plane dilation of the tethered network. This in turn leads to long-range interactions between the local mean and Gaussian curvatures, which are not present in symmetric tethered membranes. This interplay between asymmetry and Gaussian curvature leads to a double-spiral phase not found in symmetric tethered membranes. At temperature $T=0$, tethered membranes of arbitrarily large size are always rolled up tightly into a conjoined pair of Archimedes' spirals. At finite $T$ this spiral structure swells up significantly into algebraic spirals characterized by universal exponents, which we calculate. These spirals have long-range orientational order, and are the asymmetric analogs of statistically flat symmetric tethered membranes. We also find that sufficiently strong asymmetry can trigger a structural instability leading to crumpling of these membranes as well. This provides a mechanism for crumpling of asymmetric tethered membranes which is not present for symmetric membranes. We calculate the maximum linear extent $L_c$ beyond which the membrane crumples, and calculate the universal dependence of $L_c$ on the membrane parameters. By tuning the asymmetry parameter, $L_c$ can be continuously varied, implying a scale-dependent crumpling. Our theory can be tested in controlled experiments on lipids with artificial deposits of spectrin filaments, in in vitro experiments on red blood cell membrane extracts, and on graphene coated on one side.

Figure 1

Top: Schematic diagram of the cross section of a spiral. Bottom: Schematic diagram of the double-spiral structure of our model membrane.

Figure 2

(a) Schematic phase diagram in the $\chi \text{−}{\kappa }^{\prime }$ plane. (b) Schematic phase diagram in the ${\chi }^2\text{−}L$ plane for fixed $\kappa$. The continuous curve is the line $L=\xi \left({\chi }^2\right)$, demarcating the spiral and the crumpled phases. Notice that the curve approaches ${\chi }_L^2$ only for $L\to \infty$.

Figure 3

Schematic diagram depicting local patches of successive turns of the spiral at finite $T$. The length scale $L_H$ is the typical distance between successive points of contact of neighboring layers, while $d\sim \sqrt{\langle h^2\rangle }$ is the typical distance between successive layers.

Figure 4

Vertices for the Feynman diagrams. Left: $\frac{A}{4}{\left(P_{ij}{\mathbf{\nabla }}_{i}h{\mathbf{\nabla }}_{j}h\right)}^2$. Right: $B\left({\nabla }^{2}h\right)\left(P_{ij}{\mathbf{\nabla }}_{i}h{\mathbf{\nabla }}_{j}h\right)$.

Figure 5

One-loop Feynman diagrams that contribute to the fluctuation corrections to $\kappa$.

Figure 6

One-loop Feynman diagram for fluctuation corrections to $A$.

Figure 7

One-loop Feynman diagram for fluctuation corrections to $B$.

Figure 8

Schematic flow lines in the $g_1\text{−}{g}_2$ plane implied by the recursion relations (2.39), (2.40). The (green) cross marks the only stable FP ($g_1=2\epsilon /5,g_2=0$). The thick red curve is the separatrix $g_2=\frac{3g_1}{5}$ between the region controlled by the stable Aronovitz-Lubensky fixed point (ALFP) [24] (corresponding to a spiral membrane) and the region in which flows run off towards infinity. We believe the membrane is crumpled in this region.

Figure 9

Conjectured “Occam's razor” global RG flows in the $g_1\text{−}{g}_2$ plane.

Figure 10

Cross section of a single spiral at $T>0$.

Figure 11

A one-loop Feynman diagram that appears to correct $A$, but which in fact vanishes in the limit of zero external wave vectors.

Figure 12

A typical one-loop diagram that corrects $B$ that vanishes in the limit of zero external wave vectors.