Helicoid to Spiral Ribbon Transition

We present a continuum description for the transition between the helicoid and spiral ribbon structures of chiral materials. At a critical value of the ratio between the bending and stretching moduli, the Föppl–von Kármán number, we encounter a continuous buckling transition from a straight helicoid to a spiral ribbon. Two of the three persistence lengths of the ribbon become very short at the transition point, indicating strong thermal shape fluctuations. The transition is discontinuous if the ribbon width is treated as a free thermodynamic variable.

Figure 1

Characteristic chiral ribbon configurations. (a) Helicoid with ${\Omega }_1={\Omega }_2=0$ and ${\Omega }_3$ finite. The orthogonal $\widehat{e}$ and $\widehat{p}$ directions determine the coordinate system. (b) Isometric spiral with ${\Omega }_2=0$ and both ${\Omega }_1$ and ${\Omega }_3$ finite. (c) Intermediate configuration with a Föppl–von Kármán number close to its critical value. Inset: Components of the Kirchhoff rotation vector $\overrightarrow{\Omega }$. Only the $\widehat{e}$ direction is shown.

Figure 2

Ribbon energy minimum $E(w)$ as a function of the ribbon width $w$ (schematic). The curve marked $A$ corresponds to a larger value of the Föppl–von Kármán number $\gamma$ and $B$ to a lower value. As a function of $\gamma$, a first-order transition takes place from a spiral ribbon state with a large width, case $A$, to a helicoid state with a well-defined smaller width, case $B$.