Soft modes near the buckling transition of icosahedral shells

Icosahedral shells undergo a buckling transition as the ratio of Young’s modulus to bending stiffness increases. Strong bending stiffness favors smooth, nearly spherical shapes, while weak bending stiffness leads to a sharply faceted icosahedral shape. Based on the phonon spectrum of a simplified mass-and-spring model of the shell, we interpret the transition from smooth to faceted as a soft-mode transition. In contrast to the case of a disclinated planar network where the transition is sharply defined, the mean curvature of the sphere smooths the transition. We define elastic susceptibilities as the response to forces applied at vertices, edges, and faces of an icosahedron. At the soft-mode transition the vertex susceptibility is the largest, but as the shell becomes more faceted the edge and face susceptibilities greatly exceed the vertex susceptibility. Limiting behaviors of the susceptibilities are analyzed and related to the ridge-scaling behavior of elastic sheets. Our results apply to virus capsids, liposomes with crystalline order, and other shell-like structures with icosahedral symmetry.

Figure 1

(Color online) Triangulated network of $P=Q=2$. Colors identify local environments with fivefold vertices shown in blue.

Figure 2

(Color online) Shell shape above the buckling transition for $P=128$, $Q=0$ shell with $k_s=1$ and $k_b=16$, yielding Foppl–von Karman number $\gamma =930$. Color coding is logarithmic according to total elastic energy ($\text{violet}=\text{low}$, $\text{red}=\text{high}$).

Figure 3

(Color online) Triangulated network of radius $8a$ and spring constant $k_s=1$ with a single fivefold disclination at center. (Top) Eigenvalue spectrum color coded according to degeneracy. Note the nondegenerate $1X$ mode that goes to zero at the buckling transition. (Bottom) Energy, cone height, and susceptibility.

Figure 4

(Color online) Lowest-frequency modes of $P=8,Q=0$ icosahedron with $N_v=642$ vertices: (top) color coded according to degeneracy; (bottom) nondegenerate modes only. Arrows locate eigenvalue ${\Lambda }_{breathing}=8\pi \sqrt{3}∕N_v$. Note that the buckling mode (red) dips close to zero near the buckling transition.

Figure 5

(Color online) Vibrational frequencies plotted versus wave number $q=l∕R$, where $l$ is the angular momentum index. Data are for $P=8,Q=0$ icosahedron with $k_s=1,k_b=0.1,\gamma =653$. The radii of the circles indicate the sizes of the various projections.

Figure 6

(Color online) Stiffness (inverse susceptibility) to forces applied with icosahedral symmetry at vertices, edges, and faces. Dashed blue lines show $15∕\text{ln}\left(0.79\sqrt{\gamma }\right)$ (vertex, offset from the best fit for clarity), $100∕\sqrt{\gamma }$ (edge, scaling expected for $\gamma >{10}^6$), and $20000∕\gamma$ (face, scaling for bending of flat facet [9]). Inset: Diametrically opposed forces applied with uniaxial symmetry. Dashed blue line shows $5∕\sqrt{\gamma }$.