Effects of scars on icosahedral crystalline shell stability under external pressure

We study how the stability of spherical crystalline shells under external pressure is influenced by the defect structure. In particular, we compare stability for shells with a minimal set of topologically required defects to shells with extended defect arrays (grain boundary “scars” with nonvanishing net disclination charge). We perform both Monte Carlo and conjugate gradient simulations to compare how shells with and without scars deform quasistatically under external hydrostatic pressure. We find that the critical pressure at which shells collapse is lowered for scarred configurations that break icosahedral symmetry and raised for scars that preserve icosahedral symmetry. The particular shapes which arise from breaking of an initial icosahedrally symmetric shell depend on the Föppl–von Kármán number.

Figure 1

(a)–(d) Shells without scars have a $(p,q)=(8,0)$ lattice structure (left) and shells with scars have twelve $5\text{−}7\text{−}5$ scars of $C_2$ symmetry (axis along the $z$ direction) (right), with $\varepsilon =1$ ($\gamma \approx 59$) (up) and $\varepsilon =8$ ($\gamma \approx 474$) (down). (e)–(h) Shells without scars have a $(p,q)=(11,0)$ structure (left) and shells with scars have 12 starlike scars of $I$ symmetry (right), with $\varepsilon =1$ ($\gamma \approx 112$) (up) and $\varepsilon =64$ ($\gamma \approx 7149$) (down). The total number of vertices is $N_1=642$ and $N_2=1212$, respectively. Fivefold disclinations are shown in red (dark) and sevenfold in yellow (bright). Snapshots were generated using the Visual Molecular Dynamics (vmd) package [33] and rendered using the Tachyon ray tracer [34].

Figure 2

Evolution of the parameter $\beta$ under the external pressure. (a) For small values of $\varepsilon$, shells collapse directly to ${\beta }_{\text{min}}$. (b) For large $\varepsilon$, shells collapse in stages. In the latter case, the critical pressure is defined as the point of the first collapse.

Figure 3

Critical pressure $P_c$ for (a) the $N_1=642$ shells and (b) the $N_2=1212$ shells. The gray lines indicate the buckling pressure of ideal spherical shells as predicted by continuum elastic theory.

Figure 4

Plots of the evolution of the system energy $F_{\text{tot}}$ for (a) $N_1=642$ shells with $\varepsilon =8$ and (b) $N_2=1212$ shells with $\varepsilon =8$. (c)–(f) Corresponding configurations for shells before collapse. (g), (h) Histograms of $Q_l/Q_0$ for the configurations.

Figure 5

(a) One of the 12 starlike scars. (b), (c) Two ways of breaking the icosahedral symmetry by flipping a bond.

Figure 6

(a)–(e) Configurations of $(8,0)$ shells at $\beta =0.8$ during collapsing, with $\varepsilon =1,2,4,8,128$, respectively. Color represents local bending energy. (f) Energies plotted vs $\varepsilon$. The reference line at $\varepsilon =8$ shows the elastic energy keeping the configuration at $\varepsilon =8$ frozen while varying $\varepsilon$.