Energy per particle of the equilibrium shell structures (+) and of the constrained-assembly shells (dots) as a function of the template radius R. The bigger dots and shaded areas indicate the range R a –R c of sphere radii where the constrained assembly shells adopt a particular equilibrium Magic Number structure. As an example, the equilibrium radius R e (32) as well as R a (32) and R c (32) are indicated for N = 32.
Implosion of incomplete shells. The assembly units are shown as spheres colored according to the energy scale shown on the left, in units of ɛ0. (a) For a template with radius equal to the N = 32 equilibrium radius R e (32), the capsid closes prematurely at N = 30 due to inward collapse of the hole in the n = 29 structure. Note that the n = 29 hole still could accommodate more subunits. (b) For larger template radii, exceeding R a (32), this premature collapse no longer takes place and a uniformly stretched version of the equilibrium N = 32 icosahedral structure is produced. (c) As the template radius is further increased, shell closure proceeds by the collapse of the fivefold symmetric hole in the n = 31 structure when a subunit is inserted. If the radius is increased beyond R c (32), then a hole at the n = 32 shell remains stable, which allows to add more subunits leading to a N > 32 shell.
Line tension of a hexagonal lattice. (a) Cut along the crystallographic direction, the missing hexagons are colored in gray. For every step of length 2σ0 two contacts of energy ɛ c are broken (dashed lines). (b) Cut along the zig-zag direction. For every step of length 3b four contacts are lost (dashed lines), where is the edge of the hexagon.
Line energy and boundary energy. E b (n, R) of a partial shell, in units of ɛ0, obtained from the assembly simulations as a function of the number of assembly units for the two radii that limit the stability region of the icosahedral structure N = 32: R a (32) (solid dots) and R c (32) (open dots). The solid and dotted lines show the results of a fit to Eq. (5), for R a (32) and R c (32), respectively. The elastic energy ΔE str stored in the closed shell is indicated for R = R a (32).
Dimensionless line tension obtained from the fits of the simulation results to Eq. (5) for all values of R. The line tensions of Magic Number structures (see Fig. 1) are shown as black dots. The horizontal lines show the dimensionless line tensions for a hexagonal layer cut along a crystallographic and zigzag axes, where Λ zig > Λ cry .
Assembly of a shell composed of N = 410 particles. Panels (a)–(f) show subsequent snapshots starting from the growth nucleus (a) up to the completed shell (f). (a)–(c) are diametrically opposite (i.e., looking from the other side) to (d)–(f). Growth-induced disorder in (f), the “closure catastrophe,” is generated by the collision of protrusions such as the ones shown in (c) and (d) by arrows. Arrows in (b) indicate fivefold disclinations. Color coding indicates the binding energy and is similar to that of Fig. 2.
Growth front protrusions of shell growth originating at the South Pole are driven apart in the Southern Hemisphere and driven together in the Northern Hemisphere. Protrusion tips collide near the North Pole.
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