Electrostatics-driven shape transitions in soft shells

Abstract

Manipulating the shape of nanoscale objects in a controllable fashion is at the heart of designing materials that act as building blocks for self-assembly or serve as targeted drug delivery carriers. Inducing shape deformations by controlling external parameters is also an important way of designing biomimetic membranes. In this paper, we demonstrate that electrostatics can be used as a tool to manipulate the shape of soft, closed membranes by tuning environmental conditions such as the electrolyte concentration in the medium. Using a molecular dynamics-based simulated annealing procedure, we investigate charged elastic shells that do not exchange material with their environment, such as elastic membranes formed in emulsions or synthetic nanocontainers. We find that by decreasing the salt concentration or increasing the total charge on the shell's surface, the spherical symmetry is broken, leading to the formation of ellipsoids, discs, and bowls. Shape changes are accompanied by a significant lowering of the electrostatic energy and a rise in the surface area of the shell. To substantiate our simulation findings, we show analytically that a uniformly charged disc has a lower Coulomb energy than a sphere of the same volume. Further, we test the robustness of our results by including the effects of charge renormalization in the analysis of the shape transitions and find the latter to be feasible for a wide range of shell volume fractions.

Keywords: elasticity; long-range interactions; morphology; nanotechnology.

Fig. 1.

Snapshots of minimum-energy conformations of charged elastic nanoshells for three different bending rigidities κ = 1, 5, and 10 (columns from left to right). In each column the electrolyte concentration c (M) decreases (rows from top to bottom) as c = 1, 0.1, 0.05, and 0.005. Different colors suggest different concentration values, with red being the highest c under study and purple corresponding to the lowest c. As the concentration is lowered, the range of electrostatic interactions is increased, leading to the variation in the shape of the nanoshell. We find that for the concentration range under investigation, softer shells tend to form bowl-like structures, wheras more rigid vesicles form ellipsoidal and disc-like shapes. All of the above nanostructures have the same total surface charge and volume, fixed to values associated with the spherical conformation.

Fig. 2.

Shell shapes that minimize free energy $\mathrm{ℱ}$ for fixed κ = 5 and c = 0.015 M as a function of increasing z = 0.3, 0.6, and 1 (from left to right). As z increases, the strength of the electrostatic interactions increase and the shell transforms from a convex, ellipsoidal form to a dimpled disc and finally to a concave bowl-like structure. All shapes correspond to the same total volume. See Results for the meaning of symbols.

Fig. 3.

Electrostatic contribution to the energy of the shell (upper plot) and shell’s surface area (lower plot) vs. salt concentration c for different lowest-energy structures. We plot the electrostatic energy, ΔE_C = E_C − E_C,S, which is measured relative to that of a spherical shell with identical parameters. Similarly, the area A of the shell is normalized by the area of a sphere with the same volume. Black symbols are spheres or ellipsoids, blue symbols are discs, and red symbols are bowl-shaped structures. Inset shows the legend for the symbols used in the plot. The large (negative) changes in Coulomb energy help drive the shape transitions.

Fig. 4.

Spatial distribution of electrostatic and elastic energies (in units of k_BT, where T is the room temperature) on the surface of the disc (upper two rows) and bowl (lower two rows). The left column shows the front view, the center column shows the angle view, and the right column shows the side view. For either shapes, the elastic energy (second and fourth rows) is concentrated in the edges. The electrostatic energy on the disc (first row) is higher in the center where the opposite faces are nearby. The five-coordinated vertices, which are visible as spots in the electrostatic energy distribution, lead to small fluctuations in the energy.