Curvature instability of chiral colloidal membranes on crystallization

Buckling and wrinkling instabilities are failure modes of elastic sheets that are avoided in the traditional material design. Recently, a new paradigm has appeared where these instabilities are instead being utilized for high-performance applications. Multiple approaches such as heterogeneous gelation, capillary stresses, and confinement have been used to shape thin macroscopic elastic sheets. However, it remains a challenge to shape two-dimensional self-assembled monolayers at colloidal or molecular length scales. Here, we show the existence of a curvature instability that arises during the crystallization of finite-sized monolayer membranes of chiral colloidal rods. While the bulk of the membrane crystallizes, its edge remains fluid like and exhibits chiral ordering. The resulting internal stresses cause the flat membrane to buckle macroscopically and wrinkle locally. Our results demonstrate an alternate pathway based on intrinsic stresses instead of the usual external ones to assemble non-Euclidean sheets at the colloidal length scale.

the solid part of the membrane show hexagonal ordering although it is unclear whether the lattice orientation differ in different grains. At least in the case when crystallization starts from one nucleation center, it is likely that the neighboring grains have very small, if not zero, misorientation angle. This is also indirectly supported by the observation that the ridges separating the grains rapidly vanish, annealing the grains, when the solidification front reach the membrane boundary. In case of more than one nucleation center, of course, more lattice orientations are expected with arbitrary misorientation angles. Quite consistently some of the ridges are observed to persist even after global buckling occurs. The grains however are clearly identified by change in the molecular orientation of the virus rods at the walls, in particular their inclination with the membrane normal.
For simplicity, we decouple the orientational degrees of freedom from the translational degree of freedom of the virus rods. We take them to be arranged on a fixed hexagonal lattice and ask, what kind of orientation patterns they exhibit. In fact this simplified model reproduces many of the experimentally observed features as we describe below. For discussion, we rewrite the energy function already discussed in the main text.
Here nn stands for sum over all nearest neighbour pairs (i, j). The nematic directorsn i , at the i−th site, are represented by their angular orientations (θ i , φ i ). The sign of the depletion term here appears different compared to Eq.1 of main text because we used (n i .ẑ) 2 = (cos θ i ) 2 . We consider total 2314 virus rods arranged on a hexagonal lattice, bounded by a circle. Here N , D and C have dimensions of energy; we work with dimensionless parameters C / N = 100, D / N = 0.6 and T ≡ k B T N . In each Monte-Carlo step our trial rotation angles are ∆θ = ∆φ = ±1 o . Whenever θ goes beyond π/2 we transformn to −n retaining nematic symmetry of the energy function.
The first term of the energy function above is borrowed from the standard Lebwohl-Lasher model [1] for uniaxial nematics with nearest neighbor interaction U ij = − P 2 (θ ij ).
Here θ ij is the angle between the directors at neighboring sites i and j, and P 2 = 3 2 cos 2 θ ij − 1 2 is the 2nd Legendre polynomial. This term accounts for both splay and bend distortions within one Frank constant approximation. Originally this was proposed as a tensorial order parameter in Maier-Saupe theory of phase transitions [2].
We impose torque free boundary condition at the membrane boundary (r = R N ) following the Ref. [3,4]. Our hexagonal lattice ends at the circular boundary. The rods at the boundary (r = R N ) are set to lie in the x-y plane (i.e., θ = π/2) and are aligned tangentially to the membrane edge (i.e., alongφ). The last but one layer of rods, at r = R N −1 , lie in the (ẑ,φ) plane maintaining a relative inclination δθ with the rods at the last layer On the the other hand, the 2nd term is minimised when preferred chiral ordering q is maintained, however it comes with an energy cost due to the 1st and the 3rd terms.
When the depletion effect dominates (high D ) over the chiral term, the state with all up (vertical) orientation wins, except at the membrane boundary where effect of the depletion is weak and chirality shows up ( Supplementary Fig. 4a,b). The interior of the grains have lower energy than the grain boundaries ( Supplementary Fig. 5 b,c). Local energy density of the system shown in Supplementary Fig. 5 b,c indicates dominance of the chiral energy.
While the chiral energy term is minimized at grain interiors and at most of the membrane edge, it has high value at the grain boundaries. Supplementary Fig. 5c shows the local energy density map contributed by only the 2nd term of the energy function. As shown in Supplementary Fig. 5d, the interior of the grains have nonzero chirality which minimizes the 2nd term but does not minimise the 1st and the 3rd terms.
The size of the grains is dictated by q (see Supplementary Fig. 2a,b,c or equivalently Supplementary Fig. 3a,b,c): higher the q smaller the domain size. Preferred chirality enforces how slowly, in space, the directors turn (i.e., orientation gradient) from their vertical orientation at the core of the grain to nearly horizontal orientation at the grain boundaries.
The grain boundaries in our simulation are π walls. In the experiment, however, the grain boundaries buckle in the third dimension and thereby the rods avoid becoming horizontal.
Our model being 2D cannot address this buckling phenomenon.
We note that the grain boundaries in the experiment grow predominantly along the radial direction away from the nucleation site. They bifurcate frequently, making nearly symmetric