The Self Assembly of Flat Sheets into Closed Surfaces

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Original entry by Joerg Fritz, AP225 Fall 2009


"Self-assembly of flat sheets into closed surfaces"

Silas Alben and Michael P. Brenner: Physical Review E, 2007, 75, pp 056113-1 to 056113-7.


Self-Assembly, Flat sheets, Buckling, Stability


Self-assembly is in general associated with the combination of small building blocks into (more or less complex) structures. This is achieved either by designing an energy landscape where the desired configuration is a global minimum or by creating asymmetry in the building blocks through patterned surfaces or similar mechanisms.

Fig.1 Example geometry of a flat sheet with magnet locations and the nearly spherical result of self-assembly.

This paper describes a completely different approach to self-assembly. Instead of utilizing small building blocks, the spontaneous folding of planar sheets into a closed surface is investigated. The motivation for this theoretical approach are experiments described in the paper "Magnetic self-assembly of three-dimensional surfaces from planar sheets" by Boncheva In these experiments elastomer sheets with small magnets as interacting elements were folded into spheres. The success of the strategy depended critically on the shape of the sheet, the location of the interacting elements and the external forcing applied to the system. The goal is to develop a model that is able to explain how different choices of sheet shape and magnet location influence the final configuration.

The design of the experiments and the choice of magnets as interacting elements makes this a theoretically complex problem, since attraction between elements at the end of open loops determines the folding dynamics and thus interactions on the scale of the sheet size itself have to be taken into account. The main result of the simulation is that despite the global nature of the experimental setup, successful attempts at self assembly (defined in this context as final geometries that are similar to spheres) have the defining characteristic that the geometrically closest elements always connect first and the final gemoetry develops through a serives of connections based on local interaction. The authors use the term "zipping" for this process.

Numerical simulation

Fig.2 Geometric representation of the flat sheet shown in figure 1, the numerical discretization and the result of the simulation in case of successful folding.

The relevant energies in this problem are those associated with the stretching and bending of the sheets and the interactions between the small magnets. To simulate this, the elastic sheets are modeled as a network of springs, where elastic energy is the sum of energy stored in the spring elements and bending energy is proportional to the angles between different adjacent springs. Together with the integration over the Coulomb forces the total energy of the system can thus be determined.

To simulated the time evolution of the system the authors use to different strategy of differing complexity. One is essentially the dynamics of an overdamped system, where the velocity of every vertex (where springs connect to each other) is proportional to the force, or the gradient of the energy function, acting on it. The second version of the simulation is essentially a simplification of the overdamped scenario using the assumption that the described behavior of "zipping" does in fact occur. This assumption makes the simulation significantly faster. The edges of the sheets are joined in small increments, taking into account only local interactions. As soon as two formerly unconnected vertices join, the rest of the sheet is assumed to relax to local equilibrium. This is essentially a quasi-static approximation of the overdamped system. The simulation of both systems shows that the most important road to instability (ans thus different final geometries, where one or more differ from the spherical form) is buckling of sheets due to out-of-plane motion.

This can be understood by doing a small experiment. If an orange is peeled very carefully, all the pieces stay connected in one small region and the whole peel can be pressed flat into a sheet. When two edges of this sheet are joined after this process, the force of joining them will at some point cause the two flat parts of the sheet to buckle, either outwards into its original form or inwards, thus making it impossible to construct a sphere again.

Buckling threshold

So the original question can be rephrased now. Instead of asking how do shape and sheet thickness influence the final geometry, we can ask how these parameters influence the buckling behavior. The buckling in this case is essentially again an energy dominated phenomenon and whether buckling occurs or not depends on the ratio between stretching and bending energy, which can be represented by the the Karman number <math>\gamma = \frac{Y R^2}{\kappa}</math> where Y is the 2D Young's modulus, <math>\kappa</math> is the bending rigidity and R is a length scale for the problem. A detailed analysis of the problem at hand shows that buckling becomes less likely for thick sheets, geometries with short edges and short tangent angles at the connection between the edges. It has to be noted that if no buckling occurs, the self-assembly requires external forces to be successful, so avoidance of buckling does not automatically lead to a higher yield of the desired final geometry.