Randomly crumpled sheets have shape memory. In order to understand the basis of this form of memory, we simulate triangular lattices of springs whose lengths are altered to create a topography with multiple potential energy minima. We then deform the
se lattices into different shapes and investigate their ability to retain the imposed shape when the energy is relaxed. The lattices are able to retain a range of curvatures. Under moderate forcing from a state of local equilibrium, the lattices deform by several percent but return to their retained shape when the forces are removed. By increasing the forcing until an irreversible motion occurs, we find that the transitions between remembered shapes show co-operativity among several springs. For fixed lattice structures, the shape memory tends to decrease as the lattice is enlarged; we propose ways to counter this decrease by modifying the lattice geometry. We survey the energy landscape by displacing individual nodes. An extensive fraction of these nodes proves to be bistable; they retain their displaced position when the energy is relaxed. Bending the lattice to a stable curved state alters the pattern of bistable nodes. We discuss this shapeability in the context of other forms of material memory and contrast it with the shapeability of plastic deformation. We outline the prospects for making real materials based on these principles.
The binding of clusters of metal nanoparticles is partly electrostatic. We address difficulties in calculating the electrostatic energy when high charging energies limit the total charge to a single quantum, entailing unequal potentials on the partic
les. We show that the energy at small separation $h$ has a strong logarithmic dependence on $h$. We give a general law for the strength of this logarithmic correction in terms of a) the energy at contact ignoring the charge quantization effects and b) an adjacency matrix specifying which spheres of the cluster are in contact and which is charged. We verify the theory by comparing the predicted energies for a tetrahedral cluster with an explicit numerical calculation.
We study theoretically the chirality of a generic rigid objects sedimentation in a fluid under gravity in the low Reynolds number regime. We represent the object as a collection of small Stokes spheres or stokeslets, and the gravitational force as a
constant point force applied at an arbitrary point of the object. For a generic configuration of stokeslets and forcing point, the motion takes a simple form in the nearly free draining limit where the stokeslet radius is arbitrarily small. In this case, the internal hydrodynamic interactions between stokeslets are weak, and the object follows a helical path while rotating at a constant angular velocity $omega$ about a fixed axis. This $omega$ is independent of initial orientation, and thus constitutes a chiral response for the object. Even though there can be no such chiral response in the absence of hydrodynamic interactions between the stokeslets, the angular velocity obtains a fixed, nonzero limit as the stokeslet radius approaches zero. We characterize empirically how $omega$ depends on the placement of the stokeslets, concentrating on three-stokeslet objects with the external force applied far from the stokeslets. Objects with the largest $omega$ are aligned along the forcing direction. In this case, the limiting $omega$ varies as the inverse square of the minimum distance between stokeslets. We illustrate the prevalence of this robust chiral motion with experiments on small macroscopic objects of arbitrary shape.
