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Heat kernel smoothing simplifying interactions for suppressed diffraction around a wavelength. A potential is constructed using (5) to penalize a certain wavelength κ and the inverse Fourier transform gives the corresponding interactions, which have long-ranged oscillations (a). By applying the heat kernel smoothing in (3) , we can successively simplify the interactions (b)–(c) until only a single minimum and maximum remains. (d)–(f) Low temperature configurations from Monte Carlo simulations verifying that the interactions cause self-organization into states with diffraction patterns with pronounced frequency gaps (insets show their diffraction patterns).
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The main results: simplified interactions designed for lattice self-assembly. The curves show the interactions from (6) for targeted self-assembly in reciprocal space (inset) and in real space of a Kagome lattice (a) and a diamond lattice (b). The gray curves are the basic construction with almost no screening or smoothing while the red curves show the optimized potentials after applying the method defined in (3) . As a reference we also show a previously published potential (blue curve) causing self-assembly of the Kagome lattice. 17
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(a) Time snapshots of a Monte Carlo simulation of a particle system that starts in a random initial configuration and relaxes to the target Kagome lattice at constant low temperature. The interaction potential used is shown in Fig. 2 . (b) Configuration at high temperature after 25 × 103 sweeps, the Kagome pattern can assemble even with large temperature induced perturbations from the exact lattice. The pattern is highlighted by joining particles with their 4 closest neighbours. (c) If the particle density does not match a space filling Kagome lattice, grains with different particle density will also form to compensate. In this example, the particle density is 0.9 times that of that the target lattice.
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We present a method that systematically simplifies isotropic interactions designed for targeted self-assembly. The uncertainty principle is used to show that an optimal simplification is achieved by a combination of heat kernel smoothing and Gaussian screening of the interaction potential in real and reciprocal space. We use this method to analytically design isotropic interactions for self-assembly of complex lattices and of materials with functional properties. The derived interactions are simple enough to narrow the gap between theory and experimental implementation of theory based designed self-assembling materials.
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