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Using the uncertainty principle to design simple interactions for targeted self-assembly
25.Defining good measures of complexity is surprisingly hard and the subject has a long and contentious history.36–38 None of the existing measures pertains to complexity in interaction potentials and we doubt one will be found. Even if we could define a quantitative measure with desirable theoretical properties, it is unlikely it will agree with realizability in the wide range of experimental systems available today.
26.Materials exhibiting this property have been called “stealth-materials.”39
28.The effect of the penalty at low wave numbers was referred to as hyper-uniformity in Ref. 27.
31.Equation (5) is defined in two dimensions, the generalization to 3D is .
36. R. Badii and A. Politi, Complexity: Hierarchical Structures and Scaling in Physics (Cambridge University Press, Cambridge, UK, 1999).
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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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