Using the uncertainty principle to design simple interactions for targeted self-assembly

Edlund, Erik; Lindgren, Oskar; Jacobi, Martin Nilsson

doi:10.1063/1.4812727

Cited by 15 publications

(14 citation statements)

References 36 publications

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“…Finally, the previous uncertainty inequalities can be still improved for various broad relevant classes of central potentials such as e.g., the convex potentials [ 120 , 121 , 122 , 123 ] and the power-law or anharmonic potentials [ 124 , 125 , 126 , 127 , 128 , 129 ]. This improvement, however, has not yet been done up until now.…”

Section: Dispersion Measures and Heisenberg-like Uncertainty For Central Potentialsmentioning

confidence: 99%

Spherical-Symmetry and Spin Effects on the Uncertainty Measures of Multidimensional Quantum Systems with Central Potentials

Dehesa

2021

Entropy

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The spreading of the stationary states of the multidimensional single-particle systems with a central potential is quantified by means of Heisenberg-like measures (radial and logarithmic expectation values) and entropy-like quantities (Fisher, Shannon, R\'enyi) of position and momentum probability densities. Since the potential is assumed to be analytically unknown, these dispersion and information-theoretical measures are given by means of inequality-type relations which are explicitly shown to depend on dimensionality and state's angular hyperquantum numbers. The spherical-symmetry and spin effects on these spreading properties are obtained by use of various integral inequalities (Daubechies--Thakkar, Lieb--Thirring, Redheffer--Weyl, ...) and a variational approach based on the extremization of entropy-like measures. Emphasis is placed on the uncertainty relations, upon which the essential reason of the probabilistic theory of quantum systems relies.

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Section: Dispersion Measures and Heisenberg-like Uncertainty For Central Potentialsmentioning

confidence: 99%

Spherical-Symmetry and Spin Effects on the Uncertainty Measures of Multidimensional Quantum Systems with Central Potentials

Dehesa

2021

Entropy

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“…Such a) Electronic mail: truskett@che.utexas.edu approaches have found various interactions that stabilize two-dimensional square, honeycomb and kagome lattices [15][16][17][18] as well as the three-dimensional diamond crystal structure. 19,20 Furthermore, it was demonstrated that particles with the optimized interactions indeed assembled into the targeted lattice phases at higher temperature using molecular simulations 15,16,20 .…”

Section: Introductionmentioning

confidence: 99%

Breadth versus depth: Interactions that stabilize particle assemblies to changes in density or temperature

Piñeros

Bâldea

Truskett

2016

The Journal of Chemical Physics

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We use inverse methods of statistical mechanics to explore trade-offs associated with designing interactions to stabilize self-assembled structures against changes in density or temperature. Specifically, we find isotropic, convex-repulsive pair potentials that maximize the density range for which a two-dimensional square lattice is the stable ground state subject to a constraint on the chemical potential advantage it exhibits over competing structures (i.e., 'depth' of the associated minimum on the chemical potential hypersurface). We formulate the design problem as a nonlinear program, which we solve numerically. This allows us to efficiently find optimized interactions for a wide range of possible chemical potential constraints. We find that assemblies designed to exhibit a large chemical potential advantage at a specified density have a smaller overall range of densities for which they are stable. This trend can be understood by considering the separation-dependent features of the pair potential and its gradient required to enhance the stability of the target structure relative to competitors. Using molecular dynamics simulations, we further show that potentials designed with larger chemical potential advantages exhibit higher melting temperatures.

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“…Highly coordinated lattices with, e.g., face-centered cubic or hexagonal symmetries in three dimensions [7] and triangular symmetry in two dimensions [23] are commonly observed in the experimental assembly of monodisperse particles with short-range, isotropic interactions. A broader array of thermodynamically stable 3D structures-including low-coordinated diamond and simple cubic (Sc) lattices of interest for technological applications [24,25]-has also been demonstrated by computer simulations of monodisperse particles with softer, repulsive potentials [26][27][28][29][30], including those that model the interactions between elastic spheres [31] or star polymers [32]. Similar interactions favor open 2D structures as well, including honeycomb and square lattices [33][34][35][36][37][38][39] with, e.g., sterically stabilized magnetic particles in the presence of an external field [40] providing one novel experimental realization.…”

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confidence: 99%

Dimensionality and Design of Isotropic Interactions that Stabilize Honeycomb, Square, Simple Cubic, and Diamond Lattices

2014

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We use inverse methods of statistical mechanics and computer simulations to investigate whether an isotropic interaction designed to stabilize a given two-dimensional lattice will also favor an analogous three-dimensional structure, and vice versa. Specifically, we determine the 3D-ordered lattices favored by isotropic potentials optimized to exhibit stable 2D honeycomb (or square) periodic structures, as well as the 2D-ordered structures favored by isotropic interactions designed to stabilize 3D diamond (or simple cubic) lattices. We find a remarkable "transferability" of isotropic potentials designed to stabilize analogous morphologies in 2D and 3D, irrespective of the exact interaction form, and we discuss the basis of this cross-dimensional behavior. Our results suggest that the discovery of interactions that drive assembly into certain 3D periodic structures of interest can be assisted by less computationally intensive optimizations targeting the analogous 2D lattices. Material properties are intimately linked to structural characteristics featured at various length scales. Thus, discovering new ways to create materials with prescribed morphologies is a key challenge in their design for specific applications. In addition to the development of top-down material fabrication strategies, there has been considerable progress in bottom-up approaches in which the primary components (molecules, nanoparticles, colloids, etc.) A critical part of any self-assembly design problem is understanding how tunable aspects of the interactions affect the thermodynamic stability of competing assembled states with different morphologies. For nanoscale to microscale particles, this understanding has been guided in part via exploratory experiments and simulations to characterize the structures that spontaneously form from systems with various particle chemistries [7,8], shapes [9][10][11][12][13][14][15], and surface properties [16][17][18][19], as well as different dispersing solvents [20] and mixtures of assembling particles [21,22]. Highly coordinated lattices with, e.g., face-centered cubic or hexagonal symmetries in three dimensions [7] and triangular symmetry in two dimensions [23] are commonly observed in the experimental assembly of monodisperse particles with short-range, isotropic interactions. A broader array of thermodynamically stable 3D structures-including low-coordinated diamond and simple cubic (Sc) lattices of interest for technological applications [24,25]

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Using the uncertainty principle to design simple interactions for targeted self-assembly

Cited by 15 publications

References 36 publications

Spherical-Symmetry and Spin Effects on the Uncertainty Measures of Multidimensional Quantum Systems with Central Potentials

Spherical-Symmetry and Spin Effects on the Uncertainty Measures of Multidimensional Quantum Systems with Central Potentials

Breadth versus depth: Interactions that stabilize particle assemblies to changes in density or temperature

Dimensionality and Design of Isotropic Interactions that Stabilize Honeycomb, Square, Simple Cubic, and Diamond Lattices

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