The asymmetric Wigner bilayer

Abstract

We present a comprehensive discussion of the so-called asymmetric Wigner bilayer system, where mobile point charges, all of the same sign, are immersed into the space left between two parallel, homogeneously charged plates (with possibly different charge densities). At vanishing temperatures, the particles are expelled from the slab interior; they necessarily stick to one of the two plates, and form there ordered sublattices. Using complementary tools (analytic and numerical) we study systematically the self-assembly of the point charges into ordered ground state configurations as the inter-layer separation and the asymmetry in the charge densities are varied. The overwhelming plethora of emerging Wigner bilayer ground states can be understood in terms of the competition of two strategies of the system: the desire to guarantee net charge neutrality on each of the plates and the effort of the particles to self-organize into commensurate sublattices. The emerging structures range from simple, highly commensurate (and thus very stable) lattices (such as staggered structures, built up by simple motives) to structures with a complicated internal structure. The combined application of our two approaches (whose results agree within remarkable accuracy) allows to study on a quantitative level phenomena such as over-and underpopulation of the plates by the mobile particles, the nature of phase transitions between the emerging phases (which pertain to two different universality classes), and the physical laws that govern the long-range behaviour of the forces acting between the plates. Extensive, complementary Monte Carlo simulations in the canonical ensemble, which have been carried out at small, but finite temperatures along selected, well-defined pathways in parameter space confirm the analytical and numerical predictions within high accuracy. The simple setup of the Wigner bilayer system offers an attractive possibility to study and to control complex scenarios and strategies of colloidal self-assembly, via the variation of two simple system parameters.arXiv:1808.07345v1 [cond-mat.soft] 22 Aug 2018 2

Two-dimensional Wigner crystals of classical Lennard-Jones particles

“…Critical exponents can be obtained analytically. In the groundstate, critical exponents are usually of mean-field type, but there are exceptions from this rule [19]. In the context of Bétermin's analysis [17] we show analytically that the second-order transitions between the rhombic/square and square/rectangular phases are of mean-field type.…”

“…We anticipate that as the number of chains goes to infinity, this transition will be of second order. The limit A → ∞ can be treated analytically and we found that the energy of the zig-zag phase (26) is well below the rectangular one (19); the rectangular phase is the energy minimizer of simple Bravais lattices for A → ∞. The corresponding results for the 3-chain and 4-chain structures (27) indicate a systematic decrease of the energy to the one of the phase separated state (the optimal hexagonal lattice) as the number of chains increases; the Padé extrapolation (28) of available n = 2, 3, 4 data implies the result (29) which is indeed close to the optimal value (14).…”

Ordered ground state configurations of the asymmetric Wigner bilayer system -- revisited: an unsupervised clustering algorithm analysis

“…We recover -not surprisingly, as anticipated by the exact results [16,17] -that the emerging structures can be classified into the five well-known clusters, each representing the well-known ordered ground state structures. With this confirmation of our procedure in mind, we proceed to the asymmetric Wigner bilayer system, where the aforementioned "by hand" classification [6][7][8] has led to 14 structural classes. Using the same 30-dimensional feature vector the principal component analysis provides evidence that the feature space can be mapped into a nine-dimensional latent space.…”

“…In the classical Wigner bilayer system, the point charges interact via the long-range Coulomb interaction with each other and with the uniformly charged plates. Further, there is a distance-dependent, but otherwise, constant plate-to-plate interaction contributing to the total (internal) electrostatic energy of the unit cell of a bilayer structure, E(r N ; A, η), which is given by [6][7][8]…”

“…Thus for a given pair (A, η) the ground state configuration is specified by the value of x and by the lattices formed on the two layers (e.g., in terms of the respective lattice vectors). In preceding contributions [6][7][8] these ground state configurations were determined via suitably adapted optimization tools (based notably on evolutionary algorithms), limiting -for numerical reasons -the number of particles per unit cell to N = 40. In a subsequent step the emerging ground state configurations were classified in terms of the emerging structures, based on suitably defined order parameters [9,10].…”