Fundamental interactions in a classical Wigner system

Wigner crystallization in macroscopic systems provides a simple avenue for studying classical many-body physics. The Coulomb repulsion of millimeter-sized conducting spheres rolling on a high potential surface enables the self-assembly of ordered structures. Prior attempts at formulating pairwise force models for Wigner islands have resulted in different functional forms with no apparent consensus. In this study, we explore fundamental particle interactions using the finite element method to solve the electrostatic problem for one-, two-, and many-sphere configurations. Analysis of charge and force shows three interaction regimes parametrized by the packing fraction, which includes many-body force behavior for compact geometries. We find that the force relationship between spheres is accurately represented by an aligned dipole model over all configurations. Both the finite element method and aligned dipole models were validated in absolute terms using an experimental setup featuring a gravity potential. Future dynamic studies, which rely on accurate force relationships, can now be treated.

Figure 1

(a) Spatial map of $\sigma$ for a single sphere whose angular dependence (b) is well matched to Fish's approximation [21]. (c) Plot of $\sigma$ as a function of $r$ for the bottom plate. (d) Spatial map shown in (a) with ${\sigma }_0$ subtracted from the bottom plate emphasizing the effective dipole charge distribution.

Figure 2

(a) Plot of $Q$ on a sphere as a function of $r$. The inset in (a) is a spatial charge map for a two-sphere system that includes plate subtraction. (b) Semilogarithmic plot of $F$ between two spheres as a function of $r$. (c) Plot of $F_{\text{AD}}$ defined in Eq. (1) on a sphere as a function of $r$. Best-fit aligned dipole function (solid black) and exponential curves (dashed black) are provided with fit coefficients. The inset in (c) is a diagram of the aligned dipole model

Figure 3

Solutions to FEM simulations with $N=37$ spheres for (a) hexagonal and (b) random distributions for various average particle spacing (dashed ovals represent maximum particle locations). $Q_{\text{AD}}$ from Eq. (2) is plotted against $Q_{\text{FEM}}$. $|F_{\text{AD}}|$ calculated from Eq. (1) are plotted against $|F_{\text{FEM}}|$, which includes both lateral components. (c) $Q_c$ and force differential (rms) using various models plotted as a function of the packing fraction (bottom axis) and $a$ (top axis). The scattering of points observed in both (a) and (b) for low force values is a result of the simulation noise limit, which is indicated by the dotted lines in (c).

Figure 4

Charge on sphere $①$ as a function of $\theta$ (bottom axis) and $r_{23}$ (top axis). Sphere configurations are illustrated for various $\theta$ values.

Figure 5

Experimental Wigner islands for (a) leveled and (b) tilted plate geometries. Histograms of electrostatic $F_x$ and $F_y$ values for all spheres taken from FEM simulations and calculated using the AD model are shown for the leveled and tilted geometries in (c) and (d), respectively.