Analytical Interaction Potentials for Disks in Two Dimensions

TL;DR

The paper develops analytical, closed-form integrated Lennard-Jones potentials for disks in two dimensions, including disk–point, disk–disk, and disk–wall interactions, by treating disks as uniform LJ media. These potentials are implemented in LAMMPS (DISK package) to simulate 2D disk suspensions with an explicit LJ solvent, enabling studies of equilibrium structure, drying-induced ordering, and bidisperse stratification. Key findings include a disorder-to-hexagonal packing transition in monodisperse disks as area fraction or drying progresses, and a pronounced small-on-top stratification in rapidly dried bidisperse suspensions, with stratification strength increasing with size disparity. The methods provide a versatile framework for modeling 2D disk systems across colloidal, granular, and disk-like nanomaterials, with broad applicability to phase behavior and drying dynamics in two dimensions.

Abstract

Compact analytical forms are derived for the interactions involving thin disks in two dimensions using an integration approach. These include interactions between a disk and a material point, between two disks, and between a disk and a wall. Each object is treated as a continuous medium of materials points interacting by the Lennard-Jones 12-6 potential. By integrating this potential in a pairwise manner, expressions for the potentials and resultant forces between extended objects are obtained. All the results are validated with numerical integrations. The analytical potentials are implemented in LAMMPS and used to simulate two-dimensional suspension of disks with an explicit solvent modeled as a Lennard-Jones liquid. In monodisperse disk suspensions, a disorder-to-order transition of disk packing is observed as the area fraction of disks is increased or as the solvent evaporates. In bidisperse disk suspensions being rapidly dried, stratification is found with the smaller disks enriched at the evaporation front. Such "small-on-top" stratification echoes the similar phenomenon occurring in three-dimensional polydisperse colloidal suspensions that undergo fast drying. These potentials can be applied to a wide range of two-dimensional systems involving disk-like objects.

Figures (17)

• Figure 1: Coexisting densities vs. temperature for the LJ liquid in 2D. The lines are the fits based on Eqs. (\ref{['eq:den_fit_one']}) and (\ref{['eq:den_fit_two']}). The black dot indicates the critical point determined from such fits.
• Figure 2: Wettability of a single disk of $10\sigma$ radius by the 2D LJ liquid at $T=0.42\epsilon/k_\text{B}$ and different Hamaker constants (a) $A_\text{ds} = 2\epsilon$ ($\epsilon_\text{ds} = 0.5\epsilon$), (b) $A_\text{ds} = 4\epsilon$ ($\epsilon_\text{ds} = 1.0\epsilon$), (c) $A_\text{ds} = 6\epsilon$ ($\epsilon_\text{ds} = 1.5\epsilon$), and (d) $A_\text{ds} = 8\epsilon$ ($\epsilon_\text{ds} = 2.0\epsilon$).
• Figure 3: First and second rows: Visualizations of disk suspensions at increasing area fractions ($\Phi$). The disk radius is $10\sigma$ and the number of disks is fixed at $256$. The number of solvent particles (i.e., LJ point masses) is tuned to realize different area fractions. Full systems are visualized in the second row, with a square of dimensions of $175\sigma \times 175\sigma$ at the center of each system magnified in the first row. The inset of the leftmost image of the first row illustrates the size ratio of the disk and a LJ particle. Third row: Voronoi tesselation analysis of the corresponding configurations of disks in the second row. Yellow cells are used to highlight disks with a high value of the orientational order parameter, i.e., $|\Phi_{6}(i)| \gtrsim 0.8$, which is a characteristic of the hexagonal packing. Fourth row: Static structure factor $S(q_x, q_y)$ in the reciprocal space, revealing the emergence and sharpening of Bragg peaks as $\Phi$ is increased. The rightmost panel clearly shows the six-fold symmetry of the hexagonal packing of disks at $\Phi=0.66$. Fifth row: Disk-disk radial distribution function, $g(r)$, for all suspensions for $r$ up to $80\sigma$. The inset shows the features of $g(r)$ at short ranges.
• Figure 4: Diffusion coefficient of disks, $D$, determined from the mean-square displacement is plotted against their area fraction, $\Phi$. A linear fit (dashed line) to the data from four systems where the disks are disorderly packed yields $D = D_0+\chi\Phi$ with $D_0= 1.91\times 10^{-2}\sigma^2/\tau$ and $\chi = -3.61\times 10^{-2}\sigma^2/\tau$.
• Figure 5: (a) Visualization of the initial state of a 2D monodisperse suspension of disks prior to evaporation. The width of the suspension is $800\sigma$ and its initial thickness is about $834\sigma$. The solvent particles are colored yellow while the disks are colored red. (b) Evolution of the disk distribution in the drying suspension at a receding speed of $v_e \simeq 1.8\times 10^{-4} \sigma/\tau$ for the liquid-vapor interface. The elapsed time since the initiation of evaporation for each frame from top to bottom is $4 \times 10^{6}\tau$, $4.2 \times 10^{6} \tau$, and $4.4 \times10^{6} \tau$, respectively. For clarity, the solvent is not shown. Each disk is assigned a color based on its value of $|\Phi_{6}|$ computed with neighboring disks determined via Voronoi tessellation analysis. Each red stepped line indicates the surface of the drying film of disks. (c) Voronoi tessellations of the disk distributions in (b).