Casimir Interaction between Polydisperse Colloids Trapped at a Fluid Interface

TL;DR

This paper investigates how size polydispersity and particle mobility affect fluctuation-induced (Casimir-like) interactions between colloids trapped at a fluid interface. Using a scattering-matrix formalism, it analyzes two- and three-body configurations under three boundary conditions (fixed, bobbing, bobbing and tilting), revealing that mobility can qualitatively switch the effect of polydispersity from suppression to enhancement, especially at long range. The key finding is that mobility filters out dominant monopole and dipole contributions, unmasking higher-order multipoles that are highly geometry- and size-dependent, leading to large, non-additive effects and complex, distance-dependent behavior. These results have implications for guiding self-assembly and pattern formation of polydisperse colloids at interfaces, though direct long-range measurements may be challenging due to the small absolute energy scales involved.

Abstract

We investigate the effect of polydispersity on fluctuation-induced interactions between colloids trapped at a fluid interface. Using the scattering-matrix formalism, we calculate the interaction energy in two- and three-body systems under three mechanical boundary conditions. We find that size asymmetry can either suppress or enhance the many-body interaction compared to a monodisperse system, with the outcome depending on colloids mobility and separation. For fixed colloids, the interaction is suppressed at short separations but enhanced at large separations. In contrast, for mobile colloids, the interaction is predominantly enhanced at large distances but exhibits a competitive behavior in the near-field. This culminates in a large, multi-order-of-magnitude amplified sensitivity {to size asymmetry} for bobbing and tilting colloids at long range, highlighting a complex interplay between geometry and colloids mobility with significant implications for self-assembly.

Figures (5)

• Figure 1: Number of multipoles $N$ required to achieve convergence of the Casimir energy between identical colloids of radius $R$, plotted as a function of the dimensionless surface-to-surface distance $h/R$. The left panel shows $N$ for the two-body interaction $\mathcal{F}^{(2)}$, while the right panel presents $N$ for the three-body contribution $\mathcal{F}_{123}$. In both plots, the dashed blue, dot-dashed red, and solid black curves correspond to scenarios (a), (b), and (c), respectively.
• Figure 2: Two-body Casimir energy deviation parameter $\Delta_{2}$ as a function of the radius ratio $r=R_{>}/R_{<}$ for three different boundary conditions, at a fixed surface-to-surface distance of $h/R_{<}=5$. The full numerical results (symbols) are compared against the large-distance asymptotic theory (lines) from Table \ref{['tab:2p-asym']}. The main plot displays the results for bobbing (squares) and bobbing+tilting (circles). The inset shows the much smaller deviation for the fixed case (triangles).
• Figure 3: Three-body deviation parameter $\Delta_{123}$ versus the radius ratio $r=R_{>}/R_{<}$ for fixed colloids (scenario a). The central colloid has radius $R_{>}$ and the outer ones have radius $R_{<}$. Lines show the large-distance asymptotic prediction from Table. \ref{['tab:3p-asym']}, while symbols represent the full numerical calculation. Triangles, squares, and circles correspond to separations of $h/R_{<}=10,$ 5, and 1, respectively.
• Figure 4: Three-body deviation parameter $\Delta_{123}$ versus $r=R_{>}/R_{<}$ for bobbing colloids (scenario b). Lines are the asymptotic forms from Table II, while symbols denote the full numerical results (triangles for $h/R_{<}=10$, squares for $h/R_{<}=5$, and circles for $h/R_{<}=1$). The main plot is presented on a logarithmic scale, while the inset uses a linear scale to detail the behavior at the shortest separation, $h/R_{<}=1$.
• Figure 5: Three-body deviation parameter $\Delta_{123}$ versus $r=R_{>}/R_{<}$ for bobbing and tilting colloids (scenario c). Lines are the asymptotic forms from Table II, while symbols denote the full numerical results (triangles for $h/R_{<}=10$, squares for $h/R_{<}=5$, and circles for $h/R_{<}=1$). The main plot is presented on a logarithmic scale, while the inset uses a linear scale to focus on the behavior at the shortest separation, $h/R_{<}=1$.