Tuning pair interactions in colloidal systems using random light fields

TL;DR

The paper addresses tuning isotropic, translation-invariant pair interactions between absorptionless colloidal particles using artificially generated random light fields. It introduces a spectral-density design framework that casts the target potential as a nonnegative linear combination of responses at multiple frequencies and solves the optimization with nonnegative least squares (NNLS). Demonstrations across electrical double-layer, Lennard-Jones, and stealthy-hyperuniform–like potentials show accurate fits and inherently sparse spectral solutions, with a dimensionality analysis revealing a rich design space (up to about $d\approx 42$) driven by interference between electric and magnetic dipoles. The approach offers a practical route to tailor colloidal interactions for controlled self-assembly and stability, with robustness to noise and frequency-line pruning, enabling flexible, spectrum-based tuning of many-body effects.

Abstract

We propose a method to tune interactions between absorptionless colloidal particle pairs. This is achieved via optimization of the spectral energy density of a homogeneous random optical field. Several standard and more exotic interaction potentials, as well as their negative counterparts, are shown to be successfully tuned. We show that the effective dimensionality of the space of potential functions that can be created by this means can reach up to several tens.

Figures (9)

• Figure 1: Pair interaction potential $U$ induced by a random field at a single frequency as a function of its wavelength $\lambda$ and the center-to-center distance $D$ of the the two dielectric particles of radius $a=230nm$ immersed in water. $T$ is 298K and the energy density of the random field is $U_E= 10^{-17} J\cdot\mu m^{-3}$.
• Figure 2: a, color map of the error of the fitting procedure for $U^{DL}\left ( D \right)$ as a function of the exclusion radius $r_e$ and the parameter $\kappa$. b, comparison of the obtained and target potentials, as a function of the surface-to-surface distance $D$, for the set of parameters leading to the smallest error (star in a, $\kappa=12.155\mu m^{-1}$, $r_e=0\mu m$). In the inset, the obtained energy density of the random field is shown. d shows the error map obtained for $-U^{DL}\left ( D \right)$, correspondingly, c compares the target potential with the best fit (star in c, $\kappa=10\mu m^{-1}$, $r_e=0.370\mu m$), together with the optimized energy density spectrum. In all cases $\Phi=33k_BT$, and $T=298K$.
• Figure 3: a,c, color map of the error in the fitting procedure for $U^{LJ}\left ( D \right)$ and $-U^{LJ}\left ( D \right)$ resp. as a function of the exclusion radius $r_e$ and the parameter $\sigma$. b,d show the best fittings (stars) in a ($\sigma=1.2\mu m$, $r_e=0.72\mu m$ ) and b ($\sigma=1.538\mu m$, $r_e=0.96\mu m$ ), compared with the corresponding target potentials. We show the optimized spectral energy in the insets. In all cases $\Phi=4k_BT$, $T=298K$.
• Figure 4: a,c, color map of the error in the fitting procedure for $U^{SHU}\left ( D \right)$ and $-U^{SHU}\left ( D \right)$ resp. as a function of the exclusion radius $r_e$ and reciprocal length $k_c$. b,d respectively show the best fittings (stars) in a ($k_c=9.58\mu m^{-1}$, $r_e=0.96\mu m$ ) and b ($k_c=10 \mu m^{-1}$, $r_e=0.96\mu m$ ), compared with the corresponding target potentials. We show the optimized spectral energy in the insets. In all cases $\Phi=40k_BT$, $T=298K$.
• Figure 5: a, estimated dimension of the space of fittable function using the SVD. c, estimated dimension of the space of fittable functions with positive coefficients only. b, same example as in Figure \ref{['fig:figure_SHU']}b, with the particles polarizabilities giving the maximum positive dimension $d=42$ (star in c). d, same example but with the particle polarizabilities yielding the minimal positive dimension $d=7$ (cross in c).