Lattice Induced Short-Range Attraction between Like-Charged Colloidal Particles

We perform experiments and computer simulations to study the effective interactions between like-charged colloidal tracers moving in a two-dimensional fluctuating background of colloidal crystal. By a counting method that properly accounts for the configurational degeneracy of tracer pairs, we extract the relative probability of finding a tracer pair in neighboring triangular cells formed by background particles. We find that this probability at the nearest neighbor cell is remarkably greater than those at cells with larger separations, implying an effective attraction between the tracers. This effective attraction weakens sharply as the background lattice constant increases. Furthermore, we clarify that the lattice-mediated effective attraction originates from the minimization of free energy increase from deformation of the crystalline background due to the presence of diffusing tracers.

Figure 1

(a) Experimental snapshot of the tracer particles diffusing in a background of quasi-two-dimensional colloidal crystal, in which the bright particles are small fluorescent probe particles. (b) Sketch of simulation system, consisting of red small tracers and a crystalline lattice background. (c) Schematic diagram for determining the neighboring cells with respect to the red triangle in the center of simulation system. The large grey particles correspond to the background particles, and the small cyan beads are the centers of triangular cells. Neighbors from 1 to 10 are marked by digits and different colors. (d) The measured pair correlation function of the triangular cell centers of the lattice corresponding to (a). Different neighboring cells are determined by the location of peaks marked by blue digits.

Figure 2

Relative probabilities of finding a tracer pair in neighbor cells from 1 to 10 by experiments (a) and simulations (b). The stiffness of potential interactions in simulations is $n_{ll}=12$ between background particles and $n_{ls}=12$ between tracer and background particles, and a soft repulsive potential between tracers extracted from experiment is used for consistency. The lower inset compares the relative probability in the fluctuating background with that in the fixed background, in which the lattice constant $L/{\sigma }_l$ is 1.078, the stiffness parameters are $n_{ll}=12$, $n_{ls}=12$, $n_{ss}=2$, and the interaction diameter between large and small particles is taken as $0.88\times ({\sigma }_l+{\sigma }_s)/2$.

Figure 3

(a) Schematic diagram of the lattice distortion. The bond length is elongated to $d=x+\delta x$ from $x$ due to the invasion of the tracer, and two elongated bonds merge into one when two tracer particles are located in the nearest-neighbor cells. (b) Drag force $f$ as a function of elongated distance $\delta x$ between two large background particles for various lattice constants $L/{\sigma }_l$. Dotted lines are fits to the simulation data, and the arrows to the $x$ axis are the corresponding integral limits for work calculation. The stiffness parameters are $n_{ll}=12$ and $n_{ls}=12$. (c) The probability ratio of two particles in the first to the tenth nearest-neighbor cells obtained from theory and simulation.

Figure 4

Relative probabilities $P$ (solid symbols) and the contributions from the elastic energy $P_U$ (open symbols) for different stiffnesses of potential interactions (a) between background particles $n_{ll}$ and (c) between tracer and background particles $n_{ls}$. The orange curve in (a) is obtained in the pure hard spheres system by Monte Carlo simulations (see the Supplemental Material for the simulation details [32]). Here, the reference parameters are $n_{ll}=12$, $n_{ls}=12$, $n_{ss}=2$, and $L/{\sigma }_l=1.052$. (b) and (d) show the entropy contributions to the relative probabilities, $P/P_U$, corresponding to (a) and (c), respectively.