Hard-particle fluids, such as rectangles, spherocylinders, ellipses, and squares are interesting building blocks for assembly as they have the ability to produce diverse structures depending on their aspect ratio, concentration, and subtle differences in their geometry. These systems produce intriguing phases without the aid of specific interactions, such as van der Waals or DNA functionalization, between particles. Interactions that arise in mixtures of particles with only hard-core repulsion are entropically driven and termed depletion. This type of interaction has been observed in colloidal systems consisting of a collection of large particles in a solution of smaller, depletant particles. Depletion interactions have been extensively studied for large spheres in small rods and large spheres in small spheres. In all of these studies, the large particles (spheres) have no rotational entropy and hence no preferred orientation as they approach one another. It is of interest to study the interplay between depletion interactions and the rotational entropy of the large particles.

In this work, we use Monte Carlo simulations to examine the depletion of two-dimensional hard squares and rods by hard disks. We determine the effect of depletant size and concentration, as well as the aspect ratio (squares vs. rectangles) of the large particles on the potential of mean force. We find that the disks are able to deplete squares and rectangles of certain aspect ratios. The strength of the attraction increases with increasing depletant concentration and depends on depletant size. By subtracting the potential of mean force for two large objects in the absence of depletant from that in the presence of depletant, we are able to separate the effect of rotational entropy from that of depletant. We also probe the relative orientation that two squares adopt as they approach each other in a sea of depletant. We observe rich behavior, in which this relative orientation varies with inter-particle separation (e.g., for some separations, particles adopt a face-to-face orientation, while for others, the vertex-vertex orientation is preferred) and depends on depletant size and concentration. This work contributes to the understanding necessary to achieve self-assembled structures that possess complex, long-range order from simple building blocks.

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