We study the melting of quasi-two-dimensional colloidal hard spheres by considering a tilted monolayer of particles in sedimentation-diffusion equilibrium. In particular, we measure the equation of state from the density profiles and use time-dependent and height-resolved correlation functions to identify the liquid, hexatic, and crystal phases. We find that the liquid-hexatic transition is first order and that the hexatic-crystal transition is continuous. Furthermore, we directly measure the width of the liquid-hexatic coexistence gap from the fluctuations of the corresponding interface, and thereby experimentally establish the full phase behavior of hard disks.

Two-dimensional hard disks are a fundamentally important many-body model system in classical statistical mechanics. Despite their significance, a comprehensive experimental data set for two-dimensional single component and binary hard disks is lacking. Here, we present a direct comparison between the full set of radial distribution functions and the contact values of a two-dimensional binary colloidal hard sphere model system and those calculated using fundamental measure theory. We find excellent quantitative agreement between our experimental data and theoretical predictions for both single component and binary hard disk systems. Our results provide a unique and fully quantitative mapping between experiments and theory, which is crucial in establishing the fundamental link between structure and dynamics in simple liquids and glass forming systems.

We compare experimental results from a quasi-two-dimensional colloidal hard sphere fluid to a Monte Carlo simulation of hard disks with small particle displacements. The experimental short-time self-diffusion coefficient D(S) scaled by the diffusion coefficient at infinite dilution, D(0), strongly depends on the area fraction, pointing to significant hydrodynamic interactions at short times in the experiment, which are absent in the simulation. In contrast, the area fraction dependence of the experimental long-time self-diffusion coefficient D(L)/D(0) is in quantitative agreement with D(L)/D(0) obtained from the simulation. This indicates that the reduction in the particle mobility at short times due to hydrodynamic interactions does not lead to a proportional reduction in the long-time self-diffusion coefficient. Furthermore, the quantitative agreement between experiment and simulation at long times indicates that hydrodynamic interactions effectively do not affect the dependence of D(L)/D(0) on the area fraction. In light of this, we discuss the link between structure and long-time self-diffusion in terms of a configurational excess entropy and do not find a simple exponential relation between these quantities for all fluid area fractions.

Membranes that selectively filter for both anions and cations are central to technological applications from clean energy generation to desalination devices. 2D materials have immense potential as these ion-selective membranes due to their thinness, mechanical strength, and tunable surface chemistry; however, currently, only cation-selective membranes have been reported.Here we demonstrate the controllable cation and anion selectivity of both monolayer graphene and hexagonal boron nitride. In particular, we measure the ionic current through membranes grown by chemical vapor deposition containing well-known defects inherent to scalably produced and wet-transferred 2D materials. We observe a striking change from cation selectivity with monovalent ions to anion selectivity by controlling the concentration of multivalent ions and inducing charge inversion on the 2D membrane. Furthermore, we find good agreement between our experimental data and theoretical predictions from the Goldman− Hodgkin−Katz equation and use this model to extract selectivity ratios. These tunable selective membranes conduct up to 500 anions for each cation and thus show potential for osmotic power generation.

Using dynamical density functional theory (DDFT), we theoretically study Brownian self-diffusion and structural relaxation of hard disks and compare to experimental results on quasi two-dimensional colloidal hard spheres. To this end, we calculate the self-van Hove correlation function and distinct van Hove correlation function by extending a recently proposed DDFT-approach for three-dimensional systems to two dimensions. We find that the theoretical results for both self-part and distinct part of the van Hove function are in very good quantitative agreement with the experiments up to relatively high fluid packing fractions of roughly 0.60. However, at even higher densities, deviations between the experiment and the theoretical approach become clearly visible. Upon increasing packing fraction, in experiments, the short-time self-diffusive behavior is strongly affected by hydrodynamic effects and leads to a significant decrease in the respective mean-squared displacement. By contrast, and in accordance with previous simulation studies, the present DDFT, which neglects hydrodynamic effects, shows no dependence on the particle density for this quantity.