Lock-and-key dimerization in dense Brownian systems of hard annular sector particles

We develop a translational-rotational cage model that describes the behavior of dense two-dimensional (2D) Brownian systems of hard annular sector particles (ASPs), resembling C shapes. At high particle densities, pairs of ASPs can form mutually interdigitating lock-and-key dimers. This cage model considers either one or two mobile central ASPs which can translate and rotate within a static cage of surrounding ASPs that mimics the system's average local structure and density. By comparing with recent measurements made on dispersions of microscale lithographic ASPs [P.Y. Wang and T.G. Mason, J. Am. Chem. Soc. 137, 15308 (2015)], we show that mobile two-particle predictions of the probability of dimerization P-dimer, equilibrium constant K, and 2D osmotic pressure Pi(2D) as a function of the particle area fraction phi(A) correspond closely to these experiments. By contrast, predictions based on only a single mobile particle do not agree well with either the two-particle predictions or the experimental data. Thus, we show that collective entropy can play an essential role in the behavior of dense Brownian systems composed of nontrivial hard shapes, such as ASPs.