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 $P_{\mathrm{dimer}}$, equilibrium constant $K$, and 2D osmotic pressure ${\mathrm{\Pi }}_{2\mathrm{D}}$ as a function of the particle area fraction ${\varphi }_{\mathrm{A}}$ 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.

Figure 1

Schematics of an annular sector particle (ASP) and the geometrical condition defining mutual lock-and-key dimerization of two ASPs. (a) Shape and size parameters characterizing an ASP: $R_{\mathrm{i}}$ and $R_{\mathrm{o}}$ are the inner and outer radii, respectively; $\psi$ is the opening angle. Coordinates $x$ and $y$ define the center of the ASP with respect to the origin of a fixed reference frame. $\theta$ is its orientation angle, with respect to the $x$ axis, based on a ray from the center of the circular arcs to a point that is midway between the ends of the ASP's two arms. (b) Two separate monomer ASPs: no portion of a first ASP intersects with the imaginary dashed line defining the opening boundary of the concave interior of a second ASP and vice versa. (c) Two proximate ASPs: a first ASP does intersect with the imaginary dashed line of a second ASP, but the second ASP does not contact the imaginary dashed line of the first ASP. (d) Two proximate ASPs in a mutually interpenetrating lock-and-key dimer configuration. Mutually interpenetrating lock-and-key dimerization occurs when a portion of the perimeter of one ASP intersects with the imaginary dashed line spanning the inner opening of the other ASP and vice versa. This geometric definition for dimerization of nonoverlapping ASPs can be readily determined using a collision detection routine. This collision detection routine is based on a combination of routines which detect the intersection of two circular arcs, a line segment with a circular arc, and two line segments.

Figure 2

Cage configurations limiting motion of the central ASPs over a range of values of particle area fractions ${\varphi }_{\mathrm{A}}$: $R_{\mathrm{i}}/R_{\mathrm{o}}=0.75$ and $\psi ={95}^{\circ }$. Static cage ASPs are shown in black. Examples of two central mobile ASPs at ${\varphi }_{\mathrm{A}}=0.263$ in (a) a monomer configuration (green or light gray) and (b) a dimer configuration (red or dark gray). Example cages generated at ${\varphi }_{\mathrm{A}}$: (c) 0.180, (d) 0.263, and (e) 0.340.

Figure 3

Accessible microstates of a first central mobile ASP (top) from the single-particle simulation for a fixed position and orientation of a second central ASP (bottom). Here, $R_{\mathrm{i}}/R_{\mathrm{o}}=0.75$, $\psi ={95}^{\circ }$, and ${\varphi }_{\mathrm{A}}=0.263$. The location and orientation of the fixed ASP, with respect to a coordinate system located at the geometric center of the cage, are given by $x=-0.05$, $y=-0.5$, and $\theta ={300}^{\circ }$, in units where $R_{\mathrm{o}}=1$. Transparency overlay plots of accessible microstates corresponding to (a) monomer configurations (green or light gray) and (b) dimer configurations (red or dark gray).

Figure 4

Calculated minimum value of the opening angle ${\psi }_{\min }$ of ASP shapes that can dimerize as a function of $R_{\mathrm{i}}/R_{\mathrm{o}}$. In the region below this curve, dimerization is impossible.

Figure 5

Predictions from the two-particle cage model simulation of the dimerization probability $P_{\mathrm{dimer}}$. (a) $P_{\mathrm{dimer}}$ as a function of ${\varphi }_{\mathrm{A}}$ at fixed $\psi ={100}^{\circ }$: $R_{\mathrm{i}}/R_{\mathrm{o}}=0.65$ (blue diamonds), $R_{\mathrm{i}}/R_{\mathrm{o}}=0.7$ (red squares), $R_{\mathrm{i}}/R_{\mathrm{o}}=0.75$ (black circles), and $R_{\mathrm{i}}/R_{\mathrm{o}}=0.8$ (orange triangles). (b) $P_{\mathrm{dimer}}$ as a function of ${\varphi }_{\mathrm{A}}$ at fixed $R_{\mathrm{i}}/R_{\mathrm{o}}=0.75$: $\psi ={70}^{\circ }$ (blue diamonds), $\psi ={90}^{\circ }$ (red squares), and $\psi ={110}^{\circ }$ (black circles).

Figure 6

Results of cage model simulations and experimental data as functions of ${\varphi }_{\mathrm{A}}$. (a) Probability of dimerization $P_{\mathrm{dimer}}$: experiments (black circles), two-particle simulations (red squares), and one-particle simulations (blue diamonds). (b) Equilibrium constant $K$ of the dimerization reaction [same symbols as in (a)]. (c) Scaled 2D osmotic pressure ${\mathrm{\Pi }}_{2\mathrm{D}}$ [same symbols as in (a)]. Line: fit to numerical results using Eq. (3) (see text).

Figure 7

Two-particle simulation results showing the dependence of the equilibrium constant $K$ of the dimerization reaction as a function of the scaled 2D osmotic pressure ${\mathrm{\Pi }}_{2\mathrm{D}}$. Solid line: fit using Eq. (4).