Figure 1

Surface density of rods in isolated membranes. Symbols are the results of simulations (Appendix ). Lines are predictions calculated from the equation of state for a two-dimensional hard disk system [56] as described in the text. The rod lengths, from top to bottom, are $L=175$, 150, 125, 100, 75, and 50. The diameter of a polymer sphere in the simulations is $\delta =1.5$ and thus a diameter $h=1$ is used for the theoretical cylinders. The dashed line indicates the freezing density of a hard disk system, $\rho =0.88$ [57, 58].Reuse & Permissions

Figure 2

Protrusion distribution in a single membrane. $z$ is the displacement of rods from the center of the membrane. Curves are for $\delta =1.5$, $L=100$, and $p_{\mathrm{s}}=0.06$ (outer lines) or $0.12$ (inner lines). The dashed lines are the distributions predicted by Eq. (7), the solid lines are simulation results, and the dotted lines show the scaling expected from an analysis based on independent rod protrusions [16]: $\rho \left(z\right)\sim \exp (-p_{\mathrm{s}}Az)$, with $A=\pi {(\sigma +\delta )}^2/4$.Reuse & Permissions

Figure 3

(a),(b) Theoretical phase diagrams as a function of osmotic pressure $p_{\mathrm{s}}$ and (a) varying aspect ratio $L$ for depletant size $\delta =1.5$ and (b) varying $\delta$ for $L=100$. The theoretical phase boundaries, calculated as described in the text, are shown as dashed lines. The depletant size is $\delta =1.5$.(c),(d) Phase behavior predicted by simulations for the same parameters as in (a),(b) respectively. Triangles $▵$ denote denote parameters that lead to nematic configurations, $\circ$ symbols correspond to isolated membranes, and $\square$ symbols correspond to smectic layers. In (c),(d) the lower solid lines are the theoretical prediction for the nematic-colloidal membrane phase boundary, while the upper solid lines are fit by eye to the the colloidal membrane-smectic phase boundary. The dashed line in (c) indicates parameter values above which rods crystallized within simulated colloidal membranes. Simulation data is from Ref. [14].Reuse & Permissions

Figure 4

Rod density $\rho \left(z\right)$ of two attractive membranes. The dashed line is the prediction of Eq. (7) and the solid line is from simulations. Note that the two peaks are separated by approximately the rod length and thus the configuration contains two closely stacked membranes. The parameter values are $\delta =1.5$, $L=100$, and $p_{\mathrm{s}}=0.12$.Reuse & Permissions

Figure 5

(a) Theoretical free energy of interacting membranes predicted by the free energy, Eq. (3), with the constraints, Eqs. (5) and (8). Curves are for parameters $\delta =1.5$ and $L=100$, with indicated values of the osmotic pressures $p_{\mathrm{s}}$. (b) Free energy of interacting membranes from simulations using umbrella sampling at the same parameters. Data is from Ref. [14].Reuse & Permissions

Figure 6

Theoretical calculation of the total free energy of the attractive basin $F$, defined by Eq. (10) with $M={10}^4$, for rod lengths $L=40$, 75, 100, and 150 (solid lines, from left to right), and $\delta =1.5$. The horizontal dashed line is $F_0$, defined by Eq. (11) with ${\rho }_{\mathrm{m}}{v}_0={10}^{-8}$.Reuse & Permissions