Phase diagrams for sticky rods in bulk and in a monolayer from a lattice free-energy functional for anisotropic particles with depletion attractions

A density functional of fundamental measure type for a lattice model of anisotropic particles with hard-core repulsions and effective attractions is derived in the spirit of the Asakura-Oosawa model. Through polymeric lattice particles of various size and shape, effective attractions of different strength and range between the colloids can be generated. The functional is applied to the determination of phase diagrams for sticky rods of length $L$ in two dimensions, in three dimensions, and in a monolayer system on a neutral substrate. In all cases, there is a competition between ordering and gas-liquid transitions. In two dimensions, this gives rise to a tricritical point, whereas in three dimensions, the isotropic–nematic transition crosses over smoothly to a gas-nematic liquid transition. The richest phase behavior is found for the monolayer system. For $L=2$, two stable critical points are found corresponding to a standard gas-liquid transition and a nematic liquid-liquid transition. For $L=3$, the gas-liquid transition becomes metastable.

Figure 1

(a) A representation of the lattice model in 2D. Particle ${\mathcal{L}}_i$ is fully specified by its size vector ${\mathbf{L}}_i=(L_x^i,L_y^i)$ and its position (blue dot) at ${\mathbf{s}}_i=(x_{\min }^i,y_{\min }^i)$. (b) Overlap of two or more particles is forbidden due to their mutual hard-core interaction. (c) Short-ranged, sticky attraction between the particles. The strength of the attractive interaction is proportional to the number of neighboring lattice sites (length of bold red lines).

Figure 2

A representation of a mixture of colloidal rods and polymers in the lattice model in 2D. Since a polymeric rod occupying only one lattice point does not induce a depletion layer, the minimum allowed polymer size is $L_{\mathrm{p}}=2$. Here we have shown two types of such polymers, ${\mathcal{L}}_{\mathrm{p},1}$ and ${\mathcal{L}}_{\mathrm{p},2}$ with ${\mathbf{L}}_{\mathrm{p},1}=(1,2)$ and ${\mathbf{L}}_{\mathrm{p},2}=(2,1)$ in (a) and ${\mathcal{L}}_{\mathrm{p},3}$ with ${\mathbf{L}}_{\mathrm{p},3}=(2,2)$ (d). Due to our convention on specifying the position of a particle, the depletion layers around the blue colloidal rods are asymmetric as shown in (b) and (e). Overlap of the excluded volumes corresponding to one polymeric rod species increases their available free volume and results in an effective attraction between colloidal rods. The effective attraction is proportional to the overlapping area as well as the corresponding polymer reservoir density ${\rho }_{\mathrm{p},j}^{\mathrm{r}}$. The anisotropic polymeric rods, i.e., ${\mathcal{L}}_{\mathrm{p},1}$ and ${\mathcal{L}}_{\mathrm{p},2}$, induce solely sticky attractions along the rod axis as shown in (c) (the corresponding overlap area is shown by the dark green area), while ${\mathcal{L}}_{\mathrm{p},3}$ induces an edge-edge attraction as well (shown in (f) by the corresponding overlap area in dark brown).

Figure 3

The four FMT weight functions in 2D for a rod with size vector $\mathbf{L}=\left(2,3\right)$. The green circles represents the lattice points at which the weight function is 1. The lattice point $\mathbf{s}$ at which each weight function is evaluated is marked with a blue dot.

Figure 4

(a) Equilibrium demixing order parameter for $L=6$ and different polymer reservoir densities. There exists a critical density above which the system demixes. The critical density shifts to lower densities by increasing effective attraction; i.e., decreasing the effective temperature $T=1/{\rho }_{\mathrm{p}}^{\mathrm{r}}$ in the AO model. (b) The phase diagram of the 2D lattice model for $L=6$. There exists a tricritical point below which the system exhibits a first order phase transition between an isotropic gas phase and a demixed state for which $S_{\mathrm{eq}}\ne 0$. The isotropic gas-liquid phase transition is unstable with respect to the former phase transition.

Figure 5

(a) Equilibrium nematic order parameter $Q_{\mathrm{eq}}$ for $L=6$ and different polymer reservoir densities ${\rho }_{\mathrm{p}}^{\mathrm{r}}=1/T$. On increasing the total colloidal density ${\rho }_{\mathrm{c}}$ the system undergoes a first-order isotropic-nematic phase transition. The critical density shifts to lower densities by increasing effective attraction; i.e., decreasing the effective temperature $T=1/{\rho }_{\mathrm{p}}^{\mathrm{r}}$ in the AO model. (b) The phase diagram of the 3D lattice model for $L=6$. For a pure hard-rod system ($T\to \infty$), there is a first-order phase transition from an isotropic gas to a nematic liquid state. By decreasing the effective temperature, the phase coexistence region becomes broader.

Figure 6

A 3D monolayer in which the particles are confined to move on a substrate (left) can be translated to a 2D system where the projection of standing-up rods is considered as a new species with size $(1,1)$ (right).

Figure 7

Equilibrium nematic order parameter and resulting surface pressure for a (2+1)D system with $L=2$ [(a) and (c)] and $L=3$ [(b) and (d)]. Panels (a) and (b): $Q_{\mathrm{eq}}$ vs. ${\rho }_{\mathrm{c}}$ (total density) for various effective temperatures $T=1/{\rho }_{\mathrm{p}}^{\mathrm{r}}$ (full lines). Dashed lines correspond to the low-density expansion [see Eq. (46)]. Panels (c) and (d): Surface pressure $\beta p={\rho }_{\mathrm{c}}^2\partial /\partial {\rho }_{\mathrm{c}}\left(f/{\rho }_{\mathrm{c}}\right)$ as a function of ${\rho }_{\mathrm{c}}$ for various $T$. Insets show the onset of the second van der Waals loop for temperatures near $T_{\mathrm{cr},2}\approx 1.2$.

Figure 8

Phase diagram of a (2+1)D system for (a) $L=2$ and (b) $L=3$. For $L=2$ two stable phase transitions occur for $T<T_{\mathrm{cr}}$ (red dashed binodal) and $T<T_{\mathrm{cr},2}$ (blue dashed binodal). These transitions become metastable below a triple temperature $T_{\mathrm{tr}}$ with respect to a phase transition between a highly ordered state at high densities and a nearly isotropic gas state (full black binodal). The purple dot-dot-dashed binodal is the metastable continuation of the second binodal. For $L=3$, the first transition for $T<T_{\mathrm{cr}}$ is stable (full black binodal). The second transition for $T<T_{\mathrm{cr},2}$ (purple dot-dot-dashed binodal) is completely metastable. Reentrant demixing in the substrate plane (red dashed line) and a discontinuous jump in $Q_{\mathrm{eq}}$ (black dash-dash-dotted line) are metastable as well. For both $L=2$ and 3 the green dotted line is the binodal of a gas-liquid transition between isotropic states.