Sedimentation profiles and phase stacking diagrams in polydisperse hard rounded rectangle fluids

Abstract

We analyze the sedimentation behavior of a polydisperse two-dimensional liquid-crystal fluid using a local density functional theory based on scaled particle theory. Polydispersity is incorporated through variations in the roundness of hard rectangular particles interacting solely via excluded area effects. Despite its simplicity, the model displays a rich phenomenology. In bulk, the fluid exhibits isotropic, nematic, and tetratic phases. In sedimentation, we obtain complex phase stacking diagrams featuring multiphasic stacking sequences with up to four stacks of different bulk phases, inverted stacking sequences such as top isotropic and bottom nematic together with top nematic and bottom isotopic, as well as stacking sequences with reentrant stacks such as tetratic and nematic stacks floating between two isotropic stacks. This phenomenology arises as a result of an intricate coupling between particle polydispersity and the effect of gravity. Our approach can be easily adapted to investigate the sedimentation behaviour of other polydisperse colloidal systems.

Figures (3)

• Figure 1: Particle model and distributions. (a) Sketch of a HRR (blue) obtained from a fixed rectangular core (green) of dimensions $L\times D$ by sliding a disk of diameter $l$ (dashed-orange) around the perimeter of the core. (b) Parent distributions, $f(l)$, as a function of the scaled roundness length $l/l_0$. The values of the parameters $\nu$ and $s$ characterizing the distributions are indicated in the figure. (c) Representative examples of the particles considered in this study: polydisperse distribution functions with mean aspect ratios $\kappa_0=2.22$ and $\kappa_0=1.75$, see Eq. \ref{['eq:kappita']}.
• Figure 2: Sedimentation profiles. (a1) Scaled density $\rho^*$ and local packing fraction $\eta$ as a function of elevation $z$ for a sample with mean packing fraction $\overline{\eta}=0.913$ and sample height $H/\left<\xi\right>=105$, corresponding to a TN stacking sequence. The mean aspect ratio is $\kappa_0=1.75$, the polydispersity coefficient is $s=0.936$, and the mean roundness is $\theta=0.3$. (b1) Order parameters $Q_2$ and $Q_4$ as a function of elevation for the same sample as in (a1). (c1) Profiles for the first and second dimensionless moments $\sigma_1$, $\sigma_2$, and maximum value ${\cal M}$ of $x(l,z)$ with respect to $l$ as a function of $z$, for the same sample as in (a1). A sketch of the sample highlighting the stacking sequence is shown in panel (c1). (d1) Local roundness distribution functions $x(l,z)$ as a function of $l$ at three selected elevations $z$, marked in panels (c1) with arrows. Panels (a2) to (d2) display the same quantities as panels (a1) to (d1) for a sample with height $H/\left<\xi\right>=105$ and mean packing fraction $\overline{\eta}=0.908$, which corresponds to an ITNT stacking sequence. Panels (a3) to (d3) display the same quantities as panels (a1) to (d1) for a sample with height $H/\left<\xi\right>=30.8$ and mean packing fraction $\overline{\eta}=0.91$, which corresponds to a NT stacking sequence. Figure \ref{['fig3']}(d), shows the location of the three samples in the stacking diagram using pentagons labeled 1 to 3.
• Figure 3: Stacking diagrams of polydisperse hard rounded rectangles in the plane of average packing fraction $\overline{\eta}$ and scaled sample height $H/\left\langle {\xi} \right\rangle$. The aspect ratio and average roundness of the parent (bulk) distributions are: (a,b) $\kappa_0=2.22$, $\theta=0.56$ and (c,d) $\kappa_0=1.75$, $\theta=0.3$. Two degrees of polydispersity are considered in each case: $s=0.5$ (a,c) and $s=0.936$ (b,d), as indicated above panels (a,b). Each gray square corresponds to a sedimentation sample that we calculated to obtain the stacking diagrams. The black solid lines mark the approximate boundaries between two different stacking sequences in the stacking diagrams. Stacking sequences are labeled from top to bottom and colored differently (see color box). Close-up views of two highlighted regions in panel (d) are shown in the two side panels. The numbered black circles in (a,b) mark the position of the three sedimentation samples sketched to the right of panel (b). The labeled pentagons in (d) and white squares mark the position in the stacking diagram of the three sedimentation samples depicted in Fig. \ref{['fig2']}. Minimization of the functional was not possible in the white region of panel (d) due to numerical instabilities.