Free-energy model for nanoparticle self-assembly by liquid crystal sorting

We modeled the experimentally observed self-assembly of nanoparticles (NPs) into shells with diameters up to 10 $\mu \mathrm{m}$, via segregation from growing nematic domains. Using field-based Monte Carlo simulations, we found the equilibrium configurations of the system by minimizing a free-energy functional that includes effects of excluded-volume interactions among NPs, orientational elasticity, and the isotropic-nematic phase-transition energy. We developed a Gaussian-profile approximation for the liquid crystal (LC) order-parameter field that provides accurate analytical values for the free energy of LC droplets and the associated microshells. This analytical model reveals a first-order transition between equilibrium states with and without microshells, governed mainly by the competition of excluded-volume and phase-transition energies. By contrast, the LC elasticity effects are much smaller and mostly confined to setting the size of the activation barrier for the transition. In conclusion, field-based thermodynamic methods provide a theoretical framework for the self-assembly of NP shells in liquid crystal hosts and suggest that field-based kinetic methods could be useful to simulate and model the time evolution of NP self-assembly coupled to phase separation.

Figure 1

The spatial organization of quantum-dot nanoparticles sorted by a liquid crystal is visualized using fluorescence microscopy. The images above were captured during the shell formation process. In the first stage (a), the NPs segregate into shrinking isotropic domains; at a later stage (b), these domains adopt a spherical shape; in the last stage (c), shells of NPs are formed when the interior of these spherical domains also becomes nematic.

Figure 2

We study a model where (a) nematic 5CB surrounds a sphere of isotropic 5CB with NPs suspended homogeneously inside it. The radius of that sphere $R_{\text{ext}}$ is fixed. Then, (b) we analyze the free energy of the system when a droplet of nematic 5CB, with varying radius $R_{\text{int}}$, confines the NPs to a shell of width $\mathrm{\Delta }R=R_{\text{ext}}-R_{\text{int}}$.

Figure 3

We performed simulated annealing to find the inner-nematic droplet configuration that minimizes its free energy. For the case with reduced inner radius $r=0.2$, $S_{\text{b}}=0.5$, and strong radial anchoring, we show (a) the initial random configuration, (b) the equilibrated configuration after the first annealing run, and (c) the final equilibrated configuration after the second (and last) annealing run. The simulation shows that the order-parameter field $S\left(\mathbf{x}\right)$ is inhomogeneous due to the presence of a central hedgehog defect.

Figure 4

The order-parameter profile $S\left(x\right)$ as a function of the radial coordinate $x$ is shown for nematic droplets with varying radius $r=0.2$ (dark gray diamonds), 0.4 (gray squares), and 0.6 (light gray circles), at $T=294$ K with bulk order parameter $S_{\text{b}}=0.6$. The solid line corresponds to the analytical model of $S\left(x\right)$ given in Eq. (8), which is independent of the droplet size.

Figure 5

The phase-transition free energy $F_{\text{IN}}$ is a decreasing function of droplet radius $r$: Simulation results are shown as circles, with solid ones indicating those used to fit Eq. (9). The analytical free energy from Eq. (11), shown with lines, remains accurate even when extrapolated to the largest simulated droplets. The range of $S_{\text{b}}$ and $\mathrm{\Delta }T$ cover the experimental range between crystallization and isotropic-nematic transition.

Figure 6

The elastic free energy $F_{\text{E}}$ is an increasing function of droplet size $r$. As in Fig. 5, the analytical free energy from Eq. (12), shown with lines, is in good agreement with simulation results (circles). Again, solid circles indicate simulations used to fit Eq. (9).

Figure 7

The excluded-volume (black dot-dashed line), elastic (blue solid line), and phase-transition (dashed line) contributions to the total free energy of the system scale very differently with inner droplet radius $r$: While the elastic contribution grows slowly with increasing $r$, the hard-sphere and phase-transition terms grow quickly and have opposite signs. The lines correspond to analytical expressions for the free energies, while Monte Carlo simulation data are shown as points, for the case with $\mathrm{\Delta }T=-0.6$ K and initial NP volume fraction ${\eta }_0=0.02$.

Figure 8

A first-order structural transition is found for temperatures sufficiently below the clearing point $T_c$: At $\mathrm{\Delta }T=(T-T_c)=-0.5$ (blue solid line) the free energy for the minimum with $r>0$ is the same as that with the system with no shell formed ($r=0$). At lower temperatures, e.g., $\mathrm{\Delta }T=-1.0$ (red dashed line) the global minimum is the state $r>0$, that is, the state with a shell is favored thermodynamically. For sufficiently high temperatures, such as $\mathrm{\Delta }T=-0.5$ (black dot-dashed line) the only minimum is at $r=0$ and no shell would be formed.

Figure 9

The equilibrium radius as a function of the temperature difference $\mathrm{\Delta }T=(T-T_c)$ decreases as the initial NP volume fraction increases (that is, the shells thicken as ${\eta }_0$ grows). The first-order transition point $\mathrm{\Delta }{T}_{\text{shell}}$ also decreases with increasing ${\eta }_0$: Lower temperatures are needed in order to form a stable shell as one increases the initial volume fraction of NPs (lower and higher volume fraction curves indicated by black solid and red dashed lines, respectively).

Figure 10

The equilibrium radius is set mainly by competition between the NP excluded-volume and LC phase-transition free energies. The total free energy (black solid line) displays an equilibrium point E and an activated state A. If one neglects the LC elasticity term (red dashed line), the new equilibrium point ${\mathrm{E}}^{\prime }$ is shifted by 8% to the right of E and the barrier to formation of shells is lowered.