Elastocapillary interaction for particles trapped at the isotropic-nematic liquid crystal interface

We present numerical simulations on pairwise interactions between particles trapped at an isotropic-nematic liquid crystal (Iso-N) interface. The particles are subject to elastocapillary interactions arising from interfacial deformations and elastic distortions of the nematic phase. We use a recent model based on a phase-field approach [see Qiu et al., Phys. Rev. E 103, 022706 (2021)] to take into account the coupling between elastic and capillary phenomena. The pair potential is computed in a two-dimensional geometry for a range of particle separations and two anchoring configurations. The first configuration leads to the presence of an anchoring conflict at the three-phase contact line, whereas such a conflict does not exist for the second one. In the first case, the results show that significant interfacial deformations and downward particle displacements occur, resulting in sizable attractive capillary interactions able to overcome repulsive elastic forces at intermediate separations. The pair potential exhibits an equilibrium distance since elastic repulsions prevail at short range and prevent the clustering of particles. However, in the absence of any anchoring conflict, the interfacial deformations are very small and the capillary forces have a negligible contribution to the pair potential, which is fully repulsive and overwhelmed by elastic forces. These results suggest that the self-assembly properties of particles floating at Iso-N interfaces might be controlled by tuning anchoring conflicts.

Figure 1

Sketch of the simulation domain. Box size: $(H,L)=(16R,28R)$. Symbols: $\theta$, contact angle; $\mathrm{\Delta }{y}_I$, interfacial deformation; $\partial {\mathrm{\Omega }}_p$, particle surface; $\partial {\mathrm{\Omega }}_w$, box wall; $\partial {\mathrm{\Omega }}_S$, symmetry plane (see text for other symbols definitions). The blue and red arrows indicate that the particles may interact either attractively, repulsively, or both. Insets: Initial bulk and anchoring conditions of the order parameter $\mathbb{Q}$. The small ellipsoids symbolize the LC molecules (not to scale). A homeotropic (H) or planar (P) anchoring is prescribed at the Iso-N interface, while a homeotropic condition is imposed at the particle surface.

Figure 2

H-H configuration. [(a)–(c)] Contour plots of $Q_{22}^2$ (grayscale) at steady state. (a) $r^{*}=r/2R=6.2$ and (b) $r^{*}=2$. The red dot-dashed line to the right indicates the symmetry plane. The full width of the simulation box is displayed in (a) and (b), but only a part of the domain is shown in the vertical direction. (c) Zoomed-in view of (b) near the particle. The gray dashed line and circle represent the initial positions of the interface and the particle, respectively. (d) Blown-up view of (c) near the contact line. The grayscale shows the concentration-weighted scalar order parameter $\frac{1-\varphi }{2}q$, with $q_e=0.81$ (Table 1). The tiny dark blurry area located inside the dashed circle signals a topological defect. The curve $\varphi =0$ marks the fluid interface. $\theta ={90}^{\circ }$.

Figure 3

H-H configuration. Interaction force ($F_{\text{int}}^{*}=F_{\text{int}}/F_s$) as a function of the center-to-center distance ($r^{*}=r/2R$) for two values of the elastic constant ($L_1^{*}=L_1/F_s$). The shaded areas around the data points represent the typical uncertainty interval in the simulations (see text for details). A small attractive well develops at intermediate range ($r^{*}<3$) for $L_1^{*}=0.5$, whereas an all-repulsive force is computed for $L_1^{*}=1$. Inset: Snapshot obtained for $L_1^{*}=1$ and $r^{*}=1.5$. The red dot-dashed line symbolizes the symmetry plane. Grayscale: $Q_{22}^2$. $\theta ={90}^{\circ }$.

Figure 4

H-H configuration. Interfacial deformation $\mathrm{\Delta }{y}_I$ (a), particle displacement $y_p$ (b), and capillary interaction force $F_{\text{cap}}$ (c) as a function of the center-to-center distance $r$ for two values of the elastic constant $L_1^{*}$. Insets in (a) and (c): verification of the scaling relations $\mathrm{\Delta }{y}_I\sim L_1^{-1/2}$ and $F_{\text{cap}}\sim L_1^{-1/2}$, respectively (see text for details).

Figure 5

P-H configuration. Interaction force ($F_{\text{int}}^{*}=F_{\text{int}}/F_s$) as a function of the center-to-center distance ($r^{*}=r/2R$). The interaction potential is repulsive for all separations. As in Fig. 3, the shaded areas around the data points represent the typical uncertainty interval in the simulations (see text for details). Insets: Snapshots obtained for $r^{*}=4$ (a) and $r^{*}=1.1$ (b). The topological defect located underneath the particle is slightly shifted towards the symmetry plane at very close range (b). The interfacial deformations and particle displacement are negligible here. The red dot-dashed lines symbolize the symmetry plane. Grayscale: $Q_{11}^2$. $\theta ={90}^{\circ }$, $L_1^{*}=1$.

Figure 6

Proportionality relationship between the capillary interaction force $F_{\text{cap}}$ and the interfacial deformation $\mathrm{\Delta }{y}_I$ for two values of the elastic constant $L_1^{*}$.