Figure 1

Angular configuration of the pressure distributions symbolically represented as gray patches centered at the directions ${\Omega }_1=(0,0)$ (north pole) and ${\Omega }_2$ on the unit sphere. The angles ${\phi }_1$ and ${\phi }_2$ represent orientations of the local coordinate frames $x^{\prime }{y}^{\prime }{z}^{\prime }$ and $x^{\prime \prime }{y}^{\prime \prime }{z}^{\prime \prime }$ associated with the pressure distributions (see main text).Reuse & Permissions

Figure 2

(a) Rescaled effective interaction potential [Eq. (24), solid line] between two capillary monopoles. The level of zero energy is shifted such that $\Delta F_{00}(\overline{\theta }=\pi )=0$. The dashed line corresponds to the expression $[\ln \left(\overline{\theta }\right)-\ln \left(\pi \right)]/\left(2\pi \right)$ representing for comparison the case of a flat interface with the equivalent spatial separation $d=R_0\overline{\theta }$ of the particles. As expected, for $\overline{\theta }\to 0$ the full curve approaches the dashed curve. The symbols correspond to the results of the numerical minimization of the free energy for droplets of various radii, with the contact angle at the particle ${\theta }_p=\pi /2$, and with the strength of the external force $Q_0=f/\left(\gamma a\right)=-1$ (upper set of symbols at a given $\overline{\theta }$) or $Q_0=f/\left(\gamma a\right)=1$ (lower set of symbols), respectively. The data terminate at an angular separation corresponding to the contact of the particles. In (b) and (c) we show snapshots of the triangulated droplet surface corresponding to $R_0/a=4$, ${\theta }_p=\pi /2$, and $Q_0=2$ (b) or $Q_0=-2$ (c) with the position of the center of mass of the liquid indicated by CM. In (b) and (c) a cross-sectional view is shown with the plane defined by CM and the centers of the particles.Reuse & Permissions

Figure 3

(a) Rescaled effective interaction potential [Eqs. (25, 26, 27, 28), solid lines] between two capillary dipoles. The dashed line corresponds to the expression $(1/{\pi }^2-1/{\overline{\theta }}^2)/\left(2\pi \right)$ representing for comparison the case of a flat interface with the equivalent spatial separation $d=R_0\overline{\theta }$ of the particles. The dotted lines represent three metastable branches of the free energy [see Eq. (28)] corresponding to three different orientations of the particles with two relevant intersection points [at ${\overline{\theta }}_1\simeq 1.83$, where $f_{+}\left({\overline{\theta }}_1\right)=0$, and ${\overline{\theta }}_0\simeq 0.38$, where $f_{-}\left({\overline{\theta }}_0\right)=0$] indicated by solid squares. In each interval $[0,{\overline{\theta }}_0]$, $[{\overline{\theta }}_0,{\overline{\theta }}_1]$, and $[{\overline{\theta }}_1,\pi ]$ the full curve corresponds to the lowest of the branches and as such describes thermal equilibrium. The symbols correspond to the numerical results with a finite-size correction (see main text) applied in the case $\leftarrow \to$. In (b)–(d) we show the snapshots of the numerically calculated droplet shapes corresponding to $R_0/a=5$ and $\overline{\theta }=\pi /3$; the unstable configuration (b) corresponds to the local maximum of the free energy with respect to the orientations of the particles, the metastable configuration (c) to the inflection point, and the stable one (d) to the local minimum. The stability of the branches changes discontinuously at the intersection points. In (b) and (c) the green parts denote the fixed area of the particle-liquid interface with a fixed three phase contact line.Reuse & Permissions

Figure 4

(a) Rescaled effective interaction potential between two capillary quadrupoles [Eq. (30), thick solid line]. The black dashed line corresponds to the expression $3(1/{\pi }^4-1/{\overline{\theta }}^4)/\left(4\pi \right)$ representing for comparison the case of a flat interface with the equivalent spatial separation $d=R_0\overline{\theta }$ of the particles. The symbols correspond to the results of the numerical minimization of the free energy for two force- and torque-free ellipsoidal particles with aspect ratio $a/b=3$, contact angle ${\theta }_p=2\pi /3$, various droplet radii $R_0$, and two different orientational configurations: tip-to-tip (full lines) and side-to-side (dashed lines), as visualized by the snapshots shown as insets. In view of large relative numerical errors for small values of $\Delta F_{22}$ the zero of the free energy is taken as a mean value of the data points with $\overline{\theta }>1.5$. The colored lines connecting the symbols are guides to the eye. The variables describing the spatial configuration of particle $i$ are explained in (b). The immersion $h_i$ and the tilt angle ${\psi }_i$ of the long axis of the particle with respect to the plane perpendicular to the direction ${\Omega }_i$ are both subjected to minimization, while the orientation ${\phi }_i$ is kept fixed. (c) Contour of the interface around a single particle at the north pole of a droplet, with $R_0/b=8$ and for ${\phi }_i=\pi /2$, quantified in terms of the radial position $r(x,y)$ of the interface (symbols) at a fixed distance (equal to $3b$) from the particle center. For comparison the corresponding undulation $(\Delta u/2)\cos \left(2\phi \right)$, with the amplitude $\Delta u/b=0.102$ chosen to be the same and with $\phi =\arctan (y/x)$, for a pure quadrupole placed at the particle center is plotted as the full red line.Reuse & Permissions

Figure 5

Particles inside thin spherical liquid films (double droplets or bubbles). (a) A trapped spherical hydrophilic particle corresponds to two capillary monopoles of positive (+) and negative (–) sign induced at the external and at the internal interface of the film, respectively. (b) An elongated particle with a hydrophilic and a hydrophobic part corresponds, accordingly, to two capillary dipoles with opposite orientations, one at each side of the film. In both cases the sum of all the monopoles or dipoles associated with the particle at both interfaces is zero due to the condition that there are no external forces or torques acting on the particle.Reuse & Permissions