# Electric-filed-induced capillary attraction between like-charges particles at liquid interfaces

**Introduction**

Charged particles near non-polar aqueous interfaces have been shown to spontaneously order themselves in the absence of confinement. Thus the particles attract one-another, despite the Coulomb repulsion that the particles share. What really causes this attraction? One possibility is that surface roughness of the particles results is deformation of the interface leading to capillary attraction. However, the authors argue that this effect would be too small for the colloidal particles used to create a substantial effect. Entropic interactions are also thought to be negligible. Another option is that the buoyancy force due to density mismatch results in a stress at the interface, creating a deformation in the surface and thus an attractive capillary force between colloidal particles. However, these effects are also considered to be too small. The authors therefore suggest that the attraction between colloidal particles near an interface is due to electrostatic stresses due to a dipolar field created by the charged particles. These interfacial stresses result in interfacial deformations, which then result in an attractive capillary force that is much like the "cheerios effect". If this theory is correct, then the observed particle-particle attraction should be tunable via the changing the polarity of the interfacial fluids.

To study this effect, the authors place colloidal particles in a large water drop that sits below oil. Figure 1 below shows a typical configuration of the colloidal particles. Images were taken using fluorescence microscopy. The particles remain at the interface and are trapped there, which suggests that they are in an energy well much deeper than <math>k_B T</math>. A small number of particles also order themselves as shown in Figure 2, and remain stable for 30 minutes. Because the particles self-seggregate into stable configurations over large distances, it is clear that there are long-range attractive interactions present.

To quantify the interaction potential, the authors use the configuration shown in Figure 2 below, and measure the distance between the center particle and each of the outer particles to obtain a pdf for the inter-particle distance (P(r)). The inter-particle potential (V(r)) is then obtained via Boltzmann statistics. The results of this analysis are shown in Figure 3 below.

<math>P(r) \propto exp{(-\frac{V(r)}{k_B T})}</math>

The potential energy of inter-particle capillary attraction of two particles near an interface, that each apply a force, F, normal to the interface is given by

<math>U_{interface}(r) = \frac{F^2}{2\pi \gamma} log(\frac{r}{r_0})</math>

where <math>{\gamma}</math> is the interfacial tension and <math>r_0</math> is an arbitrary constant. As mentioned previously, the force due to density mismatch is too small to result in the observed attractions.

Electrical stresses, however, are substantial enough to result in the observed attraction. The electrostatic energy density is given by <math>\frac{1}{2} \epsilon \epsilon_0 E^2</math>. Since the electric permitivity in oil (~2) is much smaller than the electric permittivity of water (~40) the electric field and electric energy density is about 40 times less in water than in oil. As a result, the spheres act as if they are being pulled into the water in order to minimize the free energy as shown in Figure 4 below. The authors approxiimate the dipolar field, and obtain potential between two particles (U).

<math>F \approx \frac{p^2 \epsilon_{oil}}{16 \pi \epsilon_0 {a_w}^4 {\epsilon_{water}}^2}</math>

<math>U = \frac{F^2}{2 \pi \gamma} ln(\frac{r}{r_0} + \frac{p^2}{4 \pi \epsilon_0 r^3 \frac{2 \epsilon_0}{\epsilon_{water}^2})</math> The total energy is thus due to dipolar repulsion (the first term) and capillary attraction (the second term). From this derivation, the authors obtain an expression for the equillibrium distance (r_eq) and the spring constant (k). Although there are the dipole moment (P) and the wetted area radius (a_w) are unknown, the authors fit the data to obtain values of r_eq = 5.7 um, and k = 23K_bT um^-2, which are close the values observed experimentally.