# Like-Charged Particles at Liquid Interfaces

Entry by Haifei Zhang, AP 225, Fall 2009

## Soft matter keywords

Capillary attraction, Liquid interfaces, Colloidal particle

## Overview

Nanometre- and micrometre-sized charged particles at aqueous interfaces are typically stabilized by a repulsive Coulomb interaction. If one of the phases forming the interface is a nonpolar substance (such as air or oil) that cannot sustain a charge, the particles will exhibit long-ranged dipolar repulsion; if the interface area is confined, mutual repulsion between the particles can induce ordering and even crystallization. However, particle ordering has also been observed in the absence of area confinement, suggesting that like-charged particles at interfaces can also experience attractive interactions. Interface deformations are known to cause capillary forces that attract neighbouring particles to each other, but a satisfying explanation for the origin of such distortions remains outstanding. Here we present quantitative measurements of attractive interactions between colloidal particles at an oil–water interface and show that the attraction can be explained by capillary forces that arise from a distortion of the interface shape that is due to electrostatic stresses caused by the particles’ dipolar field. This explanation, which is consistent with all reports on interfacial particle ordering so far, also suggests that the attractive interactions might be controllable: by tuning the polarity of one of the interfacial fluids, it should be possible to adjust the electrostatic stresses of the system and hence the interparticle attractions.

## Electric-field-induced capillary attraction between like-charged particles at liquid interfaces

### The origin of the attraction

Fig. 1. Interfacial colloidal particle ordering induced by repulsive interactions.
Fig. 2. Scatter plot showing positions of a seven-particle hexagonal crystallite on a water droplet of 24 mm radius. The particles maintained this configuration for more than 30 min; 5 min of data, extracted at time intervals of 1/30 s are shown. The inset shows a fluorescent microscope image of the crystallite.

The mismatch in dielectric constants of adjacent fluids can result in asymmetric charging of particles adsorbed at their interface; if one of the fluids is water, then the aqueous surface acquires a charge, which combines with the screening ions in the water to produce an effective dipole moment of the particle, leading to repulsion. By contrast, the origin of an attractive interaction is less well understood, even though interface deformation is known to give rise to capillary forces and a logarithmic attraction between neighbouring particles. For sufficiently large particles, the deformation is caused by gravity; the particle weight is balanced by surface tension, deforming the interface. This results in the clumping of breakfast cereals at the surface of a bowl of milk, and has been harnessed to self-assemble millimetre-sized particles at interfaces. However, the buoyancy mismatch of micrometre-sized colloidal particles is too small to deform the interface significantly and the resulting attractive energies are far smaller than thermal energy.

An alternative candidate for the origin of the attraction is wetting of the particles, which also deforms the interface; however, a spherical particle positions itself exactly to achieve the equilibrium contact angle without distorting the interface unless the particles are constrained to a thin layer of fluid. Another candidate for the origin of the attraction is surface roughness, which might also deform the interface; however, the roughness required is far greater than that which typically exists on colloidal particles. A final candidate for the origin of the attraction is thermal fluctuations; entropic interactions lower the free energy of the interface when two particles approach, resulting in a Casimir type of effective attraction. However, the resulting forces are too small to cause a significant attraction for colloid particles.

### Quantify the attractive interaction

To investigate this question further, we quantified the attractive interaction between like-charged particles at a water–oil interface. A typical configuration of colloidal particles at the interface of a large water drop in oil is shown in Fig. 1. The long range of the repulsive interaction is apparent from the large particle separation.Moreover, this repulsion, combined with the geometric confinement resulting from the finite area of the emulsion drop, which is completely covered with particles, causes the pronounced ordering of the particles. But even when the particle coverage is not complete, ordering can still be observed, as illustrated by the inset in Fig. 2, which shows a group of seven particles in a hexagonal crystallite. These were the only particles on the surface of a large water drop, and the crystallite remained stable for more than 30min. The persistence of this structure over long times is clear evidence of a long-ranged attractive interaction (Fig. 3). To extract the interaction potential, we exploit the symmetry of the geometry and measure the probability distribution, P(r), of the centre-to-centre distance, r, of the centre particle to each of the outer particles. To obtain the interparticle potential V(r), we invert the Boltzmann distribution:

$P(r) \propto \exp \left\{ { - \fracTemplate:V(r)Template:K B T} \right\}$

The potential has a minimum at an interparticle separation of $r_{eq}$=5.7 um, and is well described as harmonic with a spring constant of $k=23kBT um^{-2}$ (Fig. 3).

### Experimental observations

Fig. 3. Secondary interparticle potential minimum derived from experimental observations.
Fig. 4. Sketch of the equipotential lines at the fluid interface and the resulting distortion of the oil–water interface. The distortion of the interface shape is greatly enhanced for clarity.

The motion of any given particle in the crystallite is influenced by the pair interactions with all the other particles; however, by analysing only the radial distance of the outer particles to the centre particle, these many-particle effects are largely cancelled by symmetry. This was confirmed by molecular dynamics simulation of seven particles in the same configuration, interacting with the same pair potential; an uncertainty of about 10–20% in $r_{eq}$ and k was introduced by many-particle effects.

The electrical stresses arise because oil and water have very different dielectric constants: $\varepsilon _{oil} \approx 2$ and $\varepsilon _{water} \approx 80$: When field lines cross the interface, the intensity of the electric field E and the electrostatic energy density are thus roughly 40 times smaller in water than in oil. The colloidal particle therefore behaves as if pulled into the water by an external force F, as shown schematically in Fig. 4. Thus the particles, which are the source of the electric fields, tend to be surrounded by water, lowering the total electrostatic energy. This effect is analogous to electrostriction and to electrowetting.

The force F can be calculated by the integral of the electric pressure over the free surface. Both the repulsive and the attractive components of the interaction potential depend strongly on the dipole moment, $P = (\sigma /\kappa )\pi a_w^2$. This is directly proportional to the zeta potential of the particle surface, which in turn sensitively depends on surface charge, surface chemistry, and the screening length, $\kappa ^{ - 1}$, set by the ion concentration in the water. Thus, for example, if the surface charge remains constant, small changes in $\kappa ^{ - 1}$ change P a great deal, whereas if the surface charge varies and the zeta potential remains constant, there will be no variation of P with $\kappa ^{ - 1}$. The attractive interaction is sensitively dependent on P; thus, if the surface charge remains constant, the attraction becomes negligible compared to $k_BT$ even for low concentrations of added salt, as shown in the Supplementary Information. To verify the sensitivity of the attraction on P, the measurements were repeated with 5mM NaCl added to the water; only the repulsive interaction remained and its range was reduced, leading to a correspondingly lower particle stability, as shown in the Supplementary Information. These results also confirm that the particles are charged on the aqueous side, rather than the oil side. More generally, the existence of a measurable attractive interaction depends on both P and $a_w$, and thus is sensitive to surface properties, charge and pH, as well as salt concentration.

### Summary

Although the authors' theory does not fully account for the geometry of the wetted region, it nevertheless captures the essential physics: Dipolar electric fields induce surface charges that distort the interface; the dipolar interaction causes repulsion, while the interfacial distortion causes capillary attraction. This behaviour has broader implications for interfacial and colloid chemistry, because adsorption of charged particles at fluid interfaces is a common phenomenon in foods, drugs, oil recovery, and even biology.

## Soft matter details

### The original paper

Nikolaides et al. propose that the puzzling attraction that occurs between micrometre-sized particles adsorbed at an aqueous interface is caused by a distortion of the liquid interface that is due to the dipolar electric field of the particles and which induces a capillary attraction. The authors argue that this effect cannot account for the observed attraction, on the fundamental grounds that it is inconsistent with force balance.

### Different opinion raised

Pushing a sphere into water creates a dimple (left) in the surface, caused by the force on the particle being balanced by the surface tension. But if the electrostatic force pushing the sphere into the liquid is balanced by the electrostatic pressure on the liquid interface, there is no force on the rim and the surface is flattened again (right), inhibiting capillary attraction. R1, R2 are the radii of curvature of the surface.

Megens et al. think the capillary deformation of the interface contributes a mere $1.8*10^{-5}$ to the interaction potential, so it is insignificant thermodynamically. Thus, the mechanism proposed by Nikolaides et al. does not account for the observations, and the origin of the observed attraction remains enigmatic.

Nikolaides et al. propose that the puzzling attraction that occurs between micrometre-sized particles adsorbed at an aqueous interface is caused by a distortion of the liquid interface that is due to the dipolar electric field of the particles and which induces a capillary attraction. The authors argue that this effect cannot account for the observed attraction, on the fundamental grounds that it is inconsistent with force balance.

To estimate the influence of the surface deformation, the authors assume that the sum total of the electrostatic pressure acting on the liquid interface is equivalent to an external force, $F$, pushing the particle into the water. The resulting deformation of the interface would give rise to a long-range interparticle interaction energy $U(r) = (F^2 /2\pi \gamma )\ln (r/r_0 )$ where$\gamma$ is the surface tension, $r$ is the distance between particles, and $r_0$ is an arbitrary constant. However, the electrostatic force acts on the particle and the liquid interface simultaneously, so the equation does not apply. The force F on the sphere is balanced by surface tension, creating a dimple in the water surface. The shape of the dimple is governed by the Young–Laplace equation $[(1/R_1 ) + (1/R_2 )]\gamma = \Delta p$ If the force acts only on the particle, then the pressure difference across the interface $\Delta p \approx 0$ and the radii of curvature $R_1$ and $R_2$ of the surface are equal but opposite. The resulting water level around an isolated sphere would be $h(r) = F/(2\pi \gamma )\ln (r/r_0 )$ for small surface slopes.

Owing to the long range of the logarithm, the force on the sphere is passed on to the rim of the vessel containing the liquid. The situation changes fundamentally when the force F is not external but originates from a surface pressure. Now the meniscus profile is very short range. The electrostatic force F pushing the sphere into the liquid is balanced by the liquid interface and there is no force on the rim of the vessel. In contrast, by unquestioningly applying equation 1, one implicitly admits that there is also a force on the vessel, a force not balanced by any other force. This is inconsistent with Newton's third law. Capillary attraction between spheres is caused by the overlap of their dimples, which reduces the total surface area of the water. The resulting attraction energy is $U(r) = - (F^2 /\pi \gamma )(r_c /r)^6$ for large r. this interaction has a much shorter range than that shown in equation (1), even a shorter range than the dipole-dipole repulsion between like-charged particles that is proportional to 1/r^3, so overall there is no attraction at all. The capillary deformation of the interface contributes a mere $1.8*10^{ - 5}$ to the interaction potential, so it is insignificant thermodynamically. Thus, the mechanism proposed by Nikolaides et al. does not account for the observations.

### The authors' defense

The authors think that their experiment provides a clear measure of the interactions between charged particles at fluid-fluid interfaces and demonstrates that there can be a long-range attractive interaction between such particles. However, Megens and Aizenberg raise important points about the authors' interpretation of these results. The authors'calculations account for the electrostatic stresses acting on the fluid–fluid interface, but neglect the force that the electric field exerts on the particle itself. A detailed evaluation of this force, obtained by calculating the change in electrostatic energy as the particle is pushed into the fluid, shows that the interfacial force pulling the particle out of the fluid is exactly cancelled by the electrical force pushing the particle into the fluid, in agreement with the suggestion of Megens and Aizenberg.

This creates a puzzle: the data show unambiguously that there is a long-range attraction, but what is its origin? The particles have a long-range repulsive interaction, owing to their charges; this is dipolar in character, with the electrostatic energy decaying as $1/r^3$. The attractive interaction must balance this electrostatic repulsion, so, if it decays as a power law, the attractive interaction energy must decay more slowly than $1/r^3$ to create the stable energy minimum observed experimentally. This eliminates possibilities such as asymmetries in the contact line or fluctuation-induced forces, all of which have a power-law decay that is more rapid than $1/r^3$ . The most likely interaction that has sufficient range therefore remains capillary distortion of the interface.

The authors have since confirmed that the particles do have a measurable charge, even when immersed in oil.Measurements of the electrophoretic mobility and pair correlation functions of these particles indicate that their charge can be as high as 200e in oil. In the absence of additional solubilized charges, this would result in a screening length of the order of 20um for the particle volume fraction of about $10^{-4}$ used in the authors' experiments. This is larger than the particle separation, allowing the force imbalance between the particle and the interface to persist far enough for significant interfacial distortion to exist at scales comparable to the interparticle separation. The authors therefore believe that electric-field-induced capillary distortion remains the likely culprit for the attractive interactions between like-charged interfacial particles.

## References

[1] “Like-Charged Particles at Liquid Interfaces”, M. Megens, J. Aizenberg, Nature, 2003, 424, 1014.

[2] Nikolaides, M. G. et al. Electric-field-induced capillary attraction between like-charged particles at liquid interfaces, Nature 420, 299–301 (2002).