# Difference between revisions of "Long-range attraction between colloidal spheres at the air-water interface: The consequence of an irregular meniscus"

Line 19: | Line 19: | ||

Therefore, a model based on the irregular meniscus is used to explain the lateral attraction between the particles at the air-water interface. | Therefore, a model based on the irregular meniscus is used to explain the lateral attraction between the particles at the air-water interface. | ||

Such nonuniform contact lines favor certain orientations when particles are in close proximity and give rise to an attractive force which is a function of the interparticle distance (Fig. 1). | Such nonuniform contact lines favor certain orientations when particles are in close proximity and give rise to an attractive force which is a function of the interparticle distance (Fig. 1). | ||

− | [[Image: | + | [[Image:Stamou1.jpeg|thumb|300px| Fig. 1: , from [1].]] |

Experimentally, this was observed using fluorescence microscopy for particle aggregates at the air-water surface where clusters had interparticle spacings approximately twice the particle diameters. | Experimentally, this was observed using fluorescence microscopy for particle aggregates at the air-water surface where clusters had interparticle spacings approximately twice the particle diameters. | ||

Detergent was also added in later experiments to demonstrate the surface properties of the interface. | Detergent was also added in later experiments to demonstrate the surface properties of the interface. |

## Revision as of 20:36, 13 October 2012

## General Information

**Authors:** Dimitris Stamou, Claus Duschl, and Diethelm Johannsmann

**Publication:** Physical Review E, Vol 62, Issue 4, pp. 5263-5272 (2000)

**Keywords:** Wetting, Surface force, Interface, Capillarity

## Summary

This paper discusses the behavior of colloidal particles at the air-water interface and examines their long-attraction using a model based on nonuniform wetting that leads to irregularity of the meniscus.

The study of colloidal systems at the interface has many applications. As the authors mentioned, they include those in basic physics, such as phase behavior in different dimensions; engineering, such as those in nanofabrication; and industry, such as the manufacturing of emulsions and foams. They also explained that existing models do not explain the observed attractive interactions of particles at distances ~ <math>\mu</math> m. Uncharged colloidal particles at air-water interfaces tend to aggregate due to the van der Waals interaction. Here, attraction based on merging of troughs from gravity is not strong enough for the polystyrene (PS) spheres (radius ~ -0.5 <math>\mu</math> m) used in the experiment. Additionally, although immersion capillary forces are of sufficient strength to explain the results of the experiment, they're likely not present since there is likely no tho solvent film to cause aggregation.

Therefore, a model based on the irregular meniscus is used to explain the lateral attraction between the particles at the air-water interface. Such nonuniform contact lines favor certain orientations when particles are in close proximity and give rise to an attractive force which is a function of the interparticle distance (Fig. 1).

Experimentally, this was observed using fluorescence microscopy for particle aggregates at the air-water surface where clusters had interparticle spacings approximately twice the particle diameters. Detergent was also added in later experiments to demonstrate the surface properties of the interface.

The authors used fluorescently-labeled polystyrene microspheres (PS) with diameter 1.06 <math>\mu</math>m which were prepared to be uncharged for the experiment. For depositing monolayers of the particles at the air-water interface, a Langmuir trough was used. The PS were initially deposited in DI water, then the detergent Octylglucoside was added to the water at concentrations between 5 <math>\mu</math> M and 10 mM. This detergent was chosen since it is neutral and can be back-exchanged. The particles were then imaged using a fluorescence microscope and captured on a video camera.

Time series images showing the aggregation of PS are shown in Fig. 2. The particles initially cluster to interparticle distances ~2 <math>\mu</math> m (Fig. 2a, b). This is thought to be due to the attractive interactions due to the nonuniform contact line. Then, when the detergent is added, the clusters of particles dissociate due to changes in the (Fig. 2 b, c). The detergent did not change the air-water surface tension, but did adsorb to the particles with hydrophobic head groups toward the water, which affected the contact angle of the water to the particles. Finally, when the detergent has been purged, the particles re-aggregate and resume a state similar but not identical to the initial one (Fig. 2d).

The attractive interactions and nonuniform contact line was captured in theory based on nonuniform wetting. From the shorter length scales, the authors ignore effects due to gravity and postulate that there is no pressure drop across the water surface. Then, using the Young-Laplace equation: <math> \nabla h(r,\phi) = 0 </math> , the authors expand an expression for the height of water contact line into multipoles in cylindrical coordinates locally centered at each sphere: <math> h(r_c, \phi) = \sum_{2}^{\infty} R_{m,0} r_c^{-m}\Phi_{m,0} \cos(m(\phi-\phi_m,0))</math> where <math> R(m,0)</math> gives the solution in <math> r </math> and <math> \Phi</math> from separation of variables. Both the mono- and dipole terms are zero from the lack of external forces (e.g. gravity) or torques that would rotate spheres from the equilibrium positions on the water surface. Focusing on the dominant quadrupole term which is proportional to <math>r^{-2}</math>, the "self energy" (the difference between the contact area and that from a projection onto the surface times the surface energy) and has typical values of <math> 4 \times 10^{-16}</math> J or <math> \approx 10^5 \text{k T}</math>. Then, with a similar approach, the interaction energy <math>\delta </math> for two particles whose centers are separated by <math>L</math> is given by: <math>\delta E_{AB} = \gamma(\delta S_{AB} - \delta_A - \delta S_{B})</math> where <math>\delta S_{AB}</math> is the surface area surrounding the interacting particles, <math>\delta S_A, \delta S_B</math> is the surface area of the isolated particles. After carefully accounting for the boundary conditions, using deviations of the ideal contact line of <math>50 \text{nm} </math> and the experimental values for the particle size and their observed interparticle spacings, <math> \delta E_{AB}</math> was found to be <math> 5 \times 10^4 k T </math>. The authors describe such solutions much akin to the solutions to electrostatics problems and draw analogies between attraction of particles to areas of high surface curvature to that of interactions of electric multiples to gradients of the electric field. The model described above applies only to large interparticle spacings, which ignored higher multiple terms from the quadrupole. For much shorter distances, higher multipole terms must be taken into account and other effects may influence the behavior of the particles.

## Results and Discussion

Hence, the interactions between two spheres is due to electrostatics and capillarity, depending often on the distance scale. The repulsive dipole-dipole interaction is proportional to <math> L^{-3} </math> where <math> L </math> is the distance between the particles and dominates capillarity at longer distances. This interaction exceeds that due to capillarity at shorter distances, which is attractive and proportional to <math> L^{-4} </math>. Such an inflection gives an activation barrier given in Fig. 3. Anisotropy was also observed with the formation of strings and irregular clusters by the particle aggregates. A simple qualitative explanation was proposed for three particles which favor the formation of strings rather than clusters.

When compared with the results of the experiment, the fundamental assumption the formation of irregular menisci was satisfied. Furthermore, results from the interaction energy suggested that the interaction strength is proportional to <math> R^4 </math> where <math> R </math> is the particle radius. This was observed experimentally where there was less clustering for smaller particles. The activation barrier from the combination of interactions due to electrostatics and capillarity was also observed in the particles that did not aggregate around clusters and remained so. The quadrupolar interaction model also suggested frustration for certain clustering geometries such as hexagonal array, which were indeed rarely observed; as well as the propensity for the formation of linear aggregates.

The authors also explained why the two particle correlation function was not computed for the interaction potentials since the observed disorder was static. They also cautioned that nonuniform wetting does not fully explain all aspects of the interaction although the main features of the experiments were explained with the given model. In conclusion, in this study, the authors proposed a model for the lateral attraction of colloidal particles in the air-water interface based on that of irregular meniscus shapes and the consequent distortion of the liquid surface. A net attraction proportional to the inverse fourth power of the particle pair separation. Experiments were conducted to test the model, which included the addition of detergent to change the surface properties of the water. The proposed model sufficiently accounted for the main experimental features, which included the lack of certain frustrated cluster geometries (e.g. hexagonal planar packing) and the large number of linear clusters. Such results are useful for other systems with particles trapped at various interfaces.

## References

[1] D. Stamou, C. Duschl, and D. Johannsmann. Long-range attraction between colloidal spheres at the air-water interface: The consequence of an irregular meniscus. Phys. Rev. E, 62:5263–5272, Oct 2000.

Entry by: Xingyu Zhang, AP225, Fall 2012