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

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Existing models do not explain the observed attractive interactions of particles at distances ~ <math>\mu</math> m. | Existing models do not explain the observed attractive interactions of particles at distances ~ <math>\mu</math> m. | ||

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. | 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 strong enough to explain the results of the experiment, they're likely not present since there is likely no | + | Additionally, although immersion capillary forces are strong enough to explain the results of the experiment, they're likely not present since there is likely no 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. | 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:Stamou1.jpeg|thumb|300px| Fig. 1: The water surface around two interacting particles is not uniform, given rise to certain favored orientations (top) which reduce the slope of the water level in between the particles, unlike the one in the bottom figure, from [1].]] | + | [[Image:Stamou1.jpeg|thumb|300px|left| Fig. 1: The water surface around two interacting particles is not uniform, given rise to certain favored orientations (top) which reduce the slope of the water level in between the particles, unlike the one in the bottom figure, 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. | ||

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Time series images showing the aggregation of PS are shown in Fig. 2. | 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 | + | [[Image:Stamou2.jpeg|thumb|300px|right| Fig. 2: Fluorescence images for the particles showing initial aggregation (a(i),(ii)), then reshaping when 40 <math>\mu</math>M of detergent is added (b). At (c), 70 <math>\mu</math>M of solution is added, the average particle distance increases to ~10 <math>\mu</math>m. When the detergent is purged, the particles re-aggregate similar to (a). From [1].]] |

+ | The particles initially cluster to interparticle distances ~2 <math>\mu</math> m (Fig. 2a). | ||

This is thought to be due to the attractive interactions due to the nonuniform contact line. | This is thought to be due to the attractive interactions due to the nonuniform contact line. | ||

− | Then, | + | Then, with the addition of the detergent, the particle aggregates break up (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. | 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 | + | Finally, when the detergent is 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. | 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. | 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> | 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. | + | where <math> R(m,0)</math> gives the solution in <math> r </math> and <math> \Phi</math> in <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. | 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) | + | 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) was found to have typical values of <math> 4 \times 10^{-16}</math> J or <math> \approx 10^5 k T </math>. |

− | + | Similarly, the interaction energy <math>\delta </math> for two particles whose centers are separated by L is given by: | |

− | <math>\delta E_{AB} = \gamma(\delta S_{AB} - \ | + | <math>\delta E_{AB} = \gamma(\delta S_{AB} - \delta S_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. | 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>. | 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 authors describe such solutions as much akin to those of electrostatics problems and found analogies between the attraction of particles to areas of high surface curvature to those of interactions of electric multipoles to gradients of the electric field. |

− | + | However, this model applies only to large interparticle spacings, and ignores 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. | 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== | ==Results and Discussion== | ||

− | + | The interactions between two spheres in this experimental system 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. | 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. | ||

− | + | Interactions due to capillarity, which is attractive and proportional to <math> L^{-4} </math>, exceeds that of the dipole-dipole interaction at shorter distances. | |

− | + | This combination of interactions then give an activation barrier given in Fig. 3. | |

Anisotropy was also observed with the formation of strings and irregular clusters by the particle aggregates. | 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. | A simple qualitative explanation was proposed for three particles which favor the formation of strings rather than clusters. | ||

+ | [[Image:Stamou3.jpeg|thumb|300px|right| Fig. 3: The total potential with contributions from attraction due to capillarity which is proportional to <math> L^{-4} </math> and repulsion due to dipole-dipole interactions which is proportional to <math>L^{-3}</math> has an activation barrier. From [1].]] | ||

− | + | The results from the experiment showed that the fundamental assumption of the irregular meniscus is satisfied. | |

− | Furthermore, | + | Furthermore, the theory suggests that interaction strength is proportional to <math> R^4 </math> where <math> R </math> is the particle radius. |

− | This was | + | This was corroborated experimentally where less clustering occurred 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 | + | 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 unclustered. |

− | The quadrupolar interaction model also suggested frustration | + | The quadrupolar interaction model also suggested frustration of certain clustering geometries (e.g. hexagonal array) and the propensity for the formation of linear aggregates, which were observed experimentally. |

− | + | Although the main features of the experiments were explained with the given model, the authors do caution that nonuniform wetting does not fully explain all aspects of the interaction. | |

− | + | Nonetheless, these results are useful for other systems which involve colloidal particles trapped at various interfaces. | |

− | + | ||

− | + | ||

− | + | ||

− | + | ||

− | + | ||

==References== | ==References== |

## Revision as of 21:26, 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-range attraction using a model based on nonuniform wetting. 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, in the manufacturing of emulsions and foams. Existing models do not explain the observed attractive interactions of particles at distances ~ <math>\mu</math> m. 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 strong enough to explain the results of the experiment, they're likely not present since there is likely no 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). This is thought to be due to the attractive interactions due to the nonuniform contact line. Then, with the addition of the detergent, the particle aggregates break up (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 is 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> in <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) was found to have typical values of <math> 4 \times 10^{-16}</math> J or <math> \approx 10^5 k T </math>. Similarly, the interaction energy <math>\delta </math> for two particles whose centers are separated by L is given by: <math>\delta E_{AB} = \gamma(\delta S_{AB} - \delta S_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 as much akin to those of electrostatics problems and found analogies between the attraction of particles to areas of high surface curvature to those of interactions of electric multipoles to gradients of the electric field. However, this model applies only to large interparticle spacings, and ignores 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

The interactions between two spheres in this experimental system 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. Interactions due to capillarity, which is attractive and proportional to <math> L^{-4} </math>, exceeds that of the dipole-dipole interaction at shorter distances. This combination of interactions then give 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.

The results from the experiment showed that the fundamental assumption of the irregular meniscus is satisfied. Furthermore, the theory suggests that interaction strength is proportional to <math> R^4 </math> where <math> R </math> is the particle radius. This was corroborated experimentally where less clustering occurred 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 unclustered. The quadrupolar interaction model also suggested frustration of certain clustering geometries (e.g. hexagonal array) and the propensity for the formation of linear aggregates, which were observed experimentally. Although the main features of the experiments were explained with the given model, the authors do caution that nonuniform wetting does not fully explain all aspects of the interaction. Nonetheless, these results are useful for other systems which involve colloidal 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