Formation of two-dimensional colloidal voids, soap froths, and clusters Jaime Ruiz-Garcı ´ a, 1 Rogelio Ga ´ mez-Corrales, 1 and Boris I. Ivlev 1,2 1 Instituto de Fı ´sica, Universidad Auto´noma de San Luis Potosı ´, Alvaro Obrego´n 64, 78000 San Luis Potosı ´, Mexico 2 Landau Institute for Theoretical Physics, Kosygin Street 2, Moscow 117940, Russia ~Received 8 September 1997! We report the observation of new pattern formation by spherical polystyrene particles trapped at the air- water interface; namely, the formation of two-dimensional void, soap-froth, and cluster structures. The forma- tion of the soap-froth structure depends upon the initial surface concentration of particles. The void and soap-froth structures evolve with time. The clusters can be formed after deposition of the sample or as a result of the evolution of the soap-froth structure. The experimental observation can be explained in terms of a balance between electrostatic repulsive and attractive interactions. An optimum cluster size can be obtained from an energy analysis of the system. @S1063-651X~98!05605-0# PACS number~s!: 82.70.Dd, 83.70.Hq, 68.90.1g During the past decade, colloidal systems have been used as model systems to try to understand phenomena that occur at the atomic level. For example, quasi-two-dimensional and two-dimensional studies have been performed and have given some light to our understanding of the two- dimensional ~2D! melting transition @1#. Pieranski @2# was the first to show that charge-stabilized colloidal particles can be trapped at the air-water interface. Using 0.2 m m particles, he observed crystal-like ordering, which he attributed to a stabilization due to dipole-dipole repulsive interactions. Armstrong et al. @3# performed expansion-compression ex- periments in a Langmuir trough with latex particles, 1.01 and 2.88 m m in diameter, to study two-dimensional melting. They found evidence of the appearance of a hexatic phase. Onoda @4# used steric stabilized colloidal particles at the air- water interface, with particle diameters ranging from 1 to 15 m m. Depending on the size of the particle, he observed the formation of both reversible and irreversible clustering, which was attributed to the formation of a secondary mini- mum in the potential, due to a combination of short-range electrostatic repulsive and long-range van der Waals attrac- tive interactions. Fractal clustering has also been observed with silica particles at the air-water interface @5#, where bonding of the particles was also attributed to van der Waals interactions. In this paper we report the formation of 2D void, soap- froth, and cluster structures by charge-stabilized spherical colloidal particles at the air-water interface. Similar struc- tures appear in various systems in nature and have been of long standing interest, from experiments @6–8# to theory @9– 11# and computer simulations @12#. We studied monodis- perse fluorescent @13,14# particles with different diameters: 0.5, 1.01, 1.78, and 2.26 m m. The experiments were per- formed in a Teflon Langmuir trough and the particles were observed with a fluorescence microscope. Careful cleaning of the samples was found to be very important; we found reproducible results after 8–10 sonication-centrifugation cycles with methanol, which was used as solvent @2,3#. Re- cently, we have observed the same pattern formation with bioclean particles @14#, which have the fluorescent dye chemically attached to the particle. We carefully deposited an aliquot of the colloidal solution, drop by drop, on a clean water surface ~bioresearch grade, 18.3 MV cm of resistivity! with a 25 m l syringe. After deposition of the samples, we observed that at high overall surface density the particles are arranged in a solid- like hexagonal ~hexatic! structure @13#, as shown in Fig. 1~a!. However, at lower total surface density of particles, different structures were observed depending on the local density. Re- gions with relatively high local density developed ‘‘vacan- cies’’ that grew until they formed circular voids ~bubble cells! of different sizes, as shown in Fig. 1~b!. When bubbles became too large they deformed each other, forming the characteristic 2D soap-froth structure, as shown in Figs. 1~c! and 1~d!. The soap-froth structure was formed only with par- ticles of 1.01, 1.78, and 2.26 m m in diameter. The 0.5 m m particles did not form the soap-froth structure and only formed small short-lived and not well defined voids. The soap-froth structure evolved with the well known @15# T 1 or neighbor-switching and T 2 or face-disappearing mechanisms. In addition, we observed that a large amount of single particle rearrangement is also responsible for the evo- lution of the structure. At the beginning, many cell walls have more than one row of particles. As the foam structure evolves, the particles move toward Plateau borders, thus most of the inner cell walls thin down to one row of particles ~colloidal chain!, as seen in Figs. 1~c! and 1~d!. The forma- tion and stabilization of a colloidal chain is remarkable, since the pair interactions between particles are isotropic. Forma- tion of extended structures has been observed @6,7# and pre- dicted @16# in dipolar systems of Langmuir monolayers, as a result of a balance between dipole-dipole interactions, which favor the formation of extended structures, and the line ten- sion energy, which tries to minimize boundaries @17#. In the colloidal soap froth, the discreteness of the system may also allow the formation of metastable colloidal chains. The evolution mechanisms and the particle rearrange- ments serve also to drain the particles toward the edge of the soap-froth structure: Plateau borders and walls at the edge get thicker. However, this thickening is not uniform, and some walls at the edge or parts of them remain formed by only a chain of particles. As the soap froth continues evolv- ing, thin walls at the edge of the structure can break, allow- PHYSICAL REVIEW E JULY 1998 VOLUME 58, NUMBER 1 PRE 58 1063-651X/98/58~1!/660~4!/$15.00 660 © 1998 The American Physical Society