Cross-correlation functions in two-dimensional and three-dimensional colloidal crystals

Abstract

We have measured the mean square displacement of a particle (MSD) and mean product of displacement of a particle and that of another particle (x-MSD) in two-dimensional (2d) and three-dimensional (3d) colloidal crystals for the first time using digital video microscopy. These (x-)MSDs have been compared to an overdamped bead-spring lattice model with effective viscous drag γeff (the OBS-γeff model). The observed 3d system contained ordered structures created by sedimentation equilibrium adjacent to the internal cell wall and was regarded as the (1 1 1) surface of a face-centered cubic lattice. The observed 2d system was a single layer of hexagonal-like ordered structures which were generated in a certain region of the space between the flat surface of the internal cell wall and a convex lens. In the observed time region, the MSDs for 2d system and 3d system were in good agreement with the theoretical MSDs for 2d system and 3d system, which were predicted to logarithmically diverge and to converge, respectively. The observed x-MSDs for the 2d system were in disagreement with theoretical ones in short time behavior. For the 3d system, the disagreement between the theory and the experimental results were found in the amplitudes as well as in the short time behavior. It was assumed that these differences were caused by the hydrodynamic coupling which was not fully incorporated in the OBS-γeff, and an improved treatment, the OBS-cutoff model was introduced. The OBS-γeff elongates the relaxation time of each mode with the constant ratio of γeff/γ, while the OBS-cutoff assumes the relaxation time of the modes to be infinity if the wave number of a mode is larger than a certain cutoff wave number. For the 3d system, the MSD and x-MSD for nearest neighbors obtained from the OBS-cutoff were in excellent agreement with the observed ones. In addition, it was found that the hydrodynamic interaction was the dominant factor on the short time behavior of (x-)MSD.

Hydrodynamic Effects of Particle Motions in Colloidal Crystal

“…The bottom layer of the crystal nearest to the coverslip could be observed through the coverslip by an inverted optical microscope at room temperature. We have already shown that wall effects can be neglected concerning the motion parallel to the coverslip [5].…”

“…The digitized images were stored in the PC's hard disk. The particle recognition, the tracing of the particles, and the calculation of MSDs and x-MSDs were done on the PC [5].…”

“…In recent years, a single particle motion has been analyzed in real space by digital video microscopy [1,2,3]. To explain the dynamics of colloidal crystal, we have developed the overdamped bead-spring lattice (OBS) model [4,5].…”

Phonon dispersion curves of two-dimensional colloidal crystals: the wavelength-dependence of friction

“…Of central importance to the present paper is the work of Hurd et al [2] who starting from [1] not only present a theory of hydrodynamic interaction in colloidal crystals but also performed photoncorrelation spectroscopy measurements to study the dispersion of lattice waves. Early studies of single-particle dynamics within colloidal crystals employed scattering techniques [10,11], while the first video-microscopy experiments relevant for our question were presented by Bongers and Versmold [12] who measured the particle mean square displacement as a function of time, later theoretically analysed within the framework of the Langevin model in [13,14].…”

Dynamics of Thin-Layered Colloidal Crystals from Single Layer to Multi Layers; Crossover Behavior from 2D to 3D

“…In recent years, a single particle motion has been analyzed in real space by digital video microscopy. To explain the dynamics of colloidal crystal, we have developed the overdamped bead-spring lattice (OBS) model [1,2].…”