Colloidal crystal: bead-spring lattice immersed in viscous media

We present a report about a new approach that can be used to describe the single-particle dynamics of colloidal crystals. This approach regards the colloidal crystal as a classical bead-spring lattice immersed in viscous fluid. In this model, the mean square displacement of a particle (MSD) and the mean product of displacement of a particle and that of another particle (x-MSD) are obtained exactly using the Langevin treatmentlike method. In other words, MSD and x-MSD are, respectively, an autocorrelation function of a particle and a cross-correlation function of two particles. As the first-order approximation of hydrodynamic interaction, effective Stokes' viscous drag coefficient gamma (eff) is introduced to the model that includes all of the hydrodynamic effects due to the presence of all other particles. As a result of the viscous media, traveling phonon modes are transformed into relaxation modes, and the motion of a particle is comprehended as a superposition of these relaxation modes. The predicted MSD for face-centered-cubic lattice type crystals is in good agreement with the MSD observed by Bongers [J. Chem. Phys. 104, 1519 (1996)]. As no experimental study of x-MSD has been published to date, the validity of the predicted x-MSD remains to be evaluated. Moreover, it has been demonstrated that, in the case of d=1, d=2, and d greater than or equal to3 (where d is the dimension of the system), MSD and x-MSD diverge, logarithmically diverge and converge, respectively. The presented results show that bead-spring lattices immersed in viscous media are unstable, quasistable, and stable, in the case of d=1, d=2, and d greater than or equal to3, respectively. These properties of the model are in agreement with the widely believed notions regarding how the dimension of a system affects the stability of a crystal according to solid state physics, as well as statistical mechanics. The presented model may be utilized to account for the elastic properties of colloidal crystals, such as the bulk modulus; the single-particle dynamics of colloidal crystals are also accounted for. The presented model may therefore lead to a better understanding of various macroscopic phenomena in which the corrective motion of particles or the effects of fluctuations play key roles. (C) 2001 American Institute of Physics.