Colloidal crystal: bead-spring lattice immersed in viscous media

We present a report about a new approach that can be used to describe the s ingle-particle dynamics of colloidal crystals. This approach regards the co lloidal crystal as a classical bead-spring lattice immersed in viscous flui d. 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 p article and a cross-correlation function of two particles. As the first-ord er approximation of hydrodynamic interaction, effective Stokes' viscous dra g coefficient gamma (eff) is introduced to the model that includes all of t he hydrodynamic effects due to the presence of all other particles. As a re sult of the viscous media, traveling phonon modes are transformed into rela xation modes, and the motion of a particle is comprehended as a superpositi on of these relaxation modes. The predicted MSD for face-centered-cubic lat tice 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 evalu ated. 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 great er than or equal to3, respectively. These properties of the model are in ag reement with the widely believed notions regarding how the dimension of a s ystem affects the stability of a crystal according to solid state physics, as well as statistical mechanics. The presented model may be utilized to ac count for the elastic properties of colloidal crystals, such as the bulk mo dulus; the single-particle dynamics of colloidal crystals are also accounte d for. The presented model may therefore lead to a better understanding of various macroscopic phenomena in which the corrective motion of particles o r the effects of fluctuations play key roles. (C) 2001 American Institute o f Physics.