Dynamics in Glassy Systems

The dynamics of super-cooled and glassy liquids has a long and rich history. Perhaps the first suggestion of a connection between the roughness of a potential energy surface and complex relaxation dynamics was made by Goldstein, who distinguished two time scales in glassy systems: that of rapid vibrations about local minima in the potential energy surface and that of less-frequent jumps over significant energy barriers. These two time scales, termed the beta and alpha time scales, respectively, were speculated to be responsible for the two distinct peaks which had been observed in dielectric loss spectra.

Simple Liquids: The dynamics of glass forming simple liquids close to the glass transition is very complicated due to its collective nature. The qualitative picture of the dynamics involves different relaxation mechanisms: For short times a particle is surrounded by an effective cage which keeps the particle close to its original position while allowing efficient exchange of kinetic energy with the surrounding environment through phonon-like modes. At longer times, the cage breaks apart due to collective motions which leads to structural rearrangements and relaxation. From a theoretical perspective, such collective motions promote a feedback in density. Based on these physical ideas, mode-coupling theory (MCT) yields a self-consistent equation for density fluctuations which predicts a rapid slowing down of the fluctuations leading to non-exponential relaxation and eventually structural arrest. A number of experimental probes for examining detailed dynamical features taking place on various length and time scales have emerged over the last few years. These new approaches have the potential to provide extremely useful information on how collective motions of the system are correlated with specific statistical features of the dynamics, such as the distribution of time scales of fluctuations, the length scale and size-distribution of solid-like clusters, and cage structural relaxation rates. In particular, the emergence of multi-dimensional NMR and non-resonant, non-linear Raman techniques has generated renewed interest in the information content of higher-order correlation functions involving time correlations of dynamical quantities at multiple points and time separations. Concurrently, simulation studies probing the microscopic origin of dynamic heterogeneity in dense systems have made use of the increased information content available in multiple-point and multiple-time correlation functions.

In spite of the great interest in higher-order correlation functions, there has been little theoretical work to establish a microscopic theory for general multiple-point and multiple-time correlation functions. To address this need, we have been developing a mode-coupling theory for such quantities using a projection operator formalism to derive coupled generalized Langevin equations (GLE). We have successfully developed the formal machinery necessary to describe higher-order correlation functions in simple liquids. An interesting feature of the theory is that the non-Gaussian nature of the generalized fluctuating forces appearing in the GLE is crucial in the proper description of higher-order correlation functions. Furthermore, it appears that multiple-point and multiple-time correlation functions are related to one another, which implies that the various measures of dynamic heterogeneity are not independent. In a follow up article, we validated the MCT expressions for both multiple-point and multiple-time correlation functions in the hydrodynamic regime for a hard-sphere liquid of moderate density by computer simulation. We have now extended the analysis to describe higher-order correlation functions of tagged particle densities and are in the process of formulating good statistical measures of the string-like motions evident in dense liquids. At the same time, we are working at applying the MCT to explain dynamics on the length scales which are appropriate for molecular cages.

Dynamics at low density (474 K)
Dynamics at high density (1.1 M)

Dense Colloidal Suspensions: Another area of interest is the dynamics of quasi two-dimensional colloidal suspensions. The study of the transport properties of suspensions in restricted geometries is much more complicated than that of three-dimensional systems due to the effect of boundary conditions on the interactions of the suspended particles and the collective motions of the solvent. In spite of the differences in the nature of the short-time dynamics between the colloid and simple liquid systems, the long-time dynamics of glassy colloid systems exhibit many of the qualitative features seen in supercooled liquids, including two-step relaxation regimes. In collaboration with Stuart Rice at the University of Chicago and Andrew Marcus at the University of Oregon, we have utilized digital video microscopy techniques to probe the connection between cooperative motions and long-time relaxation in colloids. Although such cooperative motions have been observed in computer simulations, the association of collective motions of many particles with the alpha relaxation time scale in an experimental system had not be shown until a study of quasi-two-dimensional colloidal liquids. In this work, we demonstrated that the time scale of string-like cooperative motions leading to the decay of fluctuations of locally-enhanced density is precisely the same as the time scale of non-Gaussian behavior observed in the van Hove self-correlation function of the density.

Experimentally, it is well-known that concentrated colloidal suspensions exhibit a transition from a fluid phase to a glass phase similar to that in super-cooled liquids. Based on the success of mode coupling theory (MCT) in predicting the scaling behavior of the beta and alpha relaxation time scales in super-cooled liquids, various researchers have attempted to apply MCT to concentrated colloidal suspensions. One fundamental assumption in these theories is that the hydrodynamic interactions of the solvent effectively alter the dynamics in a mean-field fashion in which the colloid particles diffuse between collisions with a diffusion constant which depends on the total density. Recently Tokuyama has questioned this picture, suggesting that in fact experimental measurements are carried out in quenched meta-stable fluid states in which non-equilibrium effects can change the behavior of the relaxation process. In a series of papers, Tokuyama and co-workers proposed a qualitatively different mechanism for the critical slowing down in dense suspensions in which long-range hydrodynamic interactions are prominent. Direct simulation of colloidal systems to establish which qualitative picture is correct is difficult due to the enormous separation of time scales between motions of the solvent and motions of the massive colloidal particles. We are currently in the process of examining the importance of hydrodynamic flow in the kinetic mechanism for cooperative flow in dense suspensions using mesoscale simulation methods developed by Malevanets and Kapral. In this approach, a hybrid MD algorithm is utilized where the solvent dynamics is modeled by coarse graining the solvent into cells in which collision rules are designed to yield the correct hydrodynamic flow. Some of the questions which will be addressed in this work include: What is the correlation between the hydrodynamic flow of the solvent and the cooperative string-like motions of the colloidal particles? How does this correlation change with the relative masses and sizes of the solvent and colloidal particles? Do the short-range hydrodynamic interactions screen the direct interactions between colloid particles? If so, what determines the screening length?