Mode-coupling approach for the slow dynamics of a liquid on a spherical substrate

Strain correlation functions in isotropic elastic bodies: large wavelength limit for two-dimensional systems

Strain correlation functions in two-dimensional isotropic elastic bodies are shown both theoretically (using the general structure of isotropic tensor fields) and numerically (using a glass-forming model system) to depend on the coordinates of the field variable (position vector r in real space or wavevector q in reciprocal space) and thus on the direction of the field vector and the orientation of the coordinate system. Since the fluctuations of the longitudinal and transverse components of the strain field in reciprocal space are known in the long-wavelength limit from the equipartition theorem, all components of the correlation function tensor field are imposed and no additional physical assumptions are needed. An observed dependence on the field vector direction thus cannot be used as an indication for anisotropy or for a plastic rearrangement. This dependence is different for the associated strain response field containing also information on the localized stress perturbation.

Correlations of tensor field components in isotropic systems with an application to stress correlations in elastic bodies

Correlation functions of components of second-order tensor fields in isotropic systems can be reduced to an isotropic fourth-order tensor field characterized by a few invariant correlation functions (ICFs). It is emphasized that components of this field depend in general on the coordinates of the field vector variable and thus on the orientation of the coordinate system. These angular dependencies are distinct from those of ordinary anisotropic systems. As a simple example of the procedure to obtain the ICFs we discuss correlations of time-averaged stresses in isotropic glasses where only one ICF in reciprocal space becomes a finite constant e for large sampling times and small wave vectors. It is shown that e is set by the typical size of the frozen-in stress components normal to the wave vectors, i.e., it is caused by the symmetry breaking of the stress for each independent configuration. Using the presented general mathematical formalism for isotropic tensor fields this finding explains in turn the observed long-range stress correlations in real space. Under additional but rather general assumptions e is shown to be given by a thermodynamic quantity, the equilibrium Young modulus E. We thus relate for certain isotropic amorphous bodies the existence of finite Young or shear moduli to the symmetry breaking of a stress component in reciprocal space.

Cooperatively rearranging regions change shape near the mode-coupling crossover for colloidal liquids on a sphere

The structure and dynamics of liquids on curved surfaces are often studied through the lens of frustration-based approaches to the glass transition. Competing glass transition theories, however, remain largely untested on such surfaces and moreover, studies hitherto have been entirely theoretical/numerical. Here we carry out single particle-resolved imaging of dynamics of bi-disperse colloidal liquids confined to the surface of a sphere. We find that mode-coupling theory well captures the slowing down of dynamics in the moderate to deeply supercooled regime. Strikingly, the morphology of cooperatively rearranging regions changed from string-like to compact near the mode-coupling crossover—a prediction unique to the random first-order theory of glasses. Further, we find that in the limit of strong curvature, Mermin–Wagner long-wavelength fluctuations are irrelevant and liquids on a sphere behave like three-dimensional liquids. A comparative evaluation of competing mechanisms is thus an essential step towards uncovering the true nature of the glass transition.

The static and dynamic behavior of condensed phases residing on curved surfaces can be fundamentally different from their counterparts in Euclidean space. Singh et al. test several competing glass theories on colloidal liquids confined to the surface of a sphere and show they behave like 3D bulk liquids.

Mode-coupling theory of the glass transition for colloidal liquids in slit geometry

We provide a detailed derivation of the mode-coupling equations for a colloidal liquid confined by two parallel smooth walls. We introduce irreducible memory kernels for the different relaxation channels thereby extending the projection operator technique to colloidal liquids in slit geometry. Investigating both the collective dynamics as well as the tagged-particle motion, we prove that the mode-coupling functional assumes the same form as in the Newtonian case corroborating the universality of the glass-transition singularity with respect to the microscopic dynamics.

Glassy dynamics of dense particle assemblies on a spherical substrate

We study by molecular dynamics simulation a dense one-component system of particles confined on a spherical substrate. We more specifically investigate the evolution of the structural and dynamical properties of the system when changing the control parameters, the temperature and the curvature of the substrate. We find that the dynamics become glassy at low temperature, with a strong slowdown of the relaxation and the emergence of dynamical heterogeneity. The prevalent local 6-fold order is frustrated by curvature and we analyze in detail the role of the topological defects in the statics and the dynamics of the particle assembly.