Fluctuating hydrodynamic models for supercooled liquids and development of long relaxation times

Linking density functional and mode coupling models for supercooled liquids

We compare predictions from two familiar models of the metastable supercooled liquid, respectively, constructed with thermodynamic and dynamic approaches. In the so called density functional theory the free energy F[ρ] of the liquid is a functional of the inhomogeneous density ρ(r). The metastable state is identified as a local minimum of F[ρ]. The sharp density profile characterizing ρ(r) is identified as a single particle oscillator, whose frequency is obtained from the parameters of the optimum density function. On the other hand, a dynamic approach to supercooled liquids is taken in the mode coupling theory (MCT) which predict a sharp ergodicity-non-ergodicity transition at a critical density. The single particle dynamics in the non-ergodic state, treated approximately, represents a propagating mode whose characteristic frequency is computed from the corresponding memory function of the MCT. The mass localization parameters in the above two models (treated in their simplest forms) are obtained, respectively, in terms of the corresponding natural frequencies depicted and are shown to have comparable magnitudes.

Renormalized dynamics of the Dean-Kawasaki model

We study the model of a supercooled liquid for which the equation of motion for the coarse-grained density ρ(x,t) is the nonlinear diffusion equation originally proposed by Dean and Kawasaki, respectively, for Brownian and Newtonian dynamics of fluid particles. Using a Martin-Siggia-Rose (MSR) field theory we study the renormalization of the dynamics in a self-consistent form in terms of the so-called self-energy matrix Σ. The appropriate model for the renormalized dynamics involves an extended set of field variables {ρ,θ}, linked through a nonlinear constraint. The latter incorporates, in a nonperturbative manner, the effects of an infinite number of density nonlinearities in the dynamics. We show that the contributing element of Σ which renormalizes the bare diffusion constant D(0) to D(R) is same as that proposed by Kawasaki and Miyazima [Z. Phys. B Condens. Matter 103, 423 (1997)]. D(R) sharply decreases with increasing density. We consider the likelihood of a ergodic-nonergodic (ENE) transition in the model beyond a critical point. The transition is characterized by the long-time limit of the density correlation freezing at a nonzero value. From our analysis we identify an element of Σ which arises from the above-mentioned nonlinear constraint and is key to the viability of the ENE transition. If this self-energy would be zero, then the model supports a sharp ENE transition with D(R)=0 as predicted by Kawasaki and Miyazima. With the full model having nonzero value for this self-energy, the density autocorrelation function decays to zero in the long-time limit. Hence the ENE transition is not supported in the model.

Excess entropy scaling for the segmental and global dynamics of polyethylene melts

The range of validity of the Rosenfeld and Dzugutov excess entropy scaling laws is analyzed for unentangled linear polyethylene chains.

Time-dependent correlations in a supercooled liquid from nonlinear fluctuating hydrodynamics

We solve numerically the equations of nonlinear fluctuating hydrodynamics (NFH). A coarse graining of the density field is applied at each time step to avoid instabilities which otherwise plague the algorithm at long times. The equilibrium correlation of the density fluctuations at different times obtained directly from the solutions of the NFH equations are shown here to be in quantitative agreement with corresponding molecular dynamics simulation data. Low-order perturbative treatment of the these NFH equations obtains the mode coupling model. The latter has been widely studied for understanding the slow dynamics characteristic of the supercooled state. A crucial aspect of this theory is a rounded version of a possible ergodic-nonergodic transition in the supercooled liquid at a temperature T(c) between melting point T(m) and the glass transition temperature T(g). In the present work we demonstrate numerically the role of strongly coupled density fluctuations in giving rise to slow dynamics and how the 1/ρ nonlinearity in the NFH equations of motion is essential in restoring the ergodic behavior in the liquid. The relaxation data indicate that at moderate supercooling near T(c), the time temperature superposition holds. The relaxation gets increasingly stretched with increased supercooling. The relaxation time τ shows an initial power-law divergence approaching a transition temperature T(c) generally identified as the mode coupling temperature. From the direct solutions we obtain a value for T(c) much lower than that typically estimated from solution of low-order integral equations of mode coupling theory. This is in agreement with the trend seen in computer simulations.

Do current-density nonlinearities cut off the glass transition?

Extended mode-coupling theories for dense fluids predict that nonlinear current-density couplings cut off the singular "ideal glass transition" present in the standard mode-coupling theory where such couplings are ignored. We suggest here that, rather than allowing for activated processes as sometimes supposed, contributions from current-density couplings are always negligible close to a glass transition. We discuss in schematic terms how activated processes can nonetheless cut off the transition by causing the memory function to become linear in correlators at late times.

Nature of the entropy versus self-diffusivity plot for simple liquids

The empirical relation (D(*))(alpha) = a exp[S] between the self-diffusion coefficient D(*) and the excess entropy S of a liquid is studied here in the context of theoretical model calculation. The coefficient alpha is dependent on the interaction potential and shows a crossover at an intermediate density, where cooperative dynamics become more important. Around this density a departure from the Stokes-Einstein relation is also observed. The above relation between entropy and diffusion is also tested for the scaled total diffusion coefficient in a binary mixture.

Anomalous stretching in a simple glass-forming liquid

The frequency dependent shear modulus G(omega) for a simple liquid shows strongly stretched behavior and the stretching exponent increases with decrease of temperature. This unconventional behavior was reported earlier in Phys. Rev. Lett. 73, 963 (1994) from experiments on simple liquids. We demonstrate here that this is a feature of the characteristic two-step relaxation process of the self-consistent mode coupling theory of supercooled liquids.

Elastic behavior in a supercooled liquid: analysis of viscoelasticity using an extended mode coupling model

The transverse current correlations are analyzed using the formalism of extended mode coupling theory. The undercooled liquid can sustain shear waves up to a minimum wave number. With the increase of density this wave number decreases, indicating a growing length scale that is related to the dynamics. The speed of the propagating shear waves goes to zero approaching a critical wave number. The maximum wavelength shows an initial enhancement approaching the mode coupling transition and finally grows at a slower rate as the sharp transition is cut off.