Qualitative features at the glass crossover

Abstract

We discuss some generic features of the dynamics of glass-forming liquids close to the glass transition singularity of the idealized mode-coupling theory (MCT). The analysis is based on a recent model by one of the authors for the intermediate-time dynamics (β relaxation), derived by applying dynamical field-theory techniques to the idealized MCT. Combined with the assumption of timetemperature superposition for the slow structural (α) relaxation, the model naturally explains three prominent features of the dynamical crossover: the change from a power-law to exponential increase in the structural relaxation time, the replacement of the Stokes-Einstein relation between diffusion and viscosity by a fractional law, and two distinct growth regimes of the thermal susceptibility that has been associated to dynamical heterogeneities.PACS numbers: 64.70.QTo describe how classical liquids arrest kinetically at the glass transition, is still a controversial issue within statistical physics. Starting from the high-temperature liquid state, it is natural to focus on the dramatic slowing down in the dynamics. Here the mode-coupling theory of the glass transition (MCT) is successful [1,2]. It predicts the divergence of structural relaxation times at an ideal glass transition temperature T c . Although the singularity is "avoided" in real glass formers, it is still signaled by asymptotic, so-called β-relaxation scaling laws for the dynamics at T ≈ T c [3], or a square-root singularity in the scattering intensities [4][5][6][7][8].On the low-temperature side, the replica method describes the properties of the (metastable) glassy states below T c [9, 10] yielding fewer but similar quantitative predictions [11,12]. Random-first-order transition (RFOT) theory [13,14] builds on top of replica results by advocating entropic nucleation processes to restore ergodicity below T c and predicts a debated divergence of a correlation length below the calorimetric glass transition.Several attempts have been made at "extended MCT" to incorporate such physics below T c : introducing further relaxation channels to the MCT equations [15][16][17][18][19][20][21], taking into account higher-order factorization of manyparticle density modes [22,23], considering generalizedhydrodynamic arguments [24], leading to time-dependent coupling coefficients [25], using ideas of RFOT theory [26], or within "naïve" MCT [27][28][29]. By and large, all remained rather empirical.One of us [30] has developed a new approach to ideal MCT based on the fact that close to T c , fluctuations will become important. The study of these fluctuations in a full-fledged dynamical context provides a crucial improvement on earlier static treatments [31]. After a complex field theoretical computation the original problem is mapped onto a rather intuitive model that extends the β-scaling laws (rather than full MCT, due to saddle-point approximations in its derivation) to a spatially inhomogeneous case where the distance to T c becomes a spatially fluctuating variable. The correspo...

Dynamical Landau theory of the glass crossover

“…As usual the coefficients of the various powers of 1/M are divergent as τ → 0 and we have seen that, at any given order 1/M , the leading divergent term is the same that is generated within a similar loop expansion of SBR. Thus I have provided the proof of the result that was anticipated in [1] and further discussed in [6]. Note that in that paper only the average two-point function was discussed while we have shown here that SBR provides a description of higher-order correlations as well, e.g.…”

“…In the present case, however, the transition is avoided and we are more in the situation that occurs below the lower critical dimension, therefore the mapping between the GCT and SBR is only approximate. Furthermore, the avoided nature of the transition within SBR, as discussed in [1,6,7], is clearly due to non-perturbative effects. Therefore one would like to be sure that the same non-perturbative effects are at work in the GCT, thus establishing a connection between the two theories beyond perturbation theory.…”

“…The behavior at fixed N and increasingτ can be discussed in the context of SBR see [1,6,7] and displays a satisfactory power-law-to-exponential crossover.…”

Out-of-equilibrium dynamical equations of infinite-dimensional particle systems I. The isotropic case

“…(28), and the self-consistent equations for the three kernels in Eq. (29). A complex many-body problem with N → ∞ particles has thus been reduced to the self-consistent solution of a single d-dimensional stochastic differential equation.…”

Heterogeneous diffusion, viscosity, and the Stokes-Einstein relation in binary liquids

“…Surprisingly this transition sets in at a temperature far above the MCT critical temperature, even above the liquidus temperature. Including spatially heterogeneous relaxations in MCT again a fractional SER is found [43].…”