Dynamical Arrest of Glass Formers

On cooling, the viscosity of many super cooled liquids grows by orders of magnitude. Eventually, at low enough temperature, the material becomes so viscous it appears to be a solid, though with no crystalline structure. The material thus formed is called a glass. Except for molecular vibrations over lengths smaller than atomic diameters, the dynamics of the system is arrested. One of the longstanding problems in condensed matter physics is the development of a proper theoretical description of this arrest, and solving this problem is one of Professor Chandler's current interests. The work he is doing was begun and continues in collaboration with Nottingham physicist Juan P. Garrahan.

Figure 1. Comparison of the Garrahan-Chandler theory (solid lines) with experiment for relaxation times and viscosity as functions of temperature for various glass forming liquids [4]. GeO2 is a so-called "strong" glass former as it obeys Arrhenius temperature dependence. 3BP (i.e., 3-bromopentane) and OTP (i.e., ortho-terphenyl) are so-called "fragile" glass formers as they exhibit super-Arrhenius temperature dependence over the range of experimentally accessible conditions. Garrahan and Chandler have shown that in general, there is a crossover behavior between fragile and strong behaviors. APOC (alpha-phenyl-ortho-cresol) and SL (salol) exhibit this crossover, as made explicit by the blue and red dashed lines showing fragile and strong behaviors, respectively. This figure was taken from Ref. 4.

When a glass former approaches dynamical arrest, the time scale for its relaxation to equilibrium becomes large, so large that it is longer than an experimentalist will care to measure. The experimentalist will not wait long enough to observe equilibrium properties, but rather will measure long-lived non-equilibrium properties. Under this circumstance it is said that the system has "fallen out of equilibrium." Whether or not a system has fallen out of equilibrium is therefore an issue of experimental technique, and as such, there can be no unambiguous point at which a system changes from a super cooled fluid to a glass. The ambiguity implies that the formation of glass does not involve a thermodynamic phase transition. Nevertheless, the onset of glassy behavior is precipitous: relatively modest changes in temperature are associated with vast changes in viscosity and relaxation time. Thus, it is natural to believe that something collective is occurring in a glass former. But since it cannot be a transition between different weights of ensembles of states (the essence of a thermodynamic transition), it is natural to look instead for a transition between different weights of ensembles of trajectories. This idea is central to the development pioneered by Garrahan and Chandler.

Growing relaxation times or slowing down is generally associated with growing length scales. In the case of second order phase transitions, for instance, slowing down is associated with diverging equilibrium correlation lengths. For glass formers, however, experiments reveal no large equilibrium correlation length. Rather, the growing length scales of a glass former are manifested in the phenomenon called dynamic heterogeneity. Namely, while maintaining an equilibrium distribution of states, glass formers exhibit segregation of mobile atoms from immobile atoms. It is a purely dynamical phenomenon since the distinction between mobile and immobile particles depends upon observation time. Any atom viewed for only a very short time will appear immobile, and any atom viewed for a long enough time will appear mobile. Obviously, to view dynamic heterogeneity, one must view trajectories, and to characterize dynamic heterogeneity quantitatively, one must consider ensembles of trajectories. Consideration of ensembles of trajectories - the statistical mechanics of trajectories - is the principal perspective of transition path sampling. Indeed, transition path sampling has provided important numerical tools

From the perspective of trajectory space, dynamic heterogeneity can be understood as a direct manifestation of facilitation [1]. This terminology refers to kinetic constraints of the sort identified long ago by Glenn Fredrickson and Hans Andersen [2]. In particular, in a system where mobility is sparse, an immobile microscopic region is most likely surrounded by other immobile or jammed regions. An immobile region can become mobile only in situations where its neighboring region is already mobile or unjammed. Mobility in one region can thus facilitate mobility in an adjacent region. Under these circumstances, Garrahan and Chandler pointed out [1], trajectories necessarily form connected chains of mobile regions in space-time, and fluctuations of these chains is the origin of dynamic heterogeneity. Further, when facilitation has a directional preference, as in the so-called "East model" (see Figure 2), or further contraints of facilitation, the resulting heterogeneity is hierarchical. This behavior coincides with that imagined and described by Richard Palmer, Daniel Stein, Elihu Abrahams and Philip Anderson [3] in a Physical Review Letter appearing in the same volume as Fredrickson and Andersen's. Hierarchical heterogeneity leads to non-Arrhenius relaxation times and non-exponential relaxation functions.

Figure 2. Trajectories of a non-interacting line of spins (left), of a non-interacting line of spins on a lattice with isotropic dynamic facilitation (middle - the Fredrickson-Andersen model), and with anisotropic dynamic facilitation (right - the East model). A black spot in the pictures indicates a space-time point where a spin is up (i.e., excited). Regions that are white are those where spins are down (i.e., unexcited). Figure taken from Ref. 1.

The pictures in Fig. 2 are suggestive of phase separation – an order-disorder phenomenon in space-time, where regions of low dynamical activity are separated from regions of high dynamical activity. Garrahan, Chandler and their students have been able to formalize this idea with a combination of analysis and transition path sampling [5, 6].

Garrahan and Chandler have predicted the existence of a general strong-to-fragile glass transition on the basis of fragility, i.e., super-Arrhenius temperature dependence, being the result of extra constraints of facilitation, At low enough temperatures, persistence of hierarchical dynamics must produce a relaxation time τ_fragile that is longer than τ_strong/ƒ, where ƒ is the probability of avoiding constraints responsible for hierarchical dynamics. That is to say, hierarchical dynamics will lead to longer relaxation times than those perscribed by the free energy barriers that enforce contraints on molecular motions. As temperature is lowered, therefore, Garrhan and Chandler noted that there must be a crossover from hierarchical dynamics to diffusive dynamics at temperatures near where τ_strong/ƒ =τ_fragile [4]. Depending upon the system, the crossover can be outside the range of currently accessible experimental conditions, but it is nevertheless a universal feature of corresponding states in glass forming materials. Recently, a strong-to-fragile crossover has been observed for thin films of super-cooled water [14].

The Garrahan-Chandler approach to glass formation has many implications, some of which have been examined in recent papers by the Chandler group [7] – [13], others by the Garrahan group, and still others where future research is planned. One topic for the future is to understand the detailed molecular mechanism(s) through which facilitation is born. It is not a feature of dynamics in normal liquids. Learning precisely how and when facilitation sets in upon super cooling will likely have interesting and perhaps practical consequences. Another is to perform the range of experiments required to fully test the predictions of Garrahan and Chandler's theory. Appropriate techniques must reveal non-trivial time and length scaling, not simply a single time or a single length. As Garrahan and Chandler have argued, it is this scaling that is central to understanding glass formation. A full understanding of dynamic heterogeneity in glass formers will likely have consequences in other scientific disciplines. Signal transduction in biology is one possibility as it too can be viewed in terms of sparse assemblies of facilitating units or binary switches.

The Physics of glassy systems can be illustrated with movies of trajectories. Prof. Chandler's former students Albert Pan and Lutz Maibaum have prepared websites that present these movies. Click here to see Albert Pan's presentations. Click here to see Lutz Maibaum's.

References

[1] Garrahan, J. P. and D. Chandler, "Geometrical explanation and scaling of dynamical heterogeneities in glass forming systems," Phys. Rev. Lett. 89, 035704.1-035704.4 (2002).[PDF]

[2] G. H. Fredrickson and H. C. Andersen,"Kinetic Ising-model of the glass-transition," Phys. Rev. Lett. 53, 1244-1247 (1984).

[3] Palmer, R.G., D.L. Stein, E. Abrahams and P.W. Anderson, "Models of hierarchically constrained dynamics for glassy relaxation," Phys. Rev. Lett. 53, 958-961 (1984).

[4] Garrahan, J.P and D. Chandler, "Coarse grained microscopic model for glass formers,"Proc. Natl Acad. Sci. USA 100, 9710-9714 (2003). [PDF]

[5] Merolle, M., J.P. Garrahan and D. Chandler. " Space-time thermodynamics of the glass transition," Proc. Natl Acad. Sci. USA 102,10837-10840, (2005). [PDF]

[6] Garrahan, J.P., R.L. Jack, V. Lecomte, E. Pitard, K. can Duikvendijk, and F. van Wijland, "Dynamical First-Order Phase Transition in Kinetically Constrained Models of Glasses,"Phys. Rev. Lett. 98, 195702 (2007) [PDF]

[7] Jung, Y., J.P. Garrahan and D. Chandler. "Excitation lines and the breakdown of Stokes-Einstein relations in super-cooled liquids," Phys. Rev. E 69, 061205.1-061205.7 (2004). [PDF]

[8] Pan, A.C., " Rotational correlation and dynamic heterogeneity in a kinetically constrained lattice gas," J. Chem. Phys. 123, 164501 (2005) [PDF]

[9] Jung, Y.J., J.P. Garrahan and D. Chandler, "Dynamical exchanges in facilitated models of supercooled liquids," J.Chem. Phys. 123, 084509.1-10, (2005) [PDF]

[10] Berthier, L. D. Chandler and J.P.Garrahan. "Lenght scale for the onset of Fickian diffusion in super-cooled liquids," Euro. Phys. Lett. 69, 320-326 (2005).[PDF]

[11] Pan, A.C, J.P. Garrahan and D. Chandler, "Decoupling of self-diffusion and structural relaxation during a fragile-to-strong cross-over in a kinetically constrained lattice gas," ChemPhysChem 6, 1783-1785 (2005).[PDF]

[12] Pan, A.C, J.P. Garrahan and D. Chandler. " Heterogeneity and growing lengthscales in the dynamics of kinetically constrained lattice gases in two dimensions," Phys. Rev. E. 72, 041106 (2005).[PDF]

[13] Chandler, D., J.P. Garrahan, R.L. Jack, L. Maibaum and A.C. Pan, "Lengthscale dependence of dynamic four-point susceptibilities in glass formers," cond-mat/0605084, accepted by Phys. Rev. E (2006) [PDF]

[14] Liu, L., S.H. Chen, A. Faraone, C.W.Yen, and C.Y. Mou, "Pressure dependence of fragile-to-strong transition and a possible second critical point in supercooled confined water" Phys. Rev. Lett.95 Art. No.117802 (2005).