Glass transition imminent, resistance is futile

The immense variety of glassy materials around us makes it is easy to get caught up in excessive detail when thinking about the structural glass transition. For decades, there have been only a few attempts to describe covalently bonded, molecular, and metallic glasses on the same footing, so different do the molecular motions seem in these various classes of substances. The article by Angelini and Biroli (1) is a noteworthy attempt to use the renormalization group (RG) philosophy to uncover similarities between seemingly distinct glass-forming systems, thus spotlighting universal aspects of the structural glass transition.

The idea of renormalization goes back at least as far as the 19th century, in the context of the phenomenological theory of dielectric media. Electrostatic interactions between charges in condensed media can still be thought of as interactions between elementary charges, but with their magnitude renormalized downward—owing to the ability of molecular dipoles in the medium to rotate so as to partially screen the interaction. This coarse-grained theory implies that even in a complex, interacting system, individual localized charges remain a useful way to think about the effective degrees of freedom. Onsager (2) took this idea significantly further, in the 1930s, by relating the strength of individual molecular dipoles and the renormalized bulk dielectric response of a material in a quantitative fashion. To do this, he estimated the strength of the electrical feedback each such dipole receives from its own motions that polarize the surrounding medium. In modern parlance, Onsager estimated the renormalization of the elementary dipole moment at the second-order, one-loop level. Although identical in essence, renormalization in quantum electrodynamics (QED) comes across as being much more sophisticated because the perturbation theory can be carried out to a remarkably high order.

The quantitative success of the perturbative expansion of QED may seem surprising in view of Landau’s notion (3, 4) that the possibility of arbitrarily high order processes implies particles and antiparticles can be produced in unlimited quantities.Much as free ions in a solution lead to Debye screening, virtual particles that emerge from the vacuum would lead to an infinitely large amount of screening at large distances. The framework that allowed for such arbitrarily strong renormalization came across as mathematically adventurous when first developed in the 1940s, but its precision is now legendary and it gives one a license to work with essentially the same type of elementary excitations or particles across a broad range of length and energy scales.

Liquid-to-solid transitions pose severe challenges to the RG philosophy because those two condensed phases are usually described using fundamentally different kinds of degrees of freedom. In uniform liquids, much like in dilute gases, particles can exchange places very rapidly, implying the appropriate degrees of freedom cannot be tied to specific particles but, instead, are local fluctuations of the particle concentration. The harsh repulsion between the molecules, however, causes this translational symmetry to break at sufficiently high pressures. Each particle will trade its original ability to freely explore space for an ability to avoid the vast majority of other particles, by effectively building a wall around itself, each one using a small number of nearest neighbors. Thus, the appropriate degrees of freedom are now distances between such nearest neighbors. However, how can one use RG philosophy when the active degrees of freedom change their character so much with increasing length scales?

Spins seem to come to the rescue. Focusing on the local motions and the ability to escape the local cage, the dynamics of a glassy liquid can be mapped, approximately, onto a variety of spin models, such as the random field Ising model (5), or a six-component Heisenberg model with long-range interactions (6). Decades ago it was shown that some spin models with disordered spin–spin couplings reproduce many of the salient features of the glassy phenomenology, at least in the mean-field limit of very-long-range interactions. This phenomenology is briefly summarized in Fig. 1.

Fig. 1.

Equation of state for a liquid that fails to crystallize, at constant temperature $T$ (main graph) and pressure p (Inset); $𝝆$ stands for density and V for volume. In a typical protocol, the liquid is cooled or compressed until the translational degrees of freedom become largely arrested within an individual free energy minimum, at pressure $p_g$ (temperature $T_g$), whereas the configurational entropy $s_c$, corresponding to the translations, stays roughly stationary. The vastly degenerate free energy minima themselves emerge at pressure $p_{\mathrm{cr}}$ (temperature $T_{\mathrm{cr}}$). Note the temperature in the Inset decreases left to right: $T_{cr}>T_g>T_K$.

Below a certain temperature $T_{\mathrm{cr}}$, or above a certain pressure $p_{\mathrm{cr}}$, the motions in a liquid begin to exhibit two very distinct aspects instead of being simply translations and rotations of individual molecules characteristic of gases and uniform liquids. One can think of a liquid below $T_{\mathrm{cr}}$ or above $p_{\mathrm{cr}}$ as a solid on short length scales and for times shorter than a certain structural relaxation time ${\tau }_{\alpha }$. On longer times, the liquid flows through cooperative, rare events that allow the particles to escape from the self-built cages.

The structural relaxation time ${\tau }_{\alpha }$ grows rapidly with lowering temperature so that below a certain temperature $T_g$ or above pressure $p_g$ equilibrating the liquid takes longer than the humanly relevant timescale of seconds to hours. Continuing to cool/compress the liquid further beyond this point results in the cooperative degrees of freedom falling out of equilibrium. This is called the glass transition. We see that contrary to what its moniker suggests, the glass transition in the laboratory is not purely thermodynamic but, instead, comes from a kinetic arrest when the typical free energy barrier relative to temperature, $F^{‡}$/$k_{B}T$, exceeds the logarithm of the humanly relevant timescale ${\tau }_{\text{lab}}$ in units of vibrational times ${\tau }_0$: $F^{‡}$/$k_{B}T>$ ln(${\tau }_{\text{lab}}/{\tau }_0$). If the data are extrapolated to infinitely long equilibration times, for temperatures beyond $T_g$, the relaxation times seem to diverge at a finite temperature (7). At this so-called Kauzmann temperature, $T_K$, the system would apparently be securely trapped on any conceivable timescale. Whether there is a strict divergence is of no consequence either in the laboratory or a glass manufacturing facility. The question may also be moot, for the most rigorous among us, because the simple molecular substances will crystallize or otherwise order already above the temperature $T_K$, if given supercosmological times (8). Nevertheless, the potential existence of a true thermodynamic transition at $T_K$ is of great interest because it allows us to understand what happens on available timescales. Mathematicians may still find the question of an ultimate transition interesting because it is related to the problem of the densest packing of objects. It took hundreds of years to settle the packing problem for monodisperse spheres in 3D (9), but nobody knows what is in store for other types of objects.

Angelini and Biroli (1) use the RG framework, usually used for equilibrium phase transitions, to study the length-scale dependence of the interactions for several types of disordered spin models in finite dimensions that in mean field resemble structural glasses. They use an approximate but nonperturbative scheme. Owing to its thermodynamic character, RG cannot directly access the relaxation barriers: The transition state configurations between distinct free energy minima occupy too small a portion of the phase space. To make progress, the authors assume the kinetic barriers scale with the coupling constants, which are accessible to RG. This assumption leads to the same results as more constructive arguments that indicate the rare cooperative motions resemble nucleation processes (10).

The results of the RG transformation depend on the dimensionality $d$ of space. For a sufficiently large value of $d$ the calculations yield two nontrivial fixed points, in addition to the trivial fixed point that corresponds to the high symmetry phase, viz., the uniform liquid. The nontrivial fixed points are very special

The article by Angelini and Biroli is a noteworthy attempt to use the renormalization group (RG) philosophy to uncover similarities between seemingly distinct glass-forming systems, thus spotlighting universal aspects of the structural glass transition.

in that the typical values $v$ of the interactions, relative to temperature $T$, grow with the length scale. (The divergence of the quantity $v/T$ is sometimes formally referred to as the fixed point being a zero-temperature one.) One of the zero-$T$ fixed points is fully stable; thus, the divergence of the energy scale means that the system would become securely confined within a free energy basin whose depth would be infinite at finite temperatures, hence the title of this commentary. The concurrent divergence of the length scale means that to move one particle near the fixed point one must move an infinite number of particles. This fits nicely with the description of the putative Kauzmann ideal glass state but, more generally, is symptomatic of any rigid solid, because to separate any two atoms inside a solid one must break an infinite number of bonds or contacts.

The other zero-$T$ fixed point has an unstable direction that can either lead to the uniform liquid or to the fully stable glass. This fits nicely with the mean-field-limiting behavior of the random first-order transition theory (10, 11), according to which the free energy surface of the liquid splits into a thermodynamically large number of equivalent free energy basins below a certain temperature $T_A$, which appears as a spinodal. The multiplicity of basins accounts for the excess entropy of the liquid relative to the vibrational entropy of a single minimum; the latter entropy is approximately equal to the vibrational entropy of the corresponding crystal and so can be well approximated experimentally.

Angelini and Biroli (1) further find that in sufficiently low spatial dimensions, including the physical dimensionality of 3, both zero-$T$ fixed points disappear. Nevertheless they do leave a trace: Both the escape barrier and the corresponding length scale grow hand in hand, but instead of finally diverging they reach a high fixed value. The confinement of local motions, although no longer strict, is still sufficient for the system to appear rigid. Indeed, Angelini and Biroli (1) find that the cooperativity length $\xi$ for the 3D relic of the stable fixed point saturates at values close to 20, to be compared with the figure of 6 or so achieved in laboratory glass formers (10). Given the apparent scaling of the barriers with $\xi$ found in the laboratory (10), this would correspond with structural relaxation times of order 10⁸⁰ y! In turn, the avoidance of the unstable fixed point is strikingly similar to the behavior of actual liquids; the 3D analog of the mean-field spinodal becomes a soft cross-over at $T_{\text{cr}}$ ($p_{\text{cr}}$). Interestingly, the RG argument in ref. 1 implies the putative Kauzmann state for spin models would not actually exist in 3D. If this holds true for particle systems as well, the ground state for arbitrary rigid objects should eventually exhibit some sort of long-range order!

We finish by noting that, arguably, the most famous example of a coupling diverging with increasing length scale is in the world of strong interactions between quarks (3, 4). Similarly to atoms confined within a solid, pulling apart a pair of quarks requires breaking an infinite number of connecting gluons arranged in a string, hence the confinement. Quantum chromodynamics represents a magnificent antithesis to the aforementioned Landau paradox for QED, because the strong interactions are antiscreened, the more so the longer the distance. Interestingly, the elastic interactions between local configurations of atoms in a random assembly can also be shown (6) to antiscreen at least at the level of the Onsager cavity construction (2). In hadronic matter the stringy interactions melt out at a high temperature, originally known as the Hagedorn temperature (12). The spinodal to forming glassy basins in finite dimensions, too, has been argued to resemble mathematically this Hagedorn physics (13). It would be interesting to see whether Yang–Mills field theories have instabilities associated with the replica symmetry breaking characteristic of glassy systems.