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# Equilibrium phase diagram of a randomly pinned glass-former

Edited by Pablo G. Debenedetti, Princeton University, Princeton, NJ, and approved April 16, 2015 (received for review January 13, 2015)

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## Significance

Confirming by experiments or simulations whether or not an ideal glass transition really exists is a daunting task, because at this point the equilibration time becomes astronomically large. Recently it has been proposed that this difficulty can be bypassed by pinning a fraction of the particles in the glass-forming system. Here we study numerically a liquid with such random pinned particles and identify the ideal glass transition point

## Abstract

We use computer simulations to study the thermodynamic properties of a glass-former in which a fraction *c* of the particles has been permanently frozen. By thermodynamic integration, we determine the Kauzmann, or ideal glass transition, temperature *T*- and *c* dependence of the saddle index, and use these properties to obtain the dynamic transition temperature *c* and indicate the point at which the glass transition line ends. These findings are qualitatively consistent with the scenario proposed by the random first-order transition theory.

- ideal glass transition
- computer simulations
- random first-order transition theory
- Kauzmann temperature
- configurational entropy

Upon cooling, glass-forming liquids show a dramatic increase of their viscosities and relaxation times before they eventually fall out of equilibrium at low temperatures (1, 2). This laboratory glass transition is a purely kinetic effect because it occurs at the temperature at which the relaxation time of the system crosses the time scale imposed by the experiment, e.g., via the cooling rate. Despite the intensive theoretical, numerical, and experimental studies of the last five decades, the mechanism responsible for the slowing down and thus for the (kinetic) glass transition is still under debate and hence a topic of intense research. From a fundamental point of view the ultimate goal of these studies is to find an answer to the big question in the field: Does a finite temperature exist at which the dynamics truly freezes and, if so, is this ideal glass transition associated with a thermodynamic singularity or is it of kinetic origin (3⇓⇓–6)?

Support for the existence of a kinetic transition comes from certain lattice gas models with a “facilitated dynamics” (6). In these models, the dynamics is due to the presence of “defects” and hence for such systems the freezing is not related to any thermodynamic singularity. However, the first evidence that there does indeed exist a thermodynamic singularity goes back to Kauzmann, who found that the residual entropy (the difference of the entropy of the liquid state from that of the crystalline state) vanishes at a finite temperature

Despite all these advances, the arguments put forward in the various papers must be considered as phenomenological because compelling and undisputed experimental or numerical evidence to prove or disprove any of these theories and scenarios is still lacking. The only exception is hard spheres in infinite dimensions, for which mean-field theory should become exact (13), but even in this case some unexpected problems are present (see ref. 14). This lack of understanding is mainly due to the steep increase of the relaxation times which hampers the access to the transition point of thermally equilibrated systems, and hence most of the efforts to identifying the transition point, if it exists, resort to unreliable extrapolation.

## Randomly Pinned Systems

Recently a novel idea to bypass this difficulty has been proposed (15⇓⇓–18). By freezing, or pinning, a fraction of the degrees of freedom of the system, the ideal glass transition temperature has been predicted to rise to a point at which experiments and simulations in equilibrium are feasible, thus allowing one to probe the nature of this transition. In ref. 15 the authors have studied the effect of pinning for the case of a mean-field spin-glass model which is known to exhibit a dynamical MCT transition at a temperature *c* of the degree of freedoms of the spins (selected at random) in the equilibrated system, both *c*. Thus, by equilibrating the nonpinned system at an intermediate temperature and subsequently increasing *c*, one can access and probe the ideal glass state in thermal equilibrium [Note that changing *c* does not perturb this equilibrium (19).] It was found that at sufficiently large *c* the two lines

Very recently, it has been tested whether this approach to detect *q* and its distribution *T*–*c* plane for pinned systems (21).

Despite these results, it is not clear if the so-obtained amorphous state is the bona fide ideal glass for which the configurational entropy

Although from the simulations reported in ref. 21 the existence of *q* alone may not be conclusive to demonstrate the existence of the arrested phase, because one cannot exclude the possibility that other scenarios, such as the purely kinetic one, can also explain the observed features of

In the present work, we use computer simulation to determine the ideal glass transition temperature *c*. Furthermore, we analyze the geometrical properties of the potential energy landscape (PEL) and use it to evaluate the dynamic transition temperature

## Results

We study a standard glass-forming model: A 3D binary Lennard-Jones mixture (24). The number of particles is *Materials and Methods* for details). Throughout the present study, the system has been prepared at each temperature by randomly choosing a fraction *c* of particles from the thermally equilibrated samples and quenching their positions (see *Materials and Methods* for details).

### Entropy and Configurational Entropy.

To obtain the entropy of the pinned system *S*, we used thermodynamic integration to determine the entropy of a given configuration of pinned particles and subsequently calculated *S* by averaging over the realizations of pinned particles (see *Materials and Methods* for details).

Fig. 1*A* shows the entropy per (unpinned) particle *T*, and we recognize that with increasing *c* the entropy decreases rapidly. For all temperatures this decrease is linear at small *c* but then the curves bend at intermediate *c* and follow a weaker *c* dependence. For *T* ≲ 0.5 this bent becomes sharp, strongly indicating that a thermodynamic glass transition takes place.

This becomes more evident by evaluating the configurational entropy obtained by subtracting from *S* the vibrational entropy *A* as well (solid lines) and we see that it shows basically a linear decrease with *c*, a trend which is due to the suppression of the low-frequency modes in the density of states.

We can now estimate the configurational entropy *B* we show the *c* dependence of *c* and becomes basically zero at a finite value of *c*, indicating that the system has entered the ideal glass state in which the entropy is basically due to harmonic vibrations. [The fact that the difference

We define the ideal glass transition point *N* (see *SI Text* for details).

### Overlap Approach.

An alternative method to locate and characterize the thermodynamic transition is to study the overlap *i* in configuration *a* is 0.3 (21). RFOT predicts that at the glass transition the average value *SI Text*) and in Fig. 2 we present *c*, and the peak position shifts continuously toward larger *q* as *c* increases (Fig. 2*A*). A qualitatively different behavior is observed at *B*): *c*, but has a double-peak structure at intermediate *c*, thus signaling the coexistence of the fluid and glass phase, which indicates that the transition from the fluid phase to the glass phase is first-order–like (21).

The *c* dependence of the average overlap *C*. For high temperatures, *c*, reflecting the continuous shift of the single peak of *A*. For *c*, in agreement with the presence of the double-peak structure seen in *T*. It suggests that a first-order–like transition, rounded by finite size effect, takes place, in qualitative agreement with the result for a system of harmonic spheres (21). Note that, within the accuracy of our data, we see that the amplitude of the (smeared out) jump in

From this approach with the overlap, we can define the ideal glass transition temperature *SI Text*), and in Fig. 4 we have included the resulting *c*, we find a very good agreement between *Discussion* below. Finally, we mention that the curves

### PEL and MCT.

In the past, it has been found that the slow dynamics of glass-forming systems is closely related to the features of the PEL (27) and in the following we will use these relations to characterize the relaxation dynamics of the pinned system.

Fig. 3*A* shows the *T* dependence of the average inherent structure energy *c* increases, the value of *T* moves steadily upward, which is reasonable because the energy of the system is literally pinned at the higher energy levels due to the presence of the pinned particles. Concomitantly the cross-over temperature increases with *c* and the cross-over becomes smeared out, completely disappearing at the highest *c*. The vanishing of this cross-over with growing *c* indicates thus that the pinning qualitatively affects the nature of the PEL and of the relaxation dynamics. For instance, it is found that with increasing *c* the fragility of the system decreases and shows at high *c* an Arrhenius dependence (21, 31).

In Fig. 3*A*, *Inset*, the low-temperature behavior of *T* for all *c*. According to the energy landscape scenario, this is an indicator that the distribution of

Another important quantity that connects the glassy dynamics of a system with its PEL is the saddle index *K*, i.e., the number of negative eigenvalues of the Hessian matrix at a stationary point of the PEL. For bulk systems it has been found that *K* shows a linear dependence on *K* goes to zero (this value is often denoted as “threshold energy”

We use a standard method to determine numerically the energy and index of saddles for the pinned system (see *SI Text* for details), and in Fig. 3*B* we plot the average normalized saddle index *k* decreases linearly as a function of *c* and, together with the data of Fig. 3*A* and *c*. The resulting *c* dependence of

We have also evaluated *Material and Methods*) and by fitting *SI Text* for details). Note that one needs several fit parameters to obtain *c* and that the lines asymptotically approach the *T* side.

## Discussion

In Fig. 4 we summarize the results of the previous sections in the form of a phase diagram in the *c*–*T* plane. The ideal glass transition lines *T* and *c* the two temperatures basically coincide but around *c* dependence becomes weaker. Note that this separation occurs at the same point at which

The theoretical calculations for a mean-field spin-glass model show that *c* and that this end point is a critical point of the universality class of the random field Ising model (15⇓–17, 20). Furthermore, the theory predicts that at this end point the coexistence line

From the figure we also recognize that, beyond the end point, *T* and *c* the particles are strongly confined by the labyrinthine structure imposed by the pinned particles, i.e., the system resides mainly at the bottom of a free-energy minimum where both the configurational entropy and the number of the saddles vanish simultaneously. In contrast with

As it is evident from Fig. 1*A*, *A*), because the cross-over from the fluid to the glass phase becomes broad. To estimate the effect of this ambiguity, we have also included in Fig. 4 *B* from the low*-c* side (open circles). We see that below the end point *T* ≲ 0.5,

To the best of our knowledge, the present study is the first report of a system in finite dimensions that shows the existence of an ideal glass state in equilibrium, i.e., a state in which the configurational entropy is zero at a finite *T*. The Kauzmann temperatures reported in the past have all relied on somewhat questionable extrapolation procedures, leaving thus room for debate over the very existence of a thermodynamic transition (6, 8, 37).

Our findings are inconsistent with recent simulation studies in which the *T*- and *c* dependence of the relaxation dynamics has been studied (38). In ref. 38, the structural relaxation time *c* dependence of *c*, *c* and found that the Adam–Gibbs relation is violated (see *SI Text* for details). Thus, we conclude that in the case of pinned systems one cannot deduce the Vogel–Fulcher law from the Adam–Gibbs relation.

Because the results presented here are all obtained in thermodynamic equilibrium without referring to any kind of extrapolation, we are confident that the phase diagram presented in Fig. 4 does indeed reflect the properties of the system and is not an artifact of the analysis. Further evidence that the simulated system is really in equilibrium is the observation that the entropy obtained by thermodynamic integration from the high-temperature limit matches with that obtained from the low-temperature side (via harmonic approximation) in the glass phase. It is also reassuring that all three methods, the thermodynamic integration (vanishing entropy), the overlap distribution (the discontinuous jump of *q*), and the geometric change of the PEL, consistently point to the same end point, thus giving strong evidence that this point really exists. Also suggestive is that each combination of pairs among the three methods is compatible beyond the end point, which is reminiscent of the Widom line in the standard gas–liquid phase transition.

At this stage we can conclude that the phase diagram as predicted by the RFOT theory is confirmed at least qualitatively. What remains to be done is to probe the relaxation dynamics in the vicinity of the critical end point because one can expect that this dynamics is rather unusual (39) and to establish its universality class (20, 40). Furthermore, it will also be important to see whether the predicted phase diagram can also be observed in real experiments. Although this will be not easy, for certain systems such as colloids or granular media it should be possible.

## Materials and Methods

### Model.

The system we use is a binary mixture of Lennard-Jones particles (24). Both species *A* and *B* have the same mass and the composition ratio is

### Making Pinned Configurations.

The configuration of the pinned particles is generated by making first a replica exchange run for the bulk system, i.e.,

### Simulation Methods.

#### Thermodynamics.

To sample thermodynamic properties efficiently at low-*T* and large-*c* region, we use the replica exchange method (41). The maximum number of replicas is 24. More detail is presented in *SI Text* and in ref. 21. The total CPU time to obtain the presented results is about 580 y of single core time.

#### Dynamics.

We use the Monte Carlo (MC) dynamics simulation to calculate dynamical observables (42). The rule of the MC dynamics is the following: In an elementary move, one of the *A* for the wave vector *k* at the peak of the corresponding structure factor. We have averaged over 20–30 different realizations of pinned particles to calculate

### Entropy.

To calculate the entropy *A*, *B*, and *C*, and then carried out the thermodynamic integration analytically.

### Analysis of the Saddles.

To locate the saddles of the PEL of the system, we have made a minimization of the squared gradient potential *W* is performed by the Broyden–Fletcher–Goldfarb–Shanno algorithm (44). Similar to the minimization of *U* used to calculate the inherent structures, *W* includes the position of the pinned as well as unpinned particles, but only the position of the latter is optimized. After having located a saddle with energy *SI Text* and in Fig. 3*B* we present the average index as a function of

## Acknowledgments

We thank G. Biroli, C. Cammarota, D. Coslovich, and K. Kim for helpful discussions. M.O. acknowledges the financial support by Grant-in-Aid for Japan Society for the Promotion of Science Fellows (26.1878). W.K. acknowledges the Institut Universitaire de France. A.I. acknowledges JSPS KAKENHI 26887021. K.M. and M.O. acknowledge KAKENHI 24340098, 25103005, 25000002, and the JSPS Core-to-Core Program. The simulations have been done in Research Center for Computational Science, Okazaki, Japan, at the HPC@LR, and the CINES (Grant c2014097308).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: walter.kob{at}univ-montp2.fr.

Author contributions: M.O., W.K., A.I., and K.M. designed research, performed research, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1500730112/-/DCSupplemental.

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