Novel approach to numerical measurements of the configurational entropy in supercooled liquids

Significance

When liquids are cooled sufficiently fast, they remain trapped in an amorphous metastable state called a “glass.” Due to their highly disordered structure at the microscale, however, glasses are characterized by a large multiplicity of metastable states. The configurational entropy represents a measure of their number and can be estimated in experiments and simulations via different, and sometimes inconsistent, approximations. Here we propose a novel method to compute the configurational entropy in a computer simulation. Compared with other approaches, the proposed numerical scheme allows one to obtain an observable that is conceptually closer to the theoretical definition and at the same time yields results in better agreement with experiments.

Abstract

The configurational entropy is among the key observables to characterize experimentally the formation of a glass. Physically, it quantifies the multiplicity of metastable states in which an amorphous material can be found at a given temperature, and its temperature dependence provides a major thermodynamic signature of the glass transition, which is experimentally accessible. Measurements of the configurational entropy require, however, some approximations that have often led to ambiguities and contradictory results. Here we implement a novel numerical scheme to measure the configurational entropy Σ(T) in supercooled liquids, using a direct determination of the free-energy cost to localize the system within a single metastable state at temperature T. For two prototypical glass-forming liquids, we find that Σ(T) disappears discontinuously above a temperature T_c, which is slightly lower than the usual estimate of the onset temperature for glassy dynamics. This observation is in good agreement with theoretical expectations but contrasts sharply with alternative numerical methods. While the temperature dependence of Σ(T) correlates with the glass fragility, we show that the validity of the Adam–Gibbs relation (relating configurational entropy to structural relaxation time) established in earlier numerical studies is smaller than previously thought, potentially resolving an important conflict between experiments and simulations.

The configurational entropy (or complexity) plays an important role in descriptions of the glass transition because it quantifies the temperature evolution of the free-energy landscape accompanying changes in thermodynamic and dynamic properties of supercooled liquids. It represents both a major experimental signature of the glass transition (1) and a fundamental quantity within a number of theoretical approaches (2).

The configurational entropy Σ(T) is traditionally measured by subtracting a “vibrational” contribution to the total entropy of the system: Σ(T) ≃ S_tot(T) − S_vib(T). While S_tot(T) is well defined, the vibrational contribution requires some approximation. Experiments (1, 3–6) use, for instance, the entropy of the crystalline or glass states to estimate S_vib(T). In simulations, the above decomposition relies on the assumption that the system vibrates around a given “state,” further assumed to be equivalent to a local energy minimum, or inherent structure (7). A thermodynamic formalism was developed to determine numerically the configurational entropy, and applied to a large number of models (8–12). These studies have additionally revealed that the Adam–Gibbs relation (13)

$${\tau }_{\alpha }\left(T\right)\approx \exp \left(\frac{A}{T\mathrm{\Sigma }\left(T\right)}\right)$$ [1]

between Σ(T) and the structural relaxation time τ_α(T) is obeyed over a broad temperature window. In Eq. 1, A is an energy scale and τ₀ a microscopic timescale. Eq. 1 is an important relation for supercooled liquids, as its validity would directly establish that the viscosity increase near the glass transition is caused by the temperature evolution of a complex free-energy landscape.

Available numerical methods are, however, not fully satisfactory from both theoretical and experimental viewpoints. Firstly, the identification of metastable states with energy minima within the inherent structure formalism has been questioned (2, 14). Because energy minima exist at all T, the inherent structure Σ(T) exists at arbitrarily high temperatures, where the free-energy landscape is in fact featureless. In theoretical approaches (2, 15), Σ(T) is the entropic contribution stemming from the multiplicity of metastable states proliferating at low T. While this definition is also plagued by ambiguities, see Discussion, specific calculations show that Σ(T) appears discontinuously below a temperature corresponding (within mean-field approximations) to the mode-coupling transition temperature (2, 16). Secondly, the Adam–Gibbs relation in Eq. 1 was numerically found to be valid over the entire supercooled regime (9–11, 17). Experiments report instead that it only holds at low temperatures below the mode-coupling temperature (4), in a regime not accessible in simulations. These experimental findings are physically sensible because it is only at such low temperatures that the free-energy landscape can possibly control the dynamics, but they directly contradict simulations.

We propose and implement a novel numerical method to measure the configurational entropy, which fully resolves these issues. The proposed methodology does not require precise definitions of a free-energy landscape and metastable states. Our results show that the configurational entropy appears discontinuously at a characteristic low temperature, and that the Adam–Gibbs relation is not valid above the mode-coupling temperature. Therefore, this alternative approach provides a numerical estimate of the configurational entropy that is conceptually closer to theory, and yields quantitative results that agree better with experiments.

Results

The proposed numerical method is directly inspired by statistical mechanics approaches, where the configurational entropy can be computed from the thermodynamic properties of constrained cloned systems (2, 16, 18–20). The physical idea is that constraining a system to reside “close” to a single state has a free-energy cost Σ(T), because it represents the entropic loss due to an incomplete exploration of the configurational space. To bypass the difficulty of defining metastable states rigorously, we obtain a numerical estimate of Σ(T) by measuring a free-energy difference between two thermodynamic phases that can be well defined. In practice, we estimate Σ(T) from the thermodynamic properties of a system comprising two copies, 1 and 2, of the considered liquid thermalized at temperature T. As described in more detail in SI Appendix, we conduct equilibrium simulations of these two coupled copies and carefully measure the probability distribution function of their mutual overlap, P(Q) = 〈δ(Q − Q₁₂)〉, where brackets indicate an equilibrium average. We define the overlap as $Q_{12}=N^{-1}{\displaystyle {\sum }_{i,j}\theta \left(a-\left|{\mathbf{r}}_{1,i}-{\mathbf{r}}_{2,j}\right|\right)}$, where θ(x) is the Heaviside function, r_1,i denotes the position of particle i within configuration 1, a is a length comparable to the particle diameter σ (we take a/σ = 0.3), and N is the particle number in each copy. Note that this “collective” overlap is insensitive to particle exchanges. We define the “effective potential” $V\left(Q\right)=-\frac{T}{N}\ln P\left(Q\right)$, which is by definition the constrained equilibrium free energy of the total system when the average value of the overlap is Q (21, 22).

While V(Q) was introduced long ago in theoretical calculations (21), it was only recently realized that it can be accurately determined in computer simulations by applying tools first devised to study equilibrium phase transitions (23–25). For a particular model liquid, it was shown (24) that V(Q) is convex above a critical temperature T_c below which it develops a linear part, corresponding to a strongly non-Gaussian P(Q). This observation implies that a thermodynamic field ε conjugated to the overlap Q induces, for T < T_c, an equilibrium first-order transition between a low-Q and a high-Q phase (21). This first-order transition line ε_c(T) ends at a second-order critical point at T_c, as explicitly demonstrated in refs. 24 and 25. The existence of two phases below T_c suggests estimating the configurational entropy as:

$$\mathrm{\Sigma }\left(T\right)=\frac{1}{T}\left[V\left(Q_{\text{high}}\right)-V\left(Q_{\text{low}}\right)\right],$$ [2]

where Q_low denotes the position of the global minimum of V(Q), and Q_high is determined from the position of the peak in P(Q) at coexistence; see Fig. 1. Eq. 2 states that Σ(T) represents the free-energy difference between the low-Q phase where the two copies independently explore the free-energy landscape and the high-Q phase where they remain close to one another. This free-energy difference originates from the fact that one of the copies cannot freely explore the configuration space, and this precisely costs Σ(T). (Details pertaining to quenched and annealed complexities are given in Discussion.) While the complexity in Eq. 2 emerges naturally in mean-field calculations (21), our work is the first, to our knowledge, to implement this approach to estimate the configurational entropy in finite dimensional liquids. Notice that in finite dimensions, V(Q) must be a convex function in the thermodynamic limit (and is nearly convex even for the relatively small system sizes studied here), which is why we chose to define Q_high from the overlap distribution at coexistence, which remains bimodal even for large system sizes.

Fig. 1.

Measurement of the configurational entropy defined in Eq. 2, using the free-energy difference between the global minimum of V(Q) at Q_low and the value at Q_high, defined from the overlap distribution at coexistence. (Upper) Free-energy βV(Q) of N = 108 harmonic spheres for two temperatures above (dashed line) and below (solid line) the critical temperature T_c ∼ 10. The arrow defines Σ(T = 7). (Lower) The overlap distribution is bimodal below T_c along the first-order transition line ε_c(T) (solid line), and featureless above T_c (dashed line).

The definition [2] shows that the measurement of Σ(T) does not rely on an explicit definition of a free-energy landscape and of metastable states, and Σ(T) does not stem in the present approach from an enumeration of states. Instead, by measuring the thermodynamic properties of the high-Q localized phase, we let the system itself define the extent of a state. This provides a direct determination of Σ(T) that requires neither an approximate estimate of a vibrational contribution nor a detailed investigation of the potential energy landscape. This approach, which relies on the direct measurement of a free-energy difference, is conceptually much closer to theoretical calculations.

Another consequence of Eq. 2 is that Σ(T) is only defined when two distinct phases can be distinguished, i.e., for T ≤ T_c. For T > T_c, V(Q) is featureless and Σ(T) does not exist; see Fig. 1. In this regime, the entropy cannot be decomposed in configurational and vibrational parts. This is qualitatively consistent with specific theoretical calculations (15, 18, 19, 21). Physically, it means that the free-energy landscape of the high-temperature liquid has a simple topography for which the concept of configurational entropy is not relevant. A discontinuous emergence of the configurational entropy at low T is naturally obtained within the present calculations, whereas it is missed by previous methods (8, 10, 12).

We studied two models of glass formers using Monte Carlo simulations (26). The first model is a 50:50 binary mixture of harmonic spheres of diameter ratio 1.4 (27, 28). Within reduced units (29), this quasi-hard sphere system has an onset temperature T_on ∼ 12 (30), a mode-coupling temperature T_mct ∼ 5.2 (29), and a Vogel–Fulcher temperature T₀ ∼ 2 (30) [obtained with low reliability as the system is weakly fragile at this density (27, 28)]. We used N = 64, 108, and 256, finding that finite size effects for Σ(T) are small (see SI Appendix). We show data for N = 108. The second model is an 80:20 binary mixture of Lennard–Jones particles (31). In reduced units, the onset temperature is T_on ∼ 1.0, the mode-coupling temperature T_mct ∼ 0.435 (31), and the Vogel–Fulcher temperature T₀ ∼ 0.29 (10) (the model has intermediate fragility). We performed simulations with N = 180. As described below (see Simulation Methods and SI Appendix for detailed descriptions), we combine umbrella sampling, multihistogram reweighting, and replica exchange techniques to quantify the rare fluctuations of the global overlap that need to be studied to obtain the free-energy V(Q). We find that differences between various possible estimates of Σ(T) (20) can only be distinguished in a very narrow temperature regime near T_c that is not resolved by the present set of data, and therefore do not affect any of our conclusions.

Our central results are in Fig. 2, which displays the temperature dependence of Σ(T) obtained from Eq. 2 for two glass models. In both cases, we find that Σ(T) emerges discontinuously at a critical temperature T_c. We obtain T_c ∼ 10 for harmonic spheres (24), and T_c ∼ 0.8 for the Lennard–Jones model. Because T_c is very close to, or slightly below, the onset temperature T_on, this suggests that T_c might represent a well-defined, physically meaningful definition of the onset temperature in supercooled liquids (32). Note that T_c is significantly larger than T_mct obtained from a mode-coupling analysis of the dynamics. While T_c and T_mct are found to coincide in mean-field calculations (21), we find that T_c remains well defined in finite dimensions, whereas the mode-coupling singularity is replaced by a smooth crossover. Therefore, the present approach allows us to extend concepts derived from mean-field calculations to finite dimensional systems, and provides a method to consistently extract the configurational part of the entropy.

Fig. 2.

The configurational entropy appears discontinuously at temperatures T_c ∼ 10 and T_c ∼ 0.8, respectively, for harmonic and Lennard–Jones particles, in sharp contrast with literature data (8, 12). (We used the mapping between hard and harmonic spheres discussed in refs. 27 and 28 to convert the hard spheres data of ref. 12 into equivalent data for harmonic spheres.) Note also the steeper temperature dependence of Σ(T) in the more fragile Lennard–Jones model.

The abrupt emergence of Σ(T) at T_c stands in sharp contrast to alternative methods (8, 12), as demonstrated in Fig. 2. Therefore, the qualitative evolution of Σ(T) obtained in this work is in closer agreement with theoretical and physical expectations (see, e.g., ref. 2). Notice that such a discontinuous temperature dependence is of course not observed experimentally, because experimental methods (just as previous numerical methods) are not sensitive to the sharp emergence of metastable states that we are able to reveal here. Physically, our results simply suggest that a decomposition of the entropy in vibrational and configurational parts is not meaningful at high temperatures. This is in fact qualitatively hinted by the inherent structure approach (32), and by the Frenkel–Ladd method used for hard spheres (12). In the latter case, a caging timescale becomes ill defined at low packing fraction, making the very definition of S_vib unphysical.

The configurational entropy in Fig. 2 decreases steadily as temperature is lowered below T_c. This implies that the free-energy difference between localized and delocalized states in configuration space decreases as T gets lower, suggesting that the thermodynamic driving force to structural relaxation also decreases. A quantitative comparison with literature data in Fig. 2 shows that the temperature evolution of Σ(T) below T_c is in qualitative agreement with earlier work. However, the inherent structure formalism provides an estimate of the configurational entropy that is systematically larger than Σ(T) over the explored range. Our results for Σ(T) appear to agree better with those of ref. 12, but the absence of more extensive data at low T prevents a detailed and consistent comparison of the different approaches (33). Despite the above-mentioned shortcomings, inherent structure-based approaches might still represent a valuable approximation at low T.

A motivation to determine Σ(T) follows from Kauzmann’s study of experimentally determined Σ(T) suggesting the existence of an entropy crisis, Σ(T → T_K) = 0, possibly close to the Vogel–Fulcher temperature T₀ (3). Our data do not cover a broad enough temperature range to extrapolate an entropy crisis. However, they do support the qualitative connection between thermodynamic and dynamic fragilities found experimentally (5), since the more fragile Lennard–Jones system also has a steeper T/T_c dependence of Σ(T)/Σ(T_c), as implied by Fig. 2.

The Adam–Gibbs relation in Eq. 1 is a quantitative connection between thermodynamics and dynamics that can readily be tested once Σ(T) is known; see Fig. 3. Notice first that, by construction, this relation cannot hold above the critical temperature T_c where Σ(T) is not defined. Therefore, Eq. 1 cannot be expected to work if T is too large. In fact, we find that it does not work well except close to T_mct, although we would need more data to establish more firmly its validity at lower temperatures. Therefore, our results indicate that the validity of Eq. 1 reported in earlier simulations (9–11, 17) stems from using an alternative definition of Σ(T), for which Eq. 1 holds over a broader range. We emphasize that our results conform to the general physical expectations that relaxation dynamics in supercooled liquids becomes thermally activated when temperature is low enough, typically below the mode-coupling temperature. The results exposed in Fig. 3 are therefore physically welcome, as there exists no fundamental reason for Eq. 1 to be relevant in the weakly supercooled temperature regime. Additionally, experiments find clear deviations from this relation in the temperature window covered by simulations (4). Therefore, our results also suggest a plausible resolution to the existing discrepancy between experiments and previous simulations, although more work remains to be done, especially at lower temperatures, to fully settle this issue.

Fig. 3.

Test of Adam–Gibbs relation in Eq. 1. The data show strong deviations from a linear relation between log τ_α and (TΣ)⁻¹ if temperature is not low enough, whereas the relation is possibly satisfied when T < 6 (Upper) and T < 0.5 (Lower) (shown with arrows), where the data seem to follow the indicated dashed lines.

Discussion

Despite the above successes, we emphasize that our determination of a configurational entropy Σ(T) remains an approximation to the theoretical concept of a complexity counting the number of metastable states. A first approximation stems from the fact that we perform measurements of V(Q) with two freely evolving copies. An alternative procedure (21) consists of first drawing copy 1 from the equilibrium distribution, before studying the thermodynamics of copy 2 in the presence of the quenched disorder imposed by copy 1. This amounts to distinguishing annealed from quenched complexities (22). In the mean-field limit where rigorous calculations exist, the annealed Σ(T) is an approximation to the quenched one, but the latter is more fundamental because it exactly counts the number of metastable states. Although the quenched potential V(Q) can also be measured (24), the procedure is more demanding. Before quantitatively comparing the two complexities, one should first establish more firmly the existence of a critical temperature T_c in the quenched case. Preliminary work (24) suggests a slight depression of the critical temperature T_c, and very close values for both V(Q), but these issues need to be examined thoroughly.

A more fundamental issue concerns the interpretation of Σ(T) determined from V(Q) as an entropy associated to the number of metastable states. This is true at the mean-field level, where both V(Q) and the complexity can be rigorously defined and computed (2, 16). The situation is ambiguous in finite dimensions, where infinitely long-lived metastable states do not exist, which forbids a strict definition of a complexity associated to their number (20). Metastability can therefore only be approximately defined, for instance using finite timescales (34) or lengthscales (35), and metastable states cannot sharply emerge at the mode-coupling temperature, as they do in mean-field approximations (20, 21). By contrast, we note that V(Q) and Σ(T) defined in Eq. 2 do not suffer from these ambiguities. Therefore, our estimate of a configurational entropy is well defined in finite dimensions, even though metastable states are not. This distinction also explains that a sharp emergence of Σ(T) at T_c is found in our simulations, whereas only a weak vestige of the mode-coupling transition can be observed.

We can tentatively interpret Σ(T), as measured here, as the entropy related to the number of “metastable” states, now defined over finite lengthscales, suggesting a possible deep connection between the emergence of Σ(T) found here and the growth of a static (point-to-set) correlation length (29, 35–38). We emphasize that since Σ(T) in Eq. 2 quantifies the free-energy cost to localize the system in configuration space, its rapid decrease upon supercooling is likely related to the slowing down of the dynamics. This scenario naturally emerges in thermodynamic theories of the glass transition, such as Adam–Gibbs and random first-order transition theories. The numerical strategy proposed herein provides a sensible measure of the configurational entropy in the temperature range currently accessible to numerical simulations. Future work should explore different strategies (24, 25, 39) to extend the temperature regime where the configurational entropy can be accessed numerically, which should thus allow a stringent test of theoretical approaches in which the configurational entropy plays a central role.

Materials and Methods

Models.

Our first model is a binary mixture of harmonic spheres (27, 28), where particles of type α, β interact by a harmonic potential,

$$V_{\alpha \beta }\left(r\right)={\mathit{ϵ}}_{\alpha \beta }{\left(1-r/{\sigma }_{\alpha \beta }\right)}^2,$$

which is truncated at distance r = σ_αβ. The particle types are labeled A and B, and the interaction parameters are (σ_AA, σ_AB, σ_BB) = (1.0, 1.2, 1.4)σ, and (ε_AA, ε_AB, ε_BB) = (1.0, 1.0, 1.0)ε. We consider N = 64, 108, and 256 with N_A = N_B = N/2. We express length scales in units of σ and temperatures in units of 10⁻⁴ε and perform simulations at constant number density ρ = 0.675.

In the Kob–Andersen mixture, particles of types α, β interact by a Lennard–Jones potential

$$V_{\alpha \beta }\left(r\right)=4{\mathit{ϵ}}_{\alpha \beta }\left[{\left({\sigma }_{\alpha \beta }/r\right)}^{12}-{\left({\sigma }_{\alpha \beta }/r\right)}^6\right],$$

which is truncated and shifted at r = 2.5σ_αβ. The particle types are labeled A and B, and the interaction parameters are (σ_AA, σ_AB, σ_BB) = (1.0, 0.80, 0.88)σ and (ε_AA, ε_AB, ε_BB) = (1.0, 1.5, 0.5)ε. We consider N = 180 particles with N_A = 4N/5 and N_B = N/5. We express length scales in units of σ, temperatures in units of ε, and perform simulations at constant number density ρ = 1.2.

Simulation Methods.

Both models are studied numerically using Monte Carlo simulations (26). Measuring Σ(T) from Eq. 2 is numerically challenging as it requires the determination of V(Q) over a broad range of Q, which necessitates a quantitative analysis of atypical overlap fluctuations. This difficulty is efficiently overcome by using umbrella sampling techniques (40). Briefly, V(Q) is obtained by gathering the results of a series of n simulations biased in such a way that distinct simulations explore distinct ranges of overlap values (24). Histogram reweighting techniques are then used to reconstruct P(Q) over the complete Q range (24). Another challenge is the difficulty of ensuring proper thermalization of each simulation, which becomes serious when Q is large and T is low. This is not prohibitive in studies where only the vicinity of the critical temperature T_c is explored (24, 25). To access much lower temperatures, we have introduced replica-exchange Monte Carlo moves between the n biased simulations, borrowing techniques used in phase transition studies (40–42). Because each simulation now performs a random walk in parameter space, thermalization is greatly enhanced, and lower temperatures can be sampled. By approaching the mode-coupling temperature, we are able to measure Σ(T) over a physically significant T range. The procedure can presumably be further optimized to access even lower temperatures. It can also easily be applied to different models, including hard spheres to which the inherent structure formalism does not apply (12). Full details about these methods as well as about finite size effects are given in SI Appendix.