Error-controlled determination of the Kauzmann transition and configurational entropy of binary mixture glasses

The replica ('cloned liquid') computation of Coluzzi, Mézard, Parisi and Verrocchio (J. Chem. Phys. 111, 9039 (1999)) gave a first-principles prediction of the thermodynamic liquid-glass (Kauzmann) transition of a three-dimensional binary soft-sphere mixture: an analytic Kauzmann parameter $\Gamma_K \simeq 1.65$, to be compared with numerical estimates $\Gamma_K \simeq 1.6$ (coupled system) and $\Gamma_K \simeq 1.75$ (entropy crossing) whose errors the authors stated were difficult to evaluate and might be quite large. Later literature completed the Lennard-Jones extension (reproducing the same structure of unresolved discrepancies), found the binary replica liquid theory internally inconsistent in the one-component limit and reformulated it without implementation, showed that hypernetted-chain-level closures still distort the analytic transition temperatures, and pushed equilibrium simulations into the Kauzmann regime only for continuously polydisperse three-dimensional and for two-dimensional models. The problem is therefore still open: determine $\Gamma_K$ and the configurational entropy $\Sigma(T)$ of the three-dimensional binary mixture glass former with quantified errors - equilibrium numerics with a controlled amorphous-solid entropy, and a replica-theory computation with controlled closure and expansion errors - and confront the two determinations within their error bars.

Background

When a liquid is cooled fast enough to avoid crystallization it becomes a supercooled liquid whose relaxation time grows enormously, and it eventually falls out of equilibrium at the laboratory glass temperature. Whether an underlying equilibrium thermodynamic transition exists deeper below is a central open question of glass physics. Kauzmann observed that the entropy of the supercooled liquid, extrapolated to low temperature, would fall below that of the crystal - the 'entropy crisis' - and proposed that a true thermodynamic glass transition preempts this paradox at a temperature now called the Kauzmann temperature $T_K$. In the random-first-order-transition scenario of mean-field glass theory the transition is driven by exactly this crisis: below a dynamical temperature $T_D$ (identified with the mode-coupling crossover, the temperature at which idealized mode-coupling theory predicts a dynamical arrest) the phase space splits into exponentially many long-lived amorphous states, and at $T_K$ their degeneracy - the configurational entropy or 'complexity' $\Sigma$, the entropy per particle associated with the number of distinct amorphous states - vanishes.

Mézard and Parisi (J. Chem. Phys. 111, 1076 (1999)) turned this scenario into a first-principles statistical-mechanical scheme for particle systems: introduce $m$ replicas of the system, weakly coupled so that they remain in the same amorphous state, compute the free energy of the resulting 'molecular liquid' whose molecules are $m$-atom clusters using standard liquid-state theory, and analytically continue to $m \le 1$; the Legendre transform of the molecular-liquid free energy with respect to the cage size yields the glass free energy and $\Sigma$. Coluzzi, Mézard, Parisi and Verrocchio (J. Chem. Phys. 111, 9039 (1999)) generalized this 'replica liquid theory' to binary mixtures, the standard numerical glass formers because the size contrast frustrates crystallization. Their model is a 50:50 mixture of soft spheres with pair potential

$$V^{\epsilon\epsilon'}(r) = \left(\frac{\sigma_{\epsilon\epsilon'}}{r}\right)^{12}$$

with diameter ratio $R = \sigma_{++}/\sigma_{--} = 1.2$, arithmetic-mean cross diameter $\sigma_{+-} = (\sigma_{++} + \sigma_{--})/2$, concentration $c_+ = c_- = 1/2$, the diameter normalization $(\sigma_{++}^3 + 2\sigma_{+-}^3 + \sigma_{--}^3)/4 = 1$ (which sets the mean particle diameter to 1), and density $\rho = 1$; all equilibrium thermodynamics then depends on the single scaling variable $\Gamma \equiv \rho T^{-1/4}$, with the dynamical (mode-coupling) crossover at $\Gamma_D \simeq 1.45$. The energy scale is absorbed into the temperature, so no separate interaction-strength parameter appears.

The analytic side of the computation rests on two approximations. First, the correlation functions of the replicated ('molecular') liquid are obtained from the hypernetted chain (HNC) closure - an integral-equation approximation relating the pair correlation functions to the pair potential, estimated in the paper to be accurate to better than 10% for two-point quantities - and the three-point functions needed at second order are replaced by the superposition approximation, i.e. the three-point function is factored into a product of pair functions, with an error estimated at about 30%. Second, the glass free energy itself is evaluated in two schemes that should agree: the harmonic resummation, which sums the harmonic vibrations around amorphous minima re-expressed through liquid-state input, and the small-cage expansion, a systematic expansion of the replicated partition function in the cage radii $A^{\pm}$ that measure how strongly each species is localized in the glass. Both schemes give $\Gamma_K \simeq 1.65$.

The numerical side proceeds differently. The equilibrium liquid entropy is obtained by thermodynamic integration of simulated liquid energies - integrating the equilibrium energy along a reversible path from a reference state of known entropy - with a fitted power-law extrapolation below the lowest equilibrable temperature. The amorphous-solid entropy is obtained from the spectrum of the Hessian matrix of second derivatives of the potential energy (the 'instantaneous Hessian'), following the Frenkel-Ladd idea of computing a solid's free energy from its vibrations by coupling it to a reference solid. At the relevant temperatures the instantaneous Hessian has a non-negligible fraction of negative eigenvalues (about 4% near $\Gamma \simeq 1.5$, rising to about 20% at $\Gamma \simeq 1$), reflecting saddle-dominated regions of the energy landscape, so the harmonic solid entropy is ambiguous and different prescriptions for the negative eigenvalues differ measurably. The crossing of the liquid and solid entropies gives $\Gamma_K \simeq 1.75$. An independent numerical route couples the system to a frozen reference configuration of itself through a field $\epsilon(x - x_{\rm ref})^2$ (an overlap-field construction) and reads the complexity off the associated free-energy cost, giving $\Gamma_K \simeq 1.6$.

The three determinations - analytic $\simeq 1.65$, numerical $1.6$ and $1.75$ - are mutually consistent only within errors the authors state are 'difficult to evaluate' and 'might be quite large'. The paper closes by leaving 'for future work both more extensive numerical studies and the improvement of the analytical results, that should allow a more careful comparison', naming the Lennard-Jones binary mixture as the next application; that extension was completed by the same authors in 2000 and reproduced the same structure of results (analytic $T_K \simeq 0.32$ versus numerical estimates $0.26$-$0.30$ for the Kob-Andersen Lennard-Jones mixture, the standard 80:20 size-mismatched Lennard-Jones binary mixture introduced by Kob and Andersen as a numerical glass former), so the open issue was never the existence of the application but the control of errors in both routes.

Problem Statement

Obtain an error-controlled determination of the thermodynamic (Kauzmann) transition of the three-dimensional binary mixture glass former studied in the source work - the 50:50 soft-sphere mixture with pair potential $V^{\epsilon\epsilon'}(r) = (\sigma_{\epsilon\epsilon'}/r)^{12}$, diameter ratio $R = \sigma_{++}/\sigma_{--} = 1.2$, $\sigma_{+-} = (\sigma_{++} + \sigma_{--})/2$, $c_+ = c_- = 1/2$, the diameter normalization $(\sigma_{++}^3 + 2\sigma_{+-}^3 + \sigma_{--}^3)/4 = 1$, and density $\rho = 1$ - comprising three jointly required results that serve one objective.

(1) A numerical determination of the Kauzmann point $\Gamma_K$ and of the complexity $\Sigma(T)$ with a quantified error budget. This requires (a) liquid entropies in the deep-supercooling regime above the mode-coupling crossover $\Gamma_D \simeq 1.45$ and up to the expected $\Gamma_K$ (the region where the entropy-crossing estimate $\Gamma_K \simeq 1.75$ currently rests on a fitted power-law extrapolation from equilibrium data at $\Gamma \lesssim 1.5$), obtained with an explicitly bounded systematic error, and (b) a controlled amorphous-solid entropy in which the ambiguity caused by negative Hessian eigenvalues and anharmonicity is resolved or bounded, so that the spread between the entropy-crossing estimate ($\simeq 1.75$) and the coupled-system estimate ($\simeq 1.6$) is resolved within quoted errors, whether by confirming their agreement or by demonstrating a systematic bias in one of them, together with a finite-size analysis.

(2) An analytic replica-liquid determination of the same quantities whose approximations are systematically controlled or explicitly bounded: the closure used for the replicated liquid's correlation functions (with a controlled or explicitly bounded two-point error; the HNC closure's error was estimated below 10%), the treatment of the three-point functions (with a controlled or explicitly bounded error; the superposition approximation's error was estimated near 30%), and a controlled account of the relation between the harmonic resummation and the small-cage expansion, including whether their agreement at $\Gamma_K \simeq 1.65$ survives beyond second order.

(3) The quantitative theory-simulation comparison the source paper called a 'more careful comparison': the analytic and numerical determinations of $\Gamma_K$ and $\Sigma(T)$ must be confronted within their stated error bars, establishing whether the first-principles replica prediction is quantitatively confirmed or falsified for this model.

The common objective is to replace the current determinations, which are mutually consistent only within large unquantified errors, by one that a domain expert can accept or reject on the basis of stated errors. The determination requires controlled errors, and the problem concerns the static theory rather than the dynamical questions (viscosity, $T_g$) that this theory deliberately does not touch.

References

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