Glass-gas coexistence versus coincidence near the Kauzmann temperature in the Kob-Andersen Lennard-Jones binary mixture

The Kob-Andersen (KA) 80:20 binary Lennard-Jones mixture is the most widely simulated atomistic model of a fragile glass-forming liquid. In 2000, Coluzzi, Parisi and Verrocchio applied the replica (random first-order transition) approach to this mixture and predicted an entropy-crisis (Kauzmann) temperature $T_K ≈ 0.32$ analytically and $≈ 0.28$ numerically, in Lennard-Jones units of the A-A interaction. Strikingly, the liquid-gas coexistence temperature of the same potential, estimated both from integral-equation theory ($T_{GL} ≈ 0.25$) and from simulation extrapolation ($T_{GL} ≈ 0.3$), lies close to $T_K$, and the authors suggested — but their data could not establish — a genuine glass-gas coexistence at low temperature. The calculation rests on uncontrolled approximations: the integral-equation closure for the liquid structure (which shifts $T_{GL}$ by up to a factor of two between closures), a partial treatment of fluctuations of diagonal elements of the Hessian-related matrix, the neglect of anharmonic corrections and of activated processes (the negative-eigenvalue tail of the Hessian), and an untested identification of the static one-step replica-symmetry-breaking parameter $m^*$ with the dynamical fluctuation-dissipation ratio. No known work settles whether the near-coincidence of $T_K$ and $T_{GL}$ reflects genuine thermodynamic glass-gas coexistence or is a coincidence or artifact: equilibrium simulation of this mixture below $T ≈ 0.45$ remains impossible, crystallization intervenes in very long runs, and the replica computation has never been made controlled for this system. The problem remains open.

Background

When a liquid is supercooled below its melting temperature without crystallizing, its relaxation time grows by many orders of magnitude and it falls out of equilibrium at a laboratory glass transition. A complementary thermodynamic description, the random first-order transition (RFOT) picture, treats the glass transition as an entropy crisis: the configurational entropy $S_c$ (the entropy difference between the supercooled liquid and the vibrational entropy of a solid at the same energy, accessible in simulation via the potential-energy-landscape formalism) is extrapolated to vanish at a finite Kauzmann temperature $T_K$, below which an ideal amorphous solid would be the equilibrium state. Whether this extrapolation signals a genuine phase transition in three dimensions is itself unproven.

The standard atomistic model for this physics is the binary Lennard-Jones mixture introduced by Kob and Andersen in 1995: an 80:20 mixture of large A and small B particles with Lennard-Jones pair interactions whose cross parameters are chosen to frustrate crystallization; at number density $ρ = 1.2$ (in units of the A-A interaction) its mode-coupling crossover lies at $T_D ≈ 0.435$, and relaxation times exceed any accessible simulation time below $T ≈ 0.45$.

Coluzzi, Parisi and Verrocchio (J. Chem. Phys. 112, 2933 (2000)) analyzed this mixture with the replica approach to RFOT: $m$ copies of the system are coupled and treated as a molecular liquid, one-step replica symmetry breaking (1RSB, the mean-field scenario in which the free-energy landscape splits into an exponential number of metastable amorphous states at a dynamical temperature $T_D$, with an equilibrium glass transition at a lower $T_K$) appears as a nontrivial off-diagonal structure of the molecular liquid, and the cage size and static parameter $m^*$ are computed self-consistently from a replicated free energy expanded harmonically around a glassy minimum of the Hessian-related matrix $M$. They found $T_K ≈ 0.32$ analytically and $≈ 0.28$ from numerical thermodynamic integration.

The attractive part of the potential also gives the mixture a liquid-gas coexistence line. Because simulation near coexistence is preempted by glassy arrest, the coexistence temperature $T_{GL}$ was estimated (i) by solving the Ornstein-Zernike integral equations for the liquid structure with different closures — hypernetted chain (HNC), Percus-Yevick (PY), mean spherical approximation (MSA), and the Zerah-Hansen interpolation between HNC and MSA — which give $T_{GL}$ anywhere from $≈ 0.25$ to $≈ 0.5$, and (ii) by extrapolating simulated liquid pressures to zero along isochores, giving $T_{GL} ≈ 0.3$. The near-coincidence $T_{GL} ≈ T_K$ motivated the suggestion of a glass-gas coexistence at low temperature, which the authors explicitly left as an unresolved, intriguing possibility.

The authors also stated precisely which parts of their calculation were uncontrolled: the harmonic resummation resums off-diagonal matrix elements but misses anharmonic corrections and the thermal fluctuations of the diagonal elements of $M$; the cage expansion is truncated at second order; and activated processes — which appear as a tail of negative eigenvalues of the Hessian, with a number decreasing as $exp(−1/T)$ at low temperature — are outside the harmonic description, which is valid only where the effective temperature $T/m^*$ lies below $T_D$. Finally, by analogy with short-range spin glasses, they conjectured that the static parameter $m^*$ equals the fluctuation-dissipation ratio measured in the aging regime of the same mixture (the ratio of response to correlation that quantifies the violation of the fluctuation-dissipation theorem in an out-of-equilibrium glass; Barrat and Kob had found it piecewise-constant, 1RSB-like, in the aging KA mixture), but did not establish this equivalence.

Problem Statement

Determine the low-temperature thermodynamic phase behaviour of the standard Kob-Andersen binary Lennard-Jones mixture (80:20 A:B, number density $ρ = 1.2$, Lennard-Jones units of the A-A interaction) in the temperature region where the extrapolated configurational entropy of the supercooled liquid vanishes (Kauzmann temperature $T_K ≈ 0.28$–$0.32$) and where the extrapolated liquid-gas coexistence line of the same potential lies ($T_{GL} ≈ 0.25$–$0.3$). The overall objective has two linked components. (1) The central question: is the near-coincidence $T_K ≈ T_{GL}$ a genuine thermodynamic glass-gas coexistence (equilibrium coexistence between an amorphous solid and a dilute gas) at low temperature, or a near-coincidence of two separate phenomena, so that the entropy crisis — if real — is located elsewhere relative to the liquid-gas coexistence line? (2) The accuracy prerequisite: any replica-based answer must control or quantitatively bound the approximations of the Coluzzi-Parisi-Verrocchio calculation, namely the integral-equation closure for the liquid structure (HNC, PY, MSA and Zerah-Hansen closures shift $T_{GL}$ by up to a factor of two), the truncation of fluctuations of the diagonal elements of the Hessian-related matrix, the neglect of anharmonic corrections to the replicated free energy, and the unstable (negative-eigenvalue) Hessian modes associated with activated processes. Since the original suggestion of coexistence was itself produced by the uncontrolled approximation scheme, the reliability of a replica-based resolution of component (1) is bounded by its control of the approximations listed in (2). The source paper also conjectures, by analogy with short-range spin glasses, that the static one-step-replica-symmetry-breaking parameter $m^*$ equals the fluctuation-dissipation ratio measured in the aging regime of the same mixture; establishing or refuting this identification is a related open aspect of the replica description, but deciding component (1) does not require settling it.

References

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