^{1}, David T. Limmer

^{2}, Valeria Molinero

^{3}and Mikhail A. Anisimov

^{1,a)}

### Abstract

The thermodynamic properties of the supercooled liquid state of the mW model of water show anomalous behavior. Like in real water, the heat capacity and compressibility sharply increase upon supercooling. One of the possible explanations of these anomalies, the existence of a second (liquid–liquid) critical point, is not supported by simulations for this model. In this work, we reproduce the anomalies of the mW model with two thermodynamic scenarios: one based on a non-ideal “mixture” with two different types of local order of the water molecules, and one based on weak crystallization theory. We show that both descriptions accurately reproduce the model's basic thermodynamic properties. However, the coupling constant required for the power laws implied by weak crystallization theory is too large relative to the regular backgrounds, contradicting assumptions of weak crystallization theory. Fluctuation corrections outside the scope of this work would be necessary to fit the forms predicted by weak crystallization theory. For the two-state approach, the direct computation of the low-density fraction of molecules in the mW model is in agreement with the prediction of the phenomenological equation of state. The non-ideality of the “mixture” of the two states never becomes strong enough to cause liquid–liquid phase separation, also in agreement with simulation results.

The authors have benefited from numerous interactions with Pablo Debenedetti (Princeton University). Jan V. Sengers (University of Maryland, College Park) read the manuscript and made useful comments. M.A.A. acknowledges discussions with Efim I. Kats (Landau Institute, Russia) on weak crystallization theory. The research of V.H. and M.A.A. has been supported by the Division of Chemistry of the U.S. National Science Foundation under Grant No. CHE-1012052. D.T.L. acknowledges the Helios Solar Energy Research Center, which is supported by the Director, Office of Science, Office of Basic Energy Sciences of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. V.M. acknowledges support by the National Science Foundation through Award Nos. CHE-1012651 and CHE-1125235 and the Camille and Henry Dreyfus Foundation through a Teacher-Scholar Award.

I. INTRODUCTION

II. TWO-STATE THERMODYNAMICS OF LIQUID WATER

A. Regular solution

B. Athermal solution

C. Clustering of water molecules

D. Description of thermodynamic properties of the mW model

III. WEAK CRYSTALLIZATIONTHEORY

A. Fit to the mW data

IV. CONCLUSIONS

### Key Topics

- Crystallization
- 30.0
- Thermodynamic properties
- 22.0
- Entropy
- 19.0
- Ice
- 18.0
- Enthalpy
- 16.0

## Figures

Density ρ, isobaric heat capacity C P , isothermal compressibility κ T , and thermal expansivity α P computed from the mW model (points) compared with the results (curves) of the two-state approach for an athermal solution [Eq. (13) ]. The isobar pressures are given in the density diagram; pressures and corresponding isobar colors are the same in the four panels.

Density ρ, isobaric heat capacity C P , isothermal compressibility κ T , and thermal expansivity α P computed from the mW model (points) compared with the results (curves) of the two-state approach for an athermal solution [Eq. (13) ]. The isobar pressures are given in the density diagram; pressures and corresponding isobar colors are the same in the four panels.

Fraction x of molecules in the low-density state. Solid curves: fraction x for the (a) two-state equation of state, Eq. (13) ; (b) two-state equation accounting for hexamer clustering, Eq. (16) with N = 6. Dashed curves: fraction x obtained from simulations of mW water, calculated from the fraction of four-coordinated molecules f 4 as , to account for fractions and of four-coordinated molecules in the low- and high-temperature liquid, respectively. The inflection points on the curves are marked with circles (two-state equation) and squares (mW model). The data were collected by linearly quenching the temperature of the simulations at a rate of 10 K/ns. The data below the inflection point do not correspond to equilibrium states.

Fraction x of molecules in the low-density state. Solid curves: fraction x for the (a) two-state equation of state, Eq. (13) ; (b) two-state equation accounting for hexamer clustering, Eq. (16) with N = 6. Dashed curves: fraction x obtained from simulations of mW water, calculated from the fraction of four-coordinated molecules f 4 as , to account for fractions and of four-coordinated molecules in the low- and high-temperature liquid, respectively. The inflection points on the curves are marked with circles (two-state equation) and squares (mW model). The data were collected by linearly quenching the temperature of the simulations at a rate of 10 K/ns. The data below the inflection point do not correspond to equilibrium states.

Pressure–temperature diagram. Circles: location of the computed property data of the mW model. Solid line: line at which ln K = 0 and the low-density fraction x = 1/2 for the two-state equation. Squares: location of inflection points of the low-density fraction x for the mW model (see Fig. 2 ). Dotted line: stability-limit temperature from the fit of the weak crystallization model to the mW data, Eq. (26) . The dashed curve is a fit to the melting temperature of mW ice (uncertainty about ±3 K), obtained from free energy calculations as described in Ref. 20 .

Pressure–temperature diagram. Circles: location of the computed property data of the mW model. Solid line: line at which ln K = 0 and the low-density fraction x = 1/2 for the two-state equation. Squares: location of inflection points of the low-density fraction x for the mW model (see Fig. 2 ). Dotted line: stability-limit temperature from the fit of the weak crystallization model to the mW data, Eq. (26) . The dashed curve is a fit to the melting temperature of mW ice (uncertainty about ±3 K), obtained from free energy calculations as described in Ref. 20 .

Enthalpy ΔH (a) and entropy ΔS (b) of liquid mW water with respect to ice at 0.1 MPa (black curves, from Moore and Molinero 19 ) and their fits (dashed red curves) according to Eqs. (18) and (19) , respectively. Both ΔH and ΔS are computed at a cooling rate of 10 K/ns, which prevents crystallization, so that the values below 200 K do not correspond to an equilibrium state. The circle signals K. These results support the modeling of mW water as an athermal mixture of two states.

Enthalpy ΔH (a) and entropy ΔS (b) of liquid mW water with respect to ice at 0.1 MPa (black curves, from Moore and Molinero 19 ) and their fits (dashed red curves) according to Eqs. (18) and (19) , respectively. Both ΔH and ΔS are computed at a cooling rate of 10 K/ns, which prevents crystallization, so that the values below 200 K do not correspond to an equilibrium state. The circle signals K. These results support the modeling of mW water as an athermal mixture of two states.

Heat capacity of mW water in equilibrium (circles) and on hyperquenching at 10 K/ns (black curve, computations by Moore and Molinero 19 ). The values below 200 K do not correspond to an equilibrium state. The dashed curve is the prediction of the two-state equation with hexamer clusters.

Heat capacity of mW water in equilibrium (circles) and on hyperquenching at 10 K/ns (black curve, computations by Moore and Molinero 19 ). The values below 200 K do not correspond to an equilibrium state. The dashed curve is the prediction of the two-state equation with hexamer clusters.

Fluctuation-renormalized distance to the stability-limit temperature Δ, given by Eq. (21) , as a function of the mean-field distance to the stability limit Δ0, for two values of β (solid curves). The dashed line corresponds to Δ = Δ0 and is shown as a reference.

Fluctuation-renormalized distance to the stability-limit temperature Δ, given by Eq. (21) , as a function of the mean-field distance to the stability limit Δ0, for two values of β (solid curves). The dashed line corresponds to Δ = Δ0 and is shown as a reference.

Density ρ, isobaric heat capacity C P , isothermal compressibility κ T , and thermal expansivity α P computed from the mW model (points) compared with the fit to power laws given by weak crystallization theory (curves). The isobar pressures are given in the density diagram; pressures and corresponding isobar colors are the same in the four panels.

Density ρ, isobaric heat capacity C P , isothermal compressibility κ T , and thermal expansivity α P computed from the mW model (points) compared with the fit to power laws given by weak crystallization theory (curves). The isobar pressures are given in the density diagram; pressures and corresponding isobar colors are the same in the four panels.

Fluctuations of the crystallization order parameter from weak crystallization theory (i.e., short-wavelength density fluctuations) in arbitrary units as a function of temperature, fit with Eq. (28) (curve).

Fluctuations of the crystallization order parameter from weak crystallization theory (i.e., short-wavelength density fluctuations) in arbitrary units as a function of temperature, fit with Eq. (28) (curve).

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