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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 nonideal “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 twostate approach, the direct computation of the lowdensity fraction of molecules in the mW model is in agreement with the prediction of the phenomenological equation of state. The nonideality 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. CHE1012052. 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. DEAC0205CH11231. V.M. acknowledges support by the National Science Foundation through Award Nos. CHE1012651 and CHE1125235 and the Camille and Henry Dreyfus Foundation through a TeacherScholar Award.
I. INTRODUCTION
II. TWOSTATE 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 twostate 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.
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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 twostate 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 lowdensity state. Solid curves: fraction x for the (a) twostate equation of state, Eq. (13) ; (b) twostate 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 fourcoordinated molecules f 4 as , to account for fractions and of fourcoordinated molecules in the low and hightemperature liquid, respectively. The inflection points on the curves are marked with circles (twostate 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.
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Fraction x of molecules in the lowdensity state. Solid curves: fraction x for the (a) twostate equation of state, Eq. (13) ; (b) twostate 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 fourcoordinated molecules f 4 as , to account for fractions and of fourcoordinated molecules in the low and hightemperature liquid, respectively. The inflection points on the curves are marked with circles (twostate 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 lowdensity fraction x = 1/2 for the twostate equation. Squares: location of inflection points of the lowdensity fraction x for the mW model (see Fig. 2 ). Dotted line: stabilitylimit 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 .
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Pressure–temperature diagram. Circles: location of the computed property data of the mW model. Solid line: line at which ln K = 0 and the lowdensity fraction x = 1/2 for the twostate equation. Squares: location of inflection points of the lowdensity fraction x for the mW model (see Fig. 2 ). Dotted line: stabilitylimit 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.
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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 twostate equation with hexamer clusters.
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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 twostate equation with hexamer clusters.
Fluctuationrenormalized distance to the stabilitylimit temperature Δ, given by Eq. (21) , as a function of the meanfield 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.
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Fluctuationrenormalized distance to the stabilitylimit temperature Δ, given by Eq. (21) , as a function of the meanfield 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.
Click to view
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., shortwavelength density fluctuations) in arbitrary units as a function of temperature, fit with Eq. (28) (curve).
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Fluctuations of the crystallization order parameter from weak crystallization theory (i.e., shortwavelength density fluctuations) in arbitrary units as a function of temperature, fit with Eq. (28) (curve).
Tables
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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 nonideal “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 twostate approach, the direct computation of the lowdensity fraction of molecules in the mW model is in agreement with the prediction of the phenomenological equation of state. The nonideality of the “mixture” of the two states never becomes strong enough to cause liquid–liquid phase separation, also in agreement with simulation results.
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