^{1,a)}and Barbara Drossel

^{1}

### Abstract

We compare three different simple models for water. They all show a phase behavior and anomalies that are characteristic of water. We compare these models and their features and evaluate the phase diagram, the density anomaly, and the liquid-liquid transition line. Additionally, we show that the characteristic behavior present in all three models can be deduced from the fact that all three models include three microscopic states for nearest neighbor configurations. We therefore propose an even simpler three-state model for water that still captures the phase transitions and the density anomaly. Finally, we show that this simple three-state model shows in fact all four possible scenarios discussed in the literature for the phase behavior of liquid water, if the parameters are adjusted accordingly.

This work was supported by Deutsche Forschungsgemeinschaft (DFG) (Grant No. Dr300/11) and the Studien- stiftung des deutschen Volkes (L.H.).

I. INTRODUCTION

II. MODEL

A. Cell model in the MF approach

B. 3S3W-model

C. One-dimensional model

III. RESULTS

A. Phase diagram

B. Density anomaly

C. A simplified unifying picture

1. Common features of the three models

2. A three-state model for water

3. Four scenarios for water

IV. DISCUSSION AND CONCLUSION

### Key Topics

- Entropy
- 23.0
- Phase diagrams
- 20.0
- Phase transitions
- 20.0
- Liquid liquid phase transitions
- 17.0
- Critical point phenomena
- 15.0

## Figures

The three microscopic states in the 3S3W model. The figure is adapted from Fig. 2 in the article by Truskett and Dill. 35

The interaction potential in the Ben-Naim model.

The interaction potential in the Ben-Naim model.

Phase diagram and volume per particle (volume profile v(T, p)) for the cell model (top), the 3S3W model (center), and the one-dimensional model (bottom). The temperature and pressure are scaled such that the liquid gas critical point is at T = 1 = p in all three models. The LL-CP is at T c = 0.061 and p c = 4.22 in the cell model, at T c = 0.048 and p c = 55.34 in the 3S3W model and at T c = 0.248 and p c = 10.12 in the one-dimensional model. In the cell model, the volume at high pressure is dominated by the condition p σ ⩾ 1/q.

Phase diagram and volume per particle (volume profile v(T, p)) for the cell model (top), the 3S3W model (center), and the one-dimensional model (bottom). The temperature and pressure are scaled such that the liquid gas critical point is at T = 1 = p in all three models. The LL-CP is at T c = 0.061 and p c = 4.22 in the cell model, at T c = 0.048 and p c = 55.34 in the 3S3W model and at T c = 0.248 and p c = 10.12 in the one-dimensional model. In the cell model, the volume at high pressure is dominated by the condition p σ ⩾ 1/q.

Regions of the density anomaly (left) and examples for the dependence v(T) (right) in the cell model (top), the 3S3W model (center), and the one-dimensional model (bottom) for three different pressures. The third pressure value is chosen slightly below the liquid-gas critical point p ≲ 1.0.

Regions of the density anomaly (left) and examples for the dependence v(T) (right) in the cell model (top), the 3S3W model (center), and the one-dimensional model (bottom) for three different pressures. The third pressure value is chosen slightly below the liquid-gas critical point p ≲ 1.0.

Occupation of the three microscopic states D, HB, and G (from left to right) in the cell model (top), the 3S3W model (center), and the one-dimensional model (bottom). In the cell model, , , and q G = (1 − n) with . In the 3S3W model, the three states correspond to the three states of three neighboring water molecules, that is, dense, cage-like, and gas (see Figure 1 ). In the one-dimensional model, the three microscopic configurations correspond to the three accessible regions of the nearest neighbor potential.

Occupation of the three microscopic states D, HB, and G (from left to right) in the cell model (top), the 3S3W model (center), and the one-dimensional model (bottom). In the cell model, , , and q G = (1 − n) with . In the 3S3W model, the three states correspond to the three states of three neighboring water molecules, that is, dense, cage-like, and gas (see Figure 1 ). In the one-dimensional model, the three microscopic configurations correspond to the three accessible regions of the nearest neighbor potential.

Sketch of the three-state model. The three microscopic states are characterized by a certain energy, entropy, and volume.

Sketch of the three-state model. The three microscopic states are characterized by a certain energy, entropy, and volume.

The phase diagram (a), the volume profile (volume per particle) (b), the region where α p (T) < 0 (c), and examples for v(T) (d) of the three-state model. The liquid-liquid critical point was estimated to T c = 0.12 and p c = 2.23.

The phase diagram (a), the volume profile (volume per particle) (b), the region where α p (T) < 0 (c), and examples for v(T) (d) of the three-state model. The liquid-liquid critical point was estimated to T c = 0.12 and p c = 2.23.

Qualitative sketch of the effect of varying parameters in the three-state model on the position and slope of the HDL-LDL line. In a, s D > s HB , and (u D − u HB )/(v HB − v D ) is small. In b we have s HB > s D , and in c (u D − u HB )/(v HB − v D ) is large.

Qualitative sketch of the effect of varying parameters in the three-state model on the position and slope of the HDL-LDL line. In a, s D > s HB , and (u D − u HB )/(v HB − v D ) is small. In b we have s HB > s D , and in c (u D − u HB )/(v HB − v D ) is large.

Examples for different scenarios in the three-state model. Solid lines are phase transitions and dashed lines represent the spinodals. (a) The LLCP scenario is obtained with the standard values (see Table I ). (b) In the SF scenario, v HB has changed to v HB = 1.2 ≳ v D = 1. (c) In the CPF scenario, the volume of the dense state has been changed to v D = 0.75.

Examples for different scenarios in the three-state model. Solid lines are phase transitions and dashed lines represent the spinodals. (a) The LLCP scenario is obtained with the standard values (see Table I ). (b) In the SF scenario, v HB has changed to v HB = 1.2 ≳ v D = 1. (c) In the CPF scenario, the volume of the dense state has been changed to v D = 0.75.

## Tables

Influence of different parameters on the phase diagram. For each parameter, the standard value and the influence of an increase of this parameter (when all other parameters assume their standard values) on the phase diagram is specified. slope LL is the slope of the liquid-liquid transition line, T LLCP and p LLCP (T LGCP and p LGCP ) are temperature and pressure of the liquid-liquid (liquid-gas) critical point, which can increase (↑), decrease (↓) or not change at all (-) when the parameter is increased. CR LL and CR LG denote the size of the coexistence region, that is, the size of the region in the T-p-plane in which two phases coexist. The size of this region can also increase or decrease with increase of a parameter.

Influence of different parameters on the phase diagram. For each parameter, the standard value and the influence of an increase of this parameter (when all other parameters assume their standard values) on the phase diagram is specified. slope LL is the slope of the liquid-liquid transition line, T LLCP and p LLCP (T LGCP and p LGCP ) are temperature and pressure of the liquid-liquid (liquid-gas) critical point, which can increase (↑), decrease (↓) or not change at all (-) when the parameter is increased. CR LL and CR LG denote the size of the coexistence region, that is, the size of the region in the T-p-plane in which two phases coexist. The size of this region can also increase or decrease with increase of a parameter.

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