waterEoS 0.4.0

Equations of state for supercooled water

Overview

waterEoS provides Python implementations of three two-state equations of state (EOS), one empirical Tait-Tammann EOS, and a two-state transport properties model for supercooled water, unified under a single SeaFreeze-compatible API. The two-state models capture the thermodynamic anomalies of water by treating it as a mixture of two interconvertible local structures (low-density/tetrahedral and high-density/disordered), predicting a liquid-liquid critical point (LLCP) in the deeply supercooled regime. The Grenke & Elliott (2025) Tait-Tammann model is a direct empirical correlation without two-state decomposition. The Singh et al. (2017) model extends the two-state framework to predict dynamic transport properties (viscosity, self-diffusion, rotational correlation time).

Installation

pip install waterEoS

Quick Start

import numpy as np
from watereos import getProp

# Single point: 0.1 MPa, 300 K
PT = np.array([[0.1], [300.0]], dtype=object)
out = getProp(PT, 'duska2020')
print(f"Density: {out.rho[0,0]:.2f} kg/m³")
print(f"Cp:      {out.Cp[0,0]:.1f} J/(kg·K)")
print(f"x:       {out.x[0,0]:.4f}")

Available Models

Model key	Reference	LLCP (T, P)
'holten2014'	Holten, Sengers & Anisimov, J. Phys. Chem. Ref. Data 43, 014101 (2014)	228.2 K, 0 MPa
'caupin2019'	Caupin & Anisimov, J. Chem. Phys. 151, 034503 (2019)	218.1 K, 72.0 MPa
'duska2020'	Duska, J. Chem. Phys. 152, 174501 (2020)	220.9 K, 54.2 MPa
'grenke2025'	Grenke & Elliott, J. Phys. Chem. B 129, 1997 (2025)	-- (empirical)
'singh2017'	Singh, Issenmann & Caupin, PNAS 114, 4312 (2017)	-- (transport)
'water1'	SeaFreeze water1 (pass-through)	--
'IAPWS95'	SeaFreeze IAPWS-95 (pass-through)	--

Validity Ranges

The three two-state models accept any (T, P) input without raising errors, but results are only physically meaningful within the ranges below. The "paper-stated" range is where each model was validated by its authors; the "code-accessible" range is where the code runs without numerical failure (though results outside the paper range may be unphysical).

Model	Paper-stated validity	Code-accessible range
'holten2014'	T_H(P)–300 K, 0–400 MPa (extrap. to 1000 MPa)	Unbounded (any T, P)
'caupin2019'	~200–300 K, -140–400 MPa	Unbounded (any T, P)
'duska2020'	~200–370 K, 0–100 MPa (extrap. to 200 MPa)	Unbounded (any T, P)
'grenke2025'	200–300 K, 0.1–400 MPa	Unbounded (any T, P)
'water1'	240–501 K, 0–2300 MPa	Enforced by SeaFreeze
'IAPWS95'	240–501 K, 0–2300 MPa	Enforced by SeaFreeze

Notes:

• T_H(P) is the homogeneous ice nucleation temperature (~235 K at 0.1 MPa, ~181 K at 200 MPa).
• Duska (2020) was fitted to data at positive pressures only; negative-pressure extrapolation is unvalidated.
• Caupin (2019) is the only model explicitly validated at negative pressures (stretched water).
• Grenke (2025) is a direct empirical Tait-Tammann correlation, not a two-state model. It has no x, _A, or _B outputs.
• Singh (2017) is a transport properties model that uses Holten (2014) as its thermodynamic backbone. It returns all Holten thermodynamic properties plus eta, D, and tau_r. Its validity range matches Holten (2014).
• Outside the paper-stated ranges, models may return unphysical values (e.g., negative compressibility or heat capacity) without warning.

Usage

Grid Mode

Evaluate on a pressure x temperature grid (like SeaFreeze):

import numpy as np
from watereos import getProp

P = np.arange(0.1, 200, 10)    # pressures in MPa
T = np.arange(250, 370, 1)     # temperatures in K
PT = np.array([P, T], dtype=object)

out = getProp(PT, 'holten2014')
# out.rho has shape (len(P), len(T))

Scatter Mode

Evaluate at specific (P, T) pairs:

import numpy as np
from watereos import getProp

PT = np.empty(3, dtype=object)
PT[0] = (0.1, 273.15)    # 0.1 MPa, 273.15 K
PT[1] = (0.1, 298.15)    # 0.1 MPa, 298.15 K
PT[2] = (100.0, 250.0)   # 100 MPa, 250 K

out = getProp(PT, 'caupin2019')
# out.rho has shape (3,)

Individual Model Access

Each model can also be imported directly:

from duska_eos import getProp
from caupin_eos import getProp
from holten_eos import getProp
from grenke_eos import getProp
from singh_viscosity import getProp

List Available Models

from watereos import list_models
print(list_models())
# ['water1', 'IAPWS95', 'holten2014', 'caupin2019', 'duska2020', 'grenke2025', 'singh2017']

Output Properties

All models return an object with the following attributes (the three two-state models also include x, _A, and _B suffixed properties; grenke2025 returns only the mixture properties):

Mixture (equilibrium) properties

Attribute	Property	Units
rho	Density	kg/m³
V	Specific volume	m³/kg
Cp	Isobaric heat capacity	J/(kg·K)
Cv	Isochoric heat capacity	J/(kg·K)
Kt	Isothermal bulk modulus	MPa
Ks	Adiabatic bulk modulus	MPa
Kp	Pressure derivative of bulk modulus	--
alpha	Thermal expansivity	1/K
vel	Speed of sound	m/s
S	Specific entropy	J/(kg·K)
G	Specific Gibbs energy	J/kg
H	Specific enthalpy	J/kg
U	Specific internal energy	J/kg
A	Specific Helmholtz energy	J/kg
x	Tetrahedral (LDL) fraction	--

Per-state properties

Each property above (except x) is also available for the individual states with _A and _B suffixes:

• rho_A, Cp_A, vel_A, ... (State A: high-density / disordered)
• rho_B, Cp_B, vel_B, ... (State B: low-density / tetrahedral)

Total: 43 output properties (15 mixture + 14 state A + 14 state B).

All thermodynamic potentials (S, G, H, U, A) are aligned to the IAPWS-95 reference state.

Transport properties (singh2017 only)

Attribute	Property	Units
eta	Dynamic viscosity	Pa·s
D	Self-diffusion coefficient	m²/s
tau_r	Rotational correlation time	s
f	LDS fraction (= x from Holten backbone)	--

The singh2017 model also returns all Holten (2014) thermodynamic properties listed above.

Phase Diagram

Each model provides functions to compute the liquid-liquid phase diagram:

from duska_eos import compute_phase_diagram

result = compute_phase_diagram()
# result contains: T_LLCP, p_LLCP, T_spin_upper, p_spin_upper,
#                  T_spin_lower, p_spin_lower, T_binodal, p_binodal, ...

Available functions: find_LLCP(), compute_spinodal_curve(), compute_binodal_curve(), compute_phase_diagram().

Performance

Throughput on a 100x100 = 10,000-point grid:

Model	Time	Throughput
Holten (2014)	32 ms	317k pts/s
Caupin (2019)	18 ms	563k pts/s
Duska (2020)	49 ms	203k pts/s
Grenke (2025)	9 ms	1,116k pts/s

References

1. V. Holten, J. V. Sengers, and M. A. Anisimov, "Equation of state for supercooled water at pressures up to 400 MPa," J. Phys. Chem. Ref. Data 43, 014101 (2014). doi:10.1063/1.4895593

2. F. Caupin and M. A. Anisimov, "Thermodynamics of supercooled and stretched water: Unifying two-structure description and liquid-vapor spinodal," J. Chem. Phys. 151, 034503 (2019). doi:10.1063/1.5100228

   • Erratum: J. Chem. Phys. 163, 039902 (2025). doi:10.1063/5.0239673

3. M. Duska, "Water above the spinodal," J. Chem. Phys. 152, 174501 (2020). doi:10.1063/5.0006431

4. J. C. Grenke and J. R. Elliott, "Empirical fundamental equation of state for the metastable state of water based on the Tait-Tammann equation," J. Phys. Chem. B 129, 1997-2012 (2025). doi:10.1021/acs.jpcb.4c06847

   • Correction: J. Phys. Chem. B 129, 9850-9853 (2025). doi:10.1021/acs.jpcb.5c04618

5. L. P. Singh, B. Issenmann, and F. Caupin, "Pressure dependence of viscosity in supercooled water and a unified approach for thermodynamic and dynamic anomalies of water," Proc. Natl. Acad. Sci. U.S.A. 114, 4312-4317 (2017). doi:10.1073/pnas.1619501114

Authors

• Anthony Consiglio

License

This project is licensed under the GNU General Public License v3.0.