Proposal: An advanced two-state water EoS via Ising crossover and microscopic grounding

Context: Extending the Duška (2020) / Caupin (2019) class of two-state EoSs with two independent advances that compose into a single coherent model.
Status: Research proposal — not yet implemented.
Project: waterEoS · Author: Anthony Consiglio

1. Framing

Both axes are independent in principle but compose into a single coherent model. Axis 1 fixes a thermodynamic-rigor problem (mean-field exponents are wrong near the LLCP). Axis 2 fixes an interpretability/transferability problem (polynomial coefficients are not physical). Together they give a model with correct critical scaling, physically meaningful parameters, and far fewer knobs than Caupin (2019) or Duška (2020).

2. Axis 1 — Parametric Ising crossover

The physics

Near the LLCP, mean-field theory gives the wrong exponents:

Caupin, Duška, and Holten are all mean-field and so are systematically wrong in any ∼10 K window around the predicted LLCP. Far from C′ the differences are imperceptible (and Caupin's paper explicitly justifies mean-field at the data locations), but a model claiming to be "most advanced" should be correct in both regimes.

The standard machinery is the parametric crossover Landau expansion developed by Anisimov, Sengers, Kostko, Belyakov, Kiselev and collaborators across the 1990s–2010s. Two references for water specifically:

• Fuentevilla & Anisimov, PRL 97, 195702 (2006) — scaled parametric EoS using the Schofield revised linear model (RLM).
• Bertrand & Anisimov, J. Phys. Chem. B 115, 14099 (2011) — explicit crossover construction for water with complete scaling (field mixing of T, P, μ).

The construction in three layers

Layer A — Schofield RLM near C′ (parametric Ising)

Two variables: r (distance to C′) and θ (angular).

with β = 0.326, δ = 4.789. Property = derivative of a master Gibbs potential Ψ(r, θ) that's polynomial in θ and analytic in r. This is what you'd evaluate in a small neighborhood of C′.

Layer B — Field mixing (complete scaling for water)

Water's LLCP is a "mixed" transition, so t and h aren't pure T/P — they mix:

This is what Bertrand & Anisimov work out — and it's a real complication, because water's LLCP sits in a region where μ also matters.

Layer C — Crossover function bridging Ising to mean-field

Belyakov–Kiselev (or Tang–Sengers–Chen) introduce a crossover variable Y ∈ (0, 1]:

• Y → 1 far from C′ → recover mean-field.
• Y → rΔ near C′ → recover Ising scaling.

where Δ ≈ 0.51 is the Wegner correction exponent. The Landau coefficients get renormalized by powers of Y:

A simple closed-form for Y: solve 1/Y² = 1 + (κ/Y)2/Δ · (1 + r²/rmax²) for some Ginzburg number κ and a "max distance" r_max. The model becomes pure mean-field for r >> r_max and pure Ising for r << r_max.

waterEoS implementation

You don't need a full rewrite. The cleanest path is to leave Duška/Caupin in charge of the far-field and patch the near-LLCP region:

caupin_crossover_eos/                # new module
├── params.py                        # Ising exponents + crossover constants
├── core_ising.py                    # Schofield RLM → G, dG/dT, dG/dP, etc.
├── core_crossover.py                # crossover-renormalized Landau expansion
├── core.py                          # decides regime: pure-Caupin / crossover / pure-Ising
└── phase_diagram.py                 # binodal with correct β = 0.326 exponent

The dispatch in core.py:

def compute_batch(T_K, P_MPa, pset=PARAMS):
    r = _distance_to_llcp(T_K, P_MPa, pset)
    Y = _crossover_variable(r, pset)
    # Always evaluate the crossover-renormalized form;
    # Y handles regime smoothly.
    return _properties_from_landau_with_crossover(T_K, P_MPa, Y, pset)

The trick that makes this one function rather than three patched together: the crossover-renormalized Landau expansion is exact in both limits. As Y→1 you recover Caupin's form; as Y→0 you recover the Schofield RLM. No piecewise stitching, no derivative jumps.

What you'd add per file (rough scale):

• params: ∼15 new constants (5 Ising exponents are universal, ∼5 field-mixing, ∼3 crossover, ∼2 background).
• core: ∼300 lines numpy; ∼400 lines Rust; both need autodiff or careful manual derivatives because Ψ is composed of r(t, h), θ(t, h) via implicit equations — JAX is great here.

Effort estimate: 2–3 weeks for a working numpy implementation + validation; +1 week Rust parity; +1 week JAX/AD path for derivatives. Mostly straightforward porting from the Bertrand-Anisimov paper, which is fairly explicit. The hardest part is finding the field-mixing coefficients (a₁, a₂, b₁, …) that align Caupin's far-field with the crossover near-field — likely a constrained nonlinear fit against Caupin output + Kim 2020 κ_T data.

3. Axis 2 — Microscopic grounding (lattice minimal model)

The physics

The 2021 paper (Caupin, Anisimov, and collaborators) — "Minimal microscopic model for liquid polyamorphism" — is essentially this:

Lattice sites carry one of two species (A = HDL, B = LDL) with site energies and lattice excitations. The exact (mean-field) free energy per site is:

where μ_A° and μ_B° are the chemical potentials of pure-A and pure-B states. For water:

with U_B − U_A > 0 (LDL is higher-energy per molecule), S_A − S_B > 0 (HDL is more disordered), V_B − V_A > 0 (LDL is less dense). The cooperativity ω encodes hydrogen-bond network strength.

This is the same algebraic skeleton as Holten/Caupin — ΔGBA and ω(1−2x) show up identically — but the meaning changes:

• Where Caupin writes ΔGBA = λ(ΔT̂ + a·ΔP̂ + b·ΔT̂·ΔP̂ + d·ΔP̂² + f·ΔT̂²) as a pure polynomial fit, the microscopic model writes μB° − μA° = ΔU − T·ΔS + P·ΔV + … with each coefficient being a physical quantity.
• Where Caupin writes ω̂ = (2 + ω₀·ΔP̂)/T̂ empirically, the microscopic model derives ω from lattice bond statistics.
• The LLCP location is predicted from these microscopic parameters (set ∂f/∂x = ∂²f/∂x² = ∂³f/∂x³ = 0 — three equations in T_c, P_c, x_c).

Why this matters beyond aesthetics

1. Parameter parsimony. Holten has 25 background coefficients; Duška has 28; Caupin has ∼21. A pure microscopic model has ∼5–6 physical parameters (ΔU, ΔS, ΔV, ω₀, plus 1–2 thermal-expansion terms per state). That's a 4× reduction in parameter count.
2. Transferability. Calibrated parameters can be tested against TIP4P/2005 simulations (where you can measure ΔU, ΔS, ΔV cluster-by-cluster), giving an independent check.
3. Extensibility to mixtures. Adding a solute (glycerol, DMSO) is much easier in a lattice model — you add a third species with its own interactions — than in a fit polynomial. This is directly relevant to the cryoprotectant work.
4. The natural substrate for the glass extension. A lattice model with explicit configurational entropy and free volume is exactly what you need to bolt on a fictive-temperature freezing layer for LDA/HDA.

waterEoS implementation

This is actually simpler than the crossover work, because the form is so close to what's already there:

microlat_eos/                        # new module
├── params.py                        # ~6 microscopic params (ΔU, ΔS, ΔV, ω₀, αA, αB)
├── core.py                          # G(T, P, x) from lattice free energy
│                                    # mostly cribbed from caupin_eos/core.py
├── phase_diagram.py                 # LLCP from derivative conditions on f
└── README.md

The compute_batch function structure is identical to Caupin's — same Newton solve for x, same property derivations from G derivatives. Only the form of ΔGBA(T, P) and ω(T, P) change, and both become simpler:

def compute_GBA(dT_hat, dP_hat, pset):
    # Replace Caupin's 5-term polynomial with the physical form:
    P = pset
    return (P.delta_U / (P.R * P.Tc)) \
         - (1.0 + dT_hat) * (P.delta_S / P.R) \
         + (P.Pc + dP_hat * P.P_scale_MPa) * 1e6 * P.delta_V \
             / (P.R * P.Tc) \
         + small_thermal_terms(dT_hat, dP_hat, pset)

Effort estimate: ∼1 week for numpy + tests + phase diagram; +2–3 days Rust parity. Calibration (fitting the ∼6 microscopic params against IAPWS-95 + Holten/Caupin + Kim 2020 data) is probably the time-consuming part — call it another week.

4. Combining them: the v1 advanced model

The natural composition:

Pragmatic build order

1. Microscopic-only v0 (no crossover yet, still mean-field) — gets you a 6-parameter physically-grounded EoS. Useful immediately as microlat2026 in waterEoS.
2. Add crossover — patch on the Schofield RLM near C′ with the Belyakov-Kiselev variable. Now you have correct Ising exponents AND microscopic grounding.
3. Validate against Kim 2020. The moment of truth — does the model reproduce the experimental κ_T peak shape with γ ≈ 1.24 without retuning, or do you need to add a free background polynomial to mop up?

5. Validation strategy

Build the test suite around four benchmarks:

The last one is the killer test — if a fitted ΔU disagrees with TIP4P/2005 by 50%, the "microscopic grounding" is illusory. Worth knowing.

6. Genuine research uncertainty

• Field mixing for water is contested. Bertrand-Anisimov mix T, P, and μ. Other groups (Holten, Caupin) just mix T and P. This affects the Ising prefactors but not the exponents. Probably needs to be a fitted choice.
• The Ginzburg number κ for water's LLCP is unknown because the LLCP itself is hypothetical. Treat it as a fit parameter, with the understanding that current data only constrain it weakly.
• Whether two states is enough. If VHDA is genuinely a third state (vs an HDA polymorph), the entire family of two-state models — including this proposed advanced one — has a ceiling.

7. Honest recommendation

Key references

• Fuentevilla, D. A. & Anisimov, M. A. "Scaled equation of state for supercooled water near the liquid-liquid critical point." Phys. Rev. Lett. 97, 195702 (2006).
• Bertrand, C. E. & Anisimov, M. A. "Peculiar thermodynamics of the second critical point in supercooled water." J. Phys. Chem. B 115, 14099 (2011).
• Caupin, F. & Anisimov, M. A. "Minimal microscopic model for liquid polyamorphism and water-like anomalies" (arXiv:2104.08117, 2021).
• Caupin, F. & Anisimov, M. A. "Thermodynamics of supercooled and stretched water: unifying two-structure description and liquid-vapor spinodal." J. Chem. Phys. 151, 034503 (2019).
• Holten, V., Sengers, J. V. & Anisimov, M. A. "Equation of state for supercooled water at pressures up to 400 MPa." J. Phys. Chem. Ref. Data 43, 014101 (2014).
• Duška, M. "Two-state equation of state of water above the spinodal." J. Chem. Phys. 152, 174501 (2020).
• Kim, K. H. et al. "Experimental observation of the liquid-liquid transition in bulk supercooled water under pressure." Science 370, 978 (2020).
• Belyakov, M. Yu. & Kiselev, S. B. "Crossover behavior of the susceptibility and the specific heat near a second-order phase transition." Physica A 190, 75 (1992).