Correction for issue #3862

ComponentModifiedFurstElectrolyteEosMod2004.dFBorndN previously added FBornD() directly to the mole derivative. FBornD() is a derivative with respect to relative permittivity, not mole number, and scales with phase amount. The resulting chemical potentials and fugacity coefficients changed when all mole numbers and the phase volume were multiplied by the same factor.

The correction removes this unweighted term. It retains the 2004 variant’s existing convention of zero solvent-permittivity composition derivatives, consistent with its Born mixed derivatives. Public method signatures, ion diameters and fitted interaction parameters are unchanged. Calculated fugacity coefficients and flash results can change, especially for large phase amounts; the old dependence on the chosen amount basis was erroneous.

Dividing FBornD() by total moles would restore scaling but would still not be the chain-rule derivative. It is therefore not the correction used here.

Derivation from the implemented free energy

For composition-independent ionic diameters, the Furst-based implementations use

\[F_B=\frac{A_B}{RT}=C(T)X_B\left(\frac{1}{\varepsilon_s}-1\right),\qquad C(T)=\frac{N_Ae^2}{4\pi\varepsilon_0RT},\qquad X_B=\sum_j\frac{n_jz_j^2}{d_j}.\]

Here $n_j$ is the amount in mol, $d_j$ the diameter in m, $z_j$ the signed charge number, $T$ the temperature in K, $R$ the molar gas constant, $N_A$ Avogadro’s constant, $e$ the elementary charge and $\varepsilon_0$ the vacuum permittivity. $\varepsilon_s$ is the dimensionless solvent relative permittivity. The sum includes ions; neutral components have $z_j=0$. $F_B$ is extensive and has units of mol under this molar convention; its mole derivative is dimensionless.

Direct differentiation at fixed temperature, total volume and other amounts gives

\[\frac{\mu_i^B}{RT}=\frac{\partial F_B}{\partial n_i}=C(T)\left[\left(\frac{1}{\varepsilon_s}-1\right)\frac{z_i^2}{d_i}-\frac{X_B}{\varepsilon_s^2}\frac{\partial\varepsilon_s}{\partial n_i}\right].\]

In code this is FBornX() * XBorni + FBornD() * solventdiElectricdn. For the solvent mole-average mixing rule,

\[\varepsilon_s=\frac{\sum_{j\in S}n_j\varepsilon_j(T)}{n_S},\qquad n_S=\sum_{j\in S}n_j,\] \[\frac{\partial\varepsilon_s}{\partial n_i}=\begin{cases}(\varepsilon_i-\varepsilon_s)/n_S,&i\in S,\\0,&i\notin S.\end{cases}\]

$S$ denotes the neutral-solvent components included by that mixing rule. $X_B$ scales with amount, while the dielectric derivative scales inversely with amount. Their product is intensive. A neutral component can therefore have a nonzero Born chemical-potential contribution in a mixed solvent even though its charge is zero. For one neutral solvent, this composition derivative vanishes.

The MM implementation uses Born radii rather than diameters, with the equivalent $8\pi$ prefactor. Radius/diameter conventions must be kept consistent. If radii depend on temperature or composition, their derivatives also belong in the corresponding chain rule. A density-dependent permittivity additionally gives a Born pressure contribution through $-RT(\partial F_B/\partial V)$.

These equations are a direct derivation from NeqSim’s implemented free energy; they are not a claim that one dielectric mixing rule is universally accurate.

The audit starts from master commit a7f0e929559b27e70c841b763a77b6bf04c964ff and checks the Born implementations and their inheritance paths.

Implementation Unweighted extensive Born term? Evidence/scope
Furst Mod2004 Yes, corrected Direct derivative, Euler identity, scaling and two-phase flash regression
Base Furst No Includes the solvent-permittivity chain rule; scaling regression
Electrolyte CPA and Statoil No Inherit the base component Born derivative; scaling regression
Electrolyte CPA Advanced No Uses the inherited chain rule and overridden Born radii/sum; scaling regression
Electrolyte CPA Old No in this path Source inspection: inherits the base Born derivative
Electrolyte CPA MM No Explicit chain rule; scaling regression for molar, volume, Looyenga and Lichtenecker solvent mixing
Ordinary SRK, PR, CPA and non-electrolyte SAFT models This specific defect does not apply Do not inherit the faulty Mod2004 Born implementation
Pitzer / activity-coefficient models This specific defect does not apply Separate implementation path; not a full derivative qualification

Passing the scaling checks does not establish complete derivative consistency or accuracy against experimental electrolyte data. It answers the narrower question of whether the same extensive term contaminates the chemical potential.

What modern electrolyte-model development requires

The literature supports explicit ion-solvation contributions with a documented permittivity model, while also showing that their physical interpretation and parameterization remain active research topics:

The source claims above were checked against the publisher/institutional abstracts. The implementation recommendations below follow from differentiating the model energy and are not presented as verbatim prescriptions from those papers.

Recommended development order:

  1. Enforce extensivity of free energy and intensivity of chemical potentials over multiple amount scales, including both solvents and ions.
  2. Use one free-energy expression and differentiate the actual selected permittivity and radius models consistently with respect to $T$, $V$ and every component amount. Validate analytical derivatives against finite differences or an independent automatic-differentiation calculation.
  3. Check mixed-derivative symmetry, the Euler identity and Gibbs-Duhem behavior, including solvent derivatives. Treat scale invariance as necessary, not sufficient.
  4. Validate water activity, mean ionic activity, osmotic coefficients, densities, enthalpy/heat capacity and VLE/LLE/SLE against independent data over the intended salt, solvent, temperature and pressure ranges. Declare standard states and concentration bases explicitly.
  5. Only then compare alternative dielectric decrement, density dependence, ion-pairing or ion-specific radius models, with parameter revalidation.

Remaining Mod2004 limitation

The 2004 phase recalculates solvent permittivity from temperature and neutral composition, but its derivative code suppresses that dependence. Removing the spurious extensive term fixes issue #3862; it does not make the legacy variant the full derivative of its recomputed energy for mixed solvents or thermal perturbations. Restoring these dependencies requires coordinated changes to the Born and long-range derivatives and validation of affected properties. Do not infer modern mixed-solvent/caloric qualification from this correction.

Regression coverage

ElectrolyteBornExtensivityTest checks water/methanol/NaCl liquid states at amount multipliers from $10^{-6}$ to $10^6$. It checks extensive Born energy, intensive Born mole derivatives and full log fugacity coefficients, and the Born Euler identity for these volume-independent solvent-permittivity models. It also checks the Mod2004 Born derivative against an independent energy finite difference in a single-solvent state and repeats the methane/water/NaCl TP flash from issue #3862 at several feed amounts.

The single-solvent finite difference intentionally holds the solvent dielectric constant unchanged at the selected temperature; it does not qualify the omitted mixed-solvent or thermal chain rules described above.