Thol2016LJ <: LJPotentialModel
Thol2016LJ()

Lennard-Jones Reference equation of state. Valid from 0.5 < T/Tc < 7 and pressures up to p/pc = 500.

τᵢ = 1.32/T
δᵢ = 1/0.31V
a⁰ᵢ(δ,τ) = log(δᵢ) + 1.5log(τᵢ) - 1.515151515τᵢ + 6.262265814
a⁰(δ,τ,z) = ∑xᵢ(a⁰ᵢ + log(xᵢ))
τ = 1.32ϵ/T
δ = n(Nₐσ^3)/0.31V
aʳ(δ,τ)  = aʳ₁+ aʳ₂ + aʳ₃ + aʳ₄
aʳ₁(δ,τ)  = ∑nᵢδ^(dᵢ)τ^(tᵢ), i ∈ 1:6
aʳ₂(δ,τ)  = ∑nᵢexp(-δ^cᵢ)δ^(dᵢ)τ^(tᵢ), i ∈ 7:12
aʳ₃(δ,τ)  = ∑nᵢexp(-ηᵢ(δ - εᵢ)^2 - βᵢ(τ - γᵢ)^2)δ^(dᵢ)τ^(tᵢ), i ∈ 13:23

References

1. Thol, M., Rutkai, G., Köster, A., Lustig, R., Span, R., & Vrabec, J. (2016). Equation of state for the Lennard-Jones fluid. Journal of physical and chemical reference data, 45(2), 023101. doi:10.1063/1.4945000

Johnson1993LJ <: LJPotentialModel
Johnson1993LJ()

Description

Johnson equation of state for the Lennard-Jones fluid. Default reference LJ model for softSAFT

References

1. Johnson, J. K., Zollweg, J. A., & Gubbins, K. E. (1993). The Lennard-Jones equation of state revisited. Molecular physics, 78(3), 591–618. doi:10.1080/00268979300100411

PeTSLJ <: LJPotentialModel
PeTSLJ()

Description

Perturbed, Truncated and Shifted (PeTS) Equation of State. Unscaled model.

References

1. Heier, M., Stephan, S., Liu, J., Chapman, W. G., Hasse, H., & Langenbach, K. (2018). Equation of state for the Lennard-Jones truncated and shifted fluid with a cut-off radius of 2.5 σ based on perturbation theory and its applications to interfacial thermodynamics. Molecular Physics, 116(15–16), 2083–2094. doi:10.1080/00268976.2018.1447153

FeANNMie(λr, λa; solid = false)

Input arguments

• λa: Real - Atractive range parameter (no units)
• λr: Real - Repulsive range parameter (no units)

Keyword arguments

• solid: if set to true, it will load the weights for the Fe-ANN(s)-Mie model instead of the weights for the Fe-ANN-Mie.

Description

Free Energy Artificial Neural Network equation of state for the Mie fluid. There are two available models, that can be selected with the solid keyword argument:

• Fe-ANN-Mie: model trained on VLE data of the Mie fluid
• Fe-ANN(S)-Mie: Trained model using fluid and crystal Mie data

References

1. Chaparro, G., & Müller, E. A. (2023). Development of thermodynamically consistent machine-learning equations of state: Application to the Mie fluid. The Journal of Chemical Physics, 158(18), 184505. doi:10.1063/5.0146634
2. Chaparro, G., & Müller, E. A. (2024). Development of a Helmholtz free energy equation of state for fluid and solid phases via artificial neural networks. Communications Physics, 7(1). [doi:10.1038/s42005-024-01892-3]https://doi.org/10.1038/s42005-024-01892-3)
3. Chaparro, G., & Müller, E. A. (2025). Fitting a square peg in a round hole: parameterisation of quasi-spherical molecules employing the Mie potential. Molecular Systems Design & Engineering, 10(8), 620–634. doi:10.1039/d5me00048c

Pohl2023Mie6 <: MiePotentialModel
Pohl2023Mie6(λr)

Description

Multiparameter Equation of state for the ν-6 Mie fluid, with ν (c) between 11 and 13.

References

Pohl, S., Fingerhut, R., Thol, M., Vrabec, J., & Span, R. (2023). Equation of state for the Mie (λr,6) fluid with a repulsive exponent from 11 to 13. The Journal of Chemical Physics, 158(8), 084506. doi:10.1063/5.0133412

abstract type CorrespondingStatesModel <: EoSModel end

Abstract type containing corresponding states (CS) models. corresponding states models are defined as:

Aʳᵉˢ(T,v,x)/RT = Aʳᵉˢ₀(T₀,v₀)/R₀T₀        T₀ = T/f ,  v₀ = v/h

In particular, the code implementation is just two methods.

function a_res(model::CorrespondingStatesModel,V,T,z)
    reference = reference_model(model)
    scaling = scaling_model(model)
    return a_res_ecs(reference,scaling,V,T,z)
end

function a_res_ecs(refmodel,scalingmodel,V,T,z)
    f,h = shape_factors(scalingmodel,refmodel,T,z)
    T0 = T/f
    V0 = V/h
    return a_res(refmodel,V0,T0,z)
end

The following functions are required:

• reference_model : provides a single-component, helmholtz equation of state, used as reference for the shape their a_res V-T surface.
• scaling_model : provides a struct that defines the scaling between the real fluid and the reference fluid.
• shape_factors : the scaling model needs to define f,h = shape_factors(scaling::MyModel,reference,T,z).

Some reference models may carry its own scaling (or no scaling at all). For those cases, you may need to overload a_res_ecs

shape_factors(model::CorrespondingStatesModel,T,z=SA[1.0])
shape_factors(scaling_model,reference_model,T,z)

returns f,h scaling factors, used by subtypes of CorrespondingStatesModel. T is the scaled temperature. To calculate the shape factors given a reference temperature (T0 = T/f), use inverse_scaling_factors

NoECSScaling <: EoSModel

Input parameters

None

Description

Null scaling model, to be used along a reference model. temperature and shape factors are equal to one.

SPUNGScaling(components, reference_component, model, refmodel;
shape_userlocations = nothing,
shaperef_userlocations = nothing,
verbose = false)

SPUNGScaling(shape_model, shaperef_model)

Input models

• model: cubic equation of state that describes the fluid (or mixture) of interest, the shape equation (for example SRK or PR). A cubic built beforehand can also be passed; it is then used as it is.
• refmodel: cubic equation of state that represents the reference fluid. It must be the same equation as model: same functional form, same α(T) function and same mixing and translation options. Only the substance, and the parameters describing it, may differ (see the notes below).

Input parameters

• components: substances of the fluid or mixture of interest.
• reference_component: the reference substance, as named in the cubic-equation database.
• shape_userlocations: alternative parameter source (files or values) for the cubic of the fluid of interest. Replaces database critical constants, acentric factors or binary interaction parameters, and therefore changes the scale factors.
• shaperef_userlocations: the same, for the cubic of the reference fluid.
• verbose: prints information about the parameters used while the model is built. It does not change any result.

Description

Scaling model of the SPUNG equation of state, to be used with ECS. Its role is to say which state of the reference fluid corresponds to a given state of the fluid of interest: a state (T, v, x) is mapped to the reference state

T₀ = T/f
v₀ = v/h

where f and h are the temperature and volume scale factors. They are obtained from the cubic equation of state, written as

P = RT/(v-b) - a(T)/((v+δ₁b)(v+δ₂b))

Two states are considered corresponding when the two dimensionless groups of the cubic coincide,

Γ = a(T)/(b·R·T)       β = b/v

For a cubic these two groups fully determine the reduced residual Helmholtz energy, so equality of Γ and β means equal reduced residual Helmholtz energy. This gives

h = v/v₀ = b/b₀
f = T/T₀ = [a(T)/b] / [a₀(T₀)/b₀]

where a, b belong to the fluid of interest and a₀, b₀ to the reference fluid, both from the same cubic equation. f appears on both sides (T₀ = T/f), so it is the solution of an implicit equation. For pure fluids described by Soave-type α(T) functions (SRK, PR) the solution is available in closed form; in all other cases it is found iteratively. Both routes give the same value.

For mixtures, a and b are the mixture parameters given by the mixing rule of the cubic (van der Waals one-fluid rules by default), so the composition dependence of the scale factors, and the effect of the binary interaction parameters kᵢⱼ and lᵢⱼ, come entirely from the cubic. The scaling model supplies only the mapping; the accuracy of the final Helmholtz energy comes from the reference equation.

Notes

• If the fluid of interest is the reference fluid, and both cubics use the same parameters, f = h = 1 and the reference equation is recovered unchanged.
• The two cubics must be the same equation because equality of Γ and β implies equal Helmholtz energy only if both fluids obey the same functional form (same δ₁, δ₂, α(T) and mixing rule).

Using different cubic equations for the fluid of interest and the reference fluid is not currently supported.

References

1. Mollerup, J. (1998). Unification of the two-parameter equation of state and the principle of corresponding states. Fluid Phase Equilibria, 148(1–2), 1–19. doi:10.1016/s0378-3812(98)00230-1
2. Michelsen, M. L., & Mollerup, J. M. (2007). Thermodynamic Models: Fundamentals and Computational Aspects, 2nd ed. Tie-Line Publications.

LJScaling <: LJScalingModel
LJScaling(components;
userlocations = String[],
verbose = false)

Input parameters

• sigma: Single Parameter (Float64) - particle size [Å]
• epsilon: Single Parameter (Float64) - dispersion energy [K]
• Mw: Single Parameter (Float64) - Molecular Weight [g·mol⁻¹]
• k: Pair Parameter (Float64) (optional) - Binary Interaction Parameter (no units)
• l: Pair Parameter (Float64) (optional) - Binary Interaction Parameter (no units)

Model Parameters

• Mw: Single Parameter (Float64) - Molecular Weight [g·mol⁻¹]
• sigma: Pair Parameter (Float64) - particle size [m]
• epsilon: Pair Parameter (Float64) - dispersion energy [K]

Description

Lennard-Jones corresponding-states scaling model, used together with a Lennard-Jones reference equation of state.

Their shape factors are defined as:

σ³ = Σᵢ Σⱼ xᵢ xⱼ σᵢⱼ³
ϵmix = (1/σ³) Σᵢ Σⱼ xᵢ xⱼ σᵢⱼ³ ϵᵢⱼ
σᵢⱼ = (1 - lᵢⱼ)(σᵢ + σⱼ)/2        (Lorentz)
ϵᵢⱼ = (1 - kᵢⱼ)√(ϵᵢ ϵⱼ)           (Berthelot)
f = ϵmix
h = N_A σ³

LJScaling <: LJScalingModel
LJScaling(components;
userlocations = String[],
verbose = false)

Input parameters

• sigma: Single Parameter (Float64) - particle size [Å]
• epsilon: Single Parameter (Float64) - dispersion energy [K]
• Mw: Single Parameter (Float64) - Molecular Weight [g·mol⁻¹]
• k: Pair Parameter (Float64) (optional) - Binary Interaction Parameter (no units)
• l: Pair Parameter (Float64) (optional) - Binary Interaction Parameter (no units)
• segment: Single Parameter (Float64) - Number of segments (no units)

Model Parameters

• Mw: Single Parameter (Float64) - Molecular Weight [g·mol⁻¹]
• sigma: Pair Parameter (Float64) - particle size [m]
• segment: Single Parameter (Float64) - Number of segments (no units)
• epsilon: Pair Parameter (Float64) - dispersion energy [K]

Description

Lennard-Jones corresponding-states scaling model, used together with a Lennard-Jones reference equation of state. The volume shape factor is additionally scaled by the weighted segment parameter Their shape factors are defined as:

m̄    = Σᵢ xᵢ mᵢ
σ³   = Σᵢ Σⱼ xᵢ xⱼ σᵢⱼ³
ϵmix = (1/σ³) Σᵢ Σⱼ xᵢ xⱼ σᵢⱼ³ ϵᵢⱼ
σᵢⱼ  = (1 - lᵢⱼ)(σᵢ + σⱼ)/2        (Lorentz)
ϵᵢⱼ  = (1 - kᵢⱼ)√(ϵᵢ ϵⱼ)           (Berthelot)
f    = ϵmix
h    = m̄ N_A σ³

ExtendedCorrespondingStates <: CorrespondingStatesModel

function ECS(scaling,reference,idealmodel)

Input Models

• scaling: determines the scales of temperature and volume for the model.
• reference: determines the shape of the helmholtz surface (V,T).
• idealmodel: determines the ideal gas model.

Description

Generic instance of a CorrespondingStatesModel EoS. It can be used to mix and match scaling and reference models.

SPUNG(components;
reference_model = PropaneRef(),
shape_model = SRK,
shaperef_model = shape_model,
idealmodel = BasicIdeal,
reference_component = only(Clapeyron.component_list(reference_model)),
ideal_userlocations = String[],
shape_userlocations = String[],
shaperef_userlocations = String[],
verbose = false,
reference_state = nothing)

Input models

• reference_model: accurate pure-fluid equation of state (normally a multiparameter Helmholtz equation) used as athe reference model. It must be already created.
• shape_model: cubic equation of state that defines the mapping between the fluid of interest and the reference fluid (for example SRK or PR).
• shaperef_model: cubic equation that represents the reference fluid in the mapping. Default: the same as shape_model. It must be the same model type as shape_model. It is created with components equal to reference_component.

Description

SPUNG: State Research Program for Utilization of Natural Gas

ECS method. The residual Helmholtz energy of a pure fluid or mixture is taken from a very accurate equation of state for a pure reference fluid, evaluated at a corresponding state:

Aʳᵉˢ(T,v,x)/RT = Aʳᵉˢ₀(T₀,v₀)/R₀T₀        T₀ = T/f ,  v₀ = v/h

The scale factors f and h depend on temperature and composition, and are calculated from a cubic equation of state (by default SRK) with the SPUNGScaling model: h is the ratio of the covolumes b/b₀, and f follows from equating a(T)/(bRT) for the fluid of interest and for the reference fluid. In mixtures the parameters a and b come from the cubic's mixing rules. If the reference equation is itself the same cubic, SPUNG reduces to that cubic.

The ideal-gas part is independent of the scaling and is given by idealmodel.

Model Construction Examples

# Default: SRK shape equation, propane reference fluid
model = SPUNG(["methane","carbon dioxide"])

# Another reference fluid (any pure-fluid Helmholtz equation)
model = SPUNG(["carbon dioxide","nitrogen"], reference_model = SingleFluid("methane"))

# Another shape equation (used for the mixture and for the reference fluid)
model = SPUNG(["methane","ethane"], shape_model = PR)

# Realistic ideal-gas heat capacities, for caloric properties
model = SPUNG(["methane","ethane"], idealmodel = ReidIdeal)

# Custom parameters (e.g. SPUNG-specific interaction parameters) for the mixture's cubic
model = SPUNG(["methane","ethane"], shape_userlocations = ["path/to/my/db"])

References

1. Mollerup, J. (1998). Unification of the two-parameter equation of state and the principle of corresponding states. Fluid Phase Equilibria, 148(1–2), 1–19. doi:10.1016/s0378-3812(98)00230-1
2. Michelsen, M. L., & Mollerup, J. M. (2007). Thermodynamic Models: Fundamentals and Computational Aspects, 2nd ed. Tie-Line Publications.

KolafaNezbedaModel <: EoSModel

KolafaNezbeda(components;
idealmodel = BasicIdeal,
userlocations = String[],
ideal_userlocations = String[],
reference_state = nothing,
verbose = false)

Input parameters

• Mw: Single Parameter (Float64) - Molecular Weight [g·mol⁻¹]
• sigma: Single Parameter (Float64) - Lennard-Jones size parameter [Å]
• epsilon: Single Parameter (Float64) - Lennard-Jones energy parameter [K]
• k: Pair Parameter (Float64) (optional) - Binary Interaction Parameter (no units)

Model Parameters

• Mw: Single Parameter (Float64) - Molecular Weight [g·mol⁻¹]
• sigma: Pair Parameter (Float64) - Mixed Lennard-Jones size parameter [m]
• epsilon: Pair Parameter (Float64) - Mixed Lennard-Jones energy parameter [K]

Input models

• idealmodel: Ideal Model

Description

Kolafa-Nezbeda equation of state for the Lennard-Jones fluid. Scaled via LJScaling

References

1. Kolafa, J., & Nezbeda, I. (1994). The Lennard-Jones fluid: An accurate analytic and theoretically-based equation of state. Fluid Phase Equilibria, 100, 1–34. doi:10.1016/0378-3812(94)02573-G

PeTSModel <: EoSModel

PeTS(components; 
idealmodel = BasicIdeal,
userlocations = String[],
ideal_userlocations = String[],
reference_state = nothing,
verbose = false)

Input parameters

• Mw: Single Parameter (Float64) - Molecular Weight [g·mol⁻¹]
• segment: Single Parameter (Float64) - Number of segments (no units)
• sigma: Single Parameter (Float64) - Segment Diameter [Å]
• epsilon: Single Parameter (Float64) - Reduced dispersion energy [K]
• k: Pair Parameter (Float64) (optional) - Binary Interaction Parameter (no units)

Model Parameters

• Mw: Single Parameter (Float64) - Molecular Weight [g·mol⁻¹]
• segment: Single Parameter (Float64) - Number of segments (no units)
• sigma: Pair Parameter (Float64) - Mixed segment Diameter [m]
• epsilon: Pair Parameter (Float64) - Mixed reduced dispersion energy[K]

Input models

• idealmodel: Ideal Model

Description

Perturbed, Truncated and Shifted (PeTS) Equation of State. Scaled via LJScaling

References

1. Heier, M., Stephan, S., Liu, J., Chapman, W. G., Hasse, H., & Langenbach, K. (2018). Equation of state for the Lennard-Jones truncated and shifted fluid with a cut-off radius of 2.5 σ based on perturbation theory and its applications to interfacial thermodynamics. Molecular Physics, 116(15–16), 2083–2094. doi:10.1080/00268976.2018.1447153

LJRef <: CorrespondingStatesModel

LJRef(components; idealmodel = BasicIdeal, ljmodel = Thol2016LJ(), userlocations = String[], verbose = false)

Input parameters

• sigma: Single Parameter (Float64) - particle size [Å]
• epsilon: Single Parameter (Float64) - dispersion energy [K]
• Mw: Single Parameter (Float64) - Molecular Weight [g·mol⁻¹]
• k: Pair Parameter (Float64) (optional)

Model Parameters

• sigma: Pair Parameter (Float64) - particle size [m]
• epsilon: Pair Parameter (Float64) - dispersion energy [K]
• Mw: Single Parameter (Float64) - Molecular Weight [g·mol⁻¹]

Description

vdW-1 fluid scaling applied to a generic leonard jones model.

a_res(model,V,T,z) = a_res(reference_model(model),V/h,T/f)