Closures

BallonePastoreGalliGazzillo <: Closure

Implements the Ballone-Pastore-Galli-Gazzillo closure $b(r) = (1 + s \gamma(r))^{1/s} - \gamma(r) - 1 $. Here, $s$ is a free parameter that can be determined with thermodynamic consistency.

Example:

closure = BallonePastoreGalliGazzillo(1.5)

References:

Ballone, P., et al. "Additive and non-additive hard sphere mixtures: Monte Carlo simulation and integral equation results." Molecular Physics 59.2 (1986): 275-290.

BomontBretonnet <: Closure

Implements the Bomont-Bretonnet closure $b(r) = \sqrt{1+2\gamma^*(r) + f \gamma^*(r)^2} - \gamma^*(r) - 1 $. Here $\gamma^* = \gamma - u_{LR}$ , in which $ u_{LR}$ is the long range tail of the potential, and $f$ is a free parameter that can be determined with thermodynamic consistency.

Example:

closure = BomontBretonnet(0.5)

References:

Bomont, J. M., and J. L. Bretonnet. "A self-consistent integral equation: Bridge function and thermodynamic properties for the Lennard-Jones fluid." The Journal of chemical physics 119.4 (2003): 2188-2191.

CarbajalTinoko <: Closure

Implements the Carbajal-Tinoko closure $e(r;\alpha)\left[(2-y(r))e^{y(r)}-2-y(r)\right]/\left(e^{y(r)}-1\right)$ where $e(r;\alpha)=3\exp(\alpha r)$ for $\alpha <0$ and $e(r;\alpha) = 3+\alpha$ otherwise.

Example:

closure = CarbajalTinoko(0.4)

References:

Carbajal-Tinoco, Mauricio D. "Thermodynamically consistent integral equation for soft repulsive spheres." The Journal of chemical physics 128.18 (2008).

CharpentierJackse <: Closure

Implements the Charpentier-Jackse closure $b(r) = \frac{1}{2\alpha}\left(\sqrt{1+4\alpha\gamma^*(r) } - 2\alpha\gamma^*(r) - 1\right) $. Here $\gamma^* = \gamma - u_{LR}$ , in which $ u_{LR}$ is the long range tail of the potential, and $\alpha$ is a free parameter that can be determined with thermodynamic consistency.

Example:

closure = CharpentierJackse(0.5)

References:

Charpentier, I., and N. Jakse. "Exact numerical derivatives of the pair-correlation function of simple liquids using the tangent linear method." The Journal of Chemical Physics 114.5 (2001): 2284-2292.

ChoudhuryGhosh <: Closure

Implements the Choudhury-Ghosh closure $b(r) = -\frac{-\gamma^*(r)^2}{2(1+\alpha \gamma^*(r))} $ for $\gamma^*(r) >0$, and $b(r)=-\gamma^*(r)^2/2$ otherwise. Here $\gamma^* = \gamma - u_{LR}$ , in which $ u_{LR}$ is the long range tail of the potential, and $\alpha$ is a free parameter that is determined by an empirical relation.

Example:

α(ρ) = 1.01752 - 0.275ρ # see the reference for this empirical relation
closure = ChoudhuryGhosh(α(0.4))

References:

Choudhury, Niharendu, and Swapan K. Ghosh. "Integral equation theory of Lennard-Jones fluids: A modified Verlet bridge function approach." The Journal of chemical physics 116.19 (2002): 8517-8522.

DuhHaymet <: Closure

Implements the Duh-Haymet closure $b(r) = \frac{-\gamma^*(r)^2}{2\left[1+\left(\frac{5\gamma^*(r)+11}{7\gamma^*(r)+9}\right)\gamma^*(r)\right]} $ for $\gamma^*(r) >0$, and $b(r)=-\gamma^*(r)^2/2$ otherwise. Here $\gamma^* = \gamma - u_{LR}$ , in which $ u_{LR}$ is the long range tail of the potential,

Example:

closure = DuhHaymet()

References:

ExtendedRogersYoung <: Closure

Implements the extended Rogers-Young closure $b(r) = \ln(a\phi(r)^2 + \phi(r) + 1) - γ(r) $. Here, $\phi(r) = \frac{\exp(f(r)\gamma(r)) - 1}{f(r)}$, and $f(r)=1-\exp(-\alpha r)$, in which $\alpha$ is a free parameter, that may be chosen such that thermodynamic consistency is achieved. Example:

closure = ExtendedRogersYoung(0.5, 0.5) # order is α, a

References: J. Chem. Phys. 128, 184507 (2008)

HypernettedChain

Implements the Hypernetted Chain closure $c(r) = (f(r)+1)\exp(\gamma(r)) - \gamma(r) - 1$, or equivalently $b(r) = 0$.

Example:

closure = HypernettedChain()

Khanpour <: Closure

Implements the Khanpour closure $b(r) = \frac{1}{\alpha}\ln(1+\alpha\gamma(r)) - \gamma $. Here $\alpha$ is a free parameter that can be determined with thermodynamic consistency.

Example:

closure = Khanpour(0.5)

References:

Khanpour, Mehrdad. "A unified derivation of Percus–Yevick and hyper-netted chain integral equations in liquid state theory." Molecular Physics 120.5 (2022): e2001065.

Lee <: Closure

Implements the Lee closure $b(r) = -\frac{\zeta\gamma^*(r)^2}{2} \left( 1- \frac{\phi \alpha \gamma^*(r)}{1 + \alpha\gamma^*(r)} \right) $. Here $\gamma^* = \gamma + ρ f(r)/2$ , in which $ f(r)$ is the Mayer-f function and $\rho$ the density. Additionally, $\zeta$,$\phi$, and, $\alpha$ are free parameters that can be determined with thermodynamic consistency or zero-separation theorems.

Example:

closure = Lee(1.073, 1.816, 1.0, 0.4) # ζ, ϕ, α, ρ

References:

Lee, Lloyd L. "An accurate integral equation theory for hard spheres: Role of the zero‐separation theorems in the closure relation." The Journal of chemical physics 103.21 (1995): 9388-9396.

MartynovSarkisov <: Closure

Implements the Martynov-Sarkisov Closure b(r) = -\sqrt{1+2\gamma}-1-\gamma, which is constructed for the 3d hard-sphere liquid.

Example:

closure = MartynovSarkisov()

References:

Martynov, G. A., and G. N. Sarkisov. "Exact equations and the theory of liquids. V." Molecular Physics 49.6 (1983): 1495-1504.

ModifiedHypernettedChain <: Closure

Implements the Modified Hypernetted Chain closure $b(r) = b_{HS}(r) $. Here $b_{HS}(r/σ)=\left((a_1+a_2x)(x-a_3)(x-a_4)/(a_3 a_4)\right)^2$ for $x<a_4$ and $b_{HS}(r)=\left(A_1 \exp(-a_5(x-a_4))\sin(A_2(x-a_4))/r\right)^2$ is the hard sphere bridge function found in Malijevský & Labík. The parameters are defined as

\[x = r/σ-1\]

\[A_1 = (a_1+a_2 a_4)(a_4-a_3)(a_4+1)/(A_2 a_3 a_4)\]

\[A_2 = \pi / (a_6 - a_4 - 1)\]

\[a_1 = \eta (1.55707 - 1.85633\eta) / (1-\eta)^2\]

\[a_2 = \eta (1.28127 - 1.82134\eta) / (1-\eta)\]

\[a_3 = (0.74480 - 0.93453\eta)\]

\[a_4 = (1.17102 - 0.68230\eta)\]

\[a_5 = 0.15975/\eta^2\]

\[a_6 = (2.69757 - 0.86987\eta)\]

and $\eta$ is the volume fraction of the hard sphere reference system. This closure only works for single component systems in three dimensions. By default, $\sigma = 1.0$.

Example:

closure = ModifiedHypernettedChain(0.4)
closure = ModifiedHypernettedChain(0.4; sigma=0.8)

References:

Lado, F. "Perturbation correction for the free energy and structure of simple fluids." Physical Review A 8.5 (1973): 2548.

Malijevský, Anatol, and Stanislav Labík. "The bridge function for hard spheres." Molecular Physics 60.3 (1987): 663-669.

ModifiedVerlet <: Closure

Implements the modified Verlet Closure $b(r) = -\frac{\gamma^2(r)/2}{1+\alpha \gamma(r)}$. If $\gamma(r)<0$, the closure reads $-\gamma^2/2$. Example:

closure = ModifiedVerlet(0.2)

References:

Verlet, Loup. "Integral equations for classical fluids: I. The hard sphere case." Molecular Physics 41.1 (1980): 183-190.

PercusYevick

Implements the Percus-Yevick closure $c(r) = f(r)(1+\gamma(r))$, or equivalently $b(r) = \ln(1 + \gamma(r)) - γ(r)$.

Example:

closure = PercusYevick()

RogersYoung <: Closure

Implements the Rogers-Young closure $b(r) = \ln(\frac{\exp(f(r)\gamma(r)) - 1}{f(r)} + 1) - γ(r) $. Here $f(r)=1-\exp(-\alpha r)$, in which $\alpha$ is a free parameter, that may be chosen such that thermodynamic consistency is achieved. Example:

closure = RogersYoung(0.5)

References:

SoftCoreMeanSpherical <: Closure

Implements the soft core mean spherical closure $b(r) = \ln(\gamma^*(r) + 1) - \gamma^*(r)$. Here $\gamma^* = \gamma - u_{LR}$ , in which $u_{LR}$ is the long range tail of the potential.

Example:

closure = SoftCoreMeanSpherical()

References:

Verlet <: Closure

Implements the Verlet Closure $b(r) = -\frac{A\gamma^2(r)/2}{1+B \gamma(r)/2}$, where by default $A=1$ and $B=8/5$. These values are tuned by the virial coefficients of the 3d hard sphere liquid.

Example:

closure = Verlet()
closure = Verlet(A=3.0, B=4.0)

References:

Verlet, Loup. "Integral equations for classical fluids: I. The hard sphere case." Molecular Physics 41.1 (1980): 183-190.

VompeMartynov <: Closure

Implements the Vompe-Martynov closure $b(r) = \sqrt{1+2\gamma^*(r) } - \gamma^*(r) - 1 $. Here $\gamma^* = \gamma - u_{LR}$ , in which $ u_{LR}$ is the long range tail of the potential, and $\alpha$ is a free parameter that can be determined with thermodynamic consistency.

Example:

closure = VompeMartynov()

References:

Vompe, A. G., and G. A. Martynov. "The bridge function expansion and the self‐consistency problem of the Ornstein–Zernike equation solution." The Journal of chemical physics 100.7 (1994): 5249-5258.

ZerahHansen <: Closure

Implements the Zerah-Hansen (HMSA) (HNC-SMSA) closure $b(r) = \ln(\frac{\exp(f(r)\gamma^*(r)) - 1}{f(r)} + 1) - γ^*(r) $. Here $\gamma^* = \gamma - u_{LR}$ , in which $ u_{LR}$ is the long range tail of the potential, and $f(r)=1-\exp(-\alpha r)$, in which $\alpha$ is a free parameter, that may be chosen such that thermodynamic consistency is achieved.

Example:

closure = ZerahHansen(0.5)

References:

Zerah, Gilles, and Jean‐Pierre Hansen. "Self‐consistent integral equations for fluid pair distribution functions: Another attempt." The Journal of chemical physics 84.4 (1986): 2336-2343.