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use crate::adsorption::fea_potential::calculate_fea_potential;
use crate::functional::HelmholtzEnergyFunctional;
use crate::geometry::Geometry;
use feos_core::EosUnit;
use libc::c_double;
use ndarray::{Array1, Array2, Axis as Axis_nd};
use quantity::{QuantityArray2, QuantityScalar};
use std::f64::consts::PI;
const DELTA_STEELE: f64 = 3.35;
/// A collection of external potentials.
#[derive(Clone)]
pub enum ExternalPotential<U> {
/// Hard wall potential: $V_i^\mathrm{ext}(z)=\begin{cases}\infty&z\leq\sigma_{si}\\\\0&z>\sigma_{si}\end{cases},~~~~\sigma_{si}=\frac{1}{2}\left(\sigma_{ss}+\sigma_{ii}\right)$
HardWall { sigma_ss: f64 },
/// 9-3 Lennard-Jones potential: $V_i^\mathrm{ext}(z)=\frac{2\pi}{45} m_i\varepsilon_{si}\sigma_{si}^3\rho_s\left(2\left(\frac{\sigma_{si}}{z}\right)^9-15\left(\frac{\sigma_{si}}{z}\right)^3\right),~~~~\varepsilon_{si}=\sqrt{\varepsilon_{ss}\varepsilon_{ii}},~~~~\sigma_{si}=\frac{1}{2}\left(\sigma_{ss}+\sigma_{ii}\right)$
LJ93 {
sigma_ss: f64,
epsilon_k_ss: f64,
rho_s: f64,
},
/// Simple 9-3 Lennard-Jones potential: $V_i^\mathrm{ext}(z)=\varepsilon_{si}\left(\left(\frac{\sigma_{si}}{z}\right)^9-\left(\frac{\sigma_{si}}{z}\right)^3\right),~~~~\varepsilon_{si}=\sqrt{\varepsilon_{ss}\varepsilon_{ii}},~~~~\sigma_{si}=\frac{1}{2}\left(\sigma_{ss}+\sigma_{ii}\right)$
SimpleLJ93 { sigma_ss: f64, epsilon_k_ss: f64 },
/// Custom 9-3 Lennard-Jones potential: $V_i^\mathrm{ext}(z)=\varepsilon_{si}\left(\left(\frac{\sigma_{si}}{z}\right)^9-\left(\frac{\sigma_{si}}{z}\right)^3\right)$
CustomLJ93 {
sigma_sf: Array1<f64>,
epsilon_k_sf: Array1<f64>,
},
/// Steele potential: $V_i^\mathrm{ext}(z)=2\pi m_i\xi\varepsilon_{si}\sigma_{si}^2\Delta\rho_s\left(0.4\left(\frac{\sigma_{si}}{z}\right)^{10}-\left(\frac{\sigma_{si}}{z}\right)^4-\frac{\sigma_{si}^4}{3\Delta\left(z+0.61\Delta\right)^3}\right),~~~~\varepsilon_{si}=\sqrt{\varepsilon_{ss}\varepsilon_{ii}},~~~~\sigma_{si}=\frac{1}{2}\left(\sigma_{ss}+\sigma_{ii}\right),~~~~\Delta=3.35$
Steele {
sigma_ss: f64,
epsilon_k_ss: f64,
rho_s: f64,
xi: Option<f64>,
},
/// Double well potential: $V_i^\mathrm{ext}(z)=\mathrm{min}\left(\frac{2\pi}{45} m_i\varepsilon_{2si}\sigma_{si}^3\rho_s\left(2\left(\frac{2\sigma_{si}}{z}\right)^9-15\left(\frac{2\sigma_{si}}{z}\right)^3\right),0\right)+\frac{2\pi}{45} m_i\varepsilon_{1si}\sigma_{si}^3\rho_s\left(2\left(\frac{\sigma_{si}}{z}\right)^9-15\left(\frac{\sigma_{si}}{z}\right)^3\right),~~~~\varepsilon_{1si}=\sqrt{\varepsilon_{1ss}\varepsilon_{ii}},~~~~\varepsilon_{2si}=\sqrt{\varepsilon_{2ss}\varepsilon_{ii}},~~~~\sigma_{si}=\frac{1}{2}\left(\sigma_{ss}+\sigma_{ii}\right)$
DoubleWell {
sigma_ss: f64,
epsilon1_k_ss: f64,
epsilon2_k_ss: f64,
rho_s: f64,
},
/// Free-energy averaged potential:
FreeEnergyAveraged {
coordinates: QuantityArray2<U>,
sigma_ss: Array1<f64>,
epsilon_k_ss: Array1<f64>,
pore_center: [f64; 3],
system_size: [QuantityScalar<U>; 3],
n_grid: [usize; 2],
cutoff_radius: Option<f64>,
},
/// Custom potential
Custom(Array2<f64>),
}
/// Parameters of the fluid required to evaluate the external potential.
pub trait FluidParameters: HelmholtzEnergyFunctional {
fn epsilon_k_ff(&self) -> Array1<f64>;
fn sigma_ff(&self) -> &Array1<f64>;
}
impl<U: EosUnit> ExternalPotential<U> {
// Evaluate the external potential in cartesian coordinates for a given grid and fluid parameters.
pub fn calculate_cartesian_potential<P: FluidParameters>(
&self,
z_grid: &Array1<f64>,
fluid_parameters: &P,
temperature: f64,
) -> Array2<f64> {
if let ExternalPotential::Custom(potential) = self {
return potential.clone();
}
// Allocate external potential
let m = fluid_parameters.m();
let mut ext_pot = Array2::zeros((m.len(), z_grid.len()));
for (i, &mi) in m.iter().enumerate() {
ext_pot.index_axis_mut(Axis_nd(0), i).assign(&match self {
Self::HardWall { sigma_ss } => {
let sigma_sf = (fluid_parameters.sigma_ff()[i] + *sigma_ss) * 0.5;
z_grid.mapv(|z| if z < sigma_sf { f64::INFINITY } else { 0.0 })
}
Self::LJ93 {
sigma_ss,
epsilon_k_ss,
rho_s,
} => {
// combining rules
let epsilon_k_sf =
(fluid_parameters.epsilon_k_ff() * *epsilon_k_ss).mapv(|e| e.sqrt());
let sigma_sf = (fluid_parameters.sigma_ff() + *sigma_ss) * 0.5;
2.0 * PI * mi * epsilon_k_sf[i] * sigma_sf[i].powi(3) * rho_s / 45.0
* (2.0 * (sigma_sf[i] / z_grid).mapv(|x| x.powi(9))
- 15.0 * (sigma_sf[i] / z_grid).mapv(|x| x.powi(3)))
}
Self::SimpleLJ93 {
sigma_ss,
epsilon_k_ss,
} => {
// combining rules
let epsilon_k_sf =
(fluid_parameters.epsilon_k_ff() * *epsilon_k_ss).mapv(|e| e.sqrt());
let sigma_sf = (fluid_parameters.sigma_ff() + *sigma_ss) * 0.5;
epsilon_k_sf[i]
* ((sigma_sf[i] / z_grid).mapv(|x| x.powi(9))
- (sigma_sf[i] / z_grid).mapv(|x| x.powi(3)))
}
Self::CustomLJ93 {
sigma_sf,
epsilon_k_sf,
} => {
epsilon_k_sf[i]
* ((sigma_sf[i] / z_grid).mapv(|x| x.powi(9))
- (sigma_sf[i] / z_grid).mapv(|x| x.powi(3)))
}
Self::Steele {
sigma_ss,
epsilon_k_ss,
rho_s,
xi,
} => {
// combining rules
let epsilon_k_sf =
(fluid_parameters.epsilon_k_ff() * *epsilon_k_ss).mapv(|e| e.sqrt());
let sigma_sf = (fluid_parameters.sigma_ff() + *sigma_ss) * 0.5;
(2.0 * PI * mi * xi.unwrap_or(1.0) * epsilon_k_sf[i])
* (sigma_sf[i].powi(2) * DELTA_STEELE * rho_s)
* (0.4 * (sigma_sf[i] / z_grid).mapv(|x| x.powi(10))
- (sigma_sf[i] / z_grid).mapv(|x| x.powi(4))
- sigma_sf[i].powi(4)
/ ((3.0 * DELTA_STEELE)
* (z_grid + 0.61 * DELTA_STEELE).mapv(|x| x.powi(3))))
}
Self::DoubleWell {
sigma_ss,
epsilon1_k_ss,
epsilon2_k_ss,
rho_s,
} => {
// combining rules
let epsilon1_k_sf =
(fluid_parameters.epsilon_k_ff() * *epsilon1_k_ss).mapv(|e| e.sqrt());
let epsilon2_k_sf =
(fluid_parameters.epsilon_k_ff() * *epsilon2_k_ss).mapv(|e| e.sqrt());
let sigma_sf = (fluid_parameters.sigma_ff() + *sigma_ss) * 0.5;
let bh_tail = (2.0 * PI * mi * epsilon2_k_sf[i] * sigma_sf[i].powi(3) * rho_s
/ 45.0
* (2.0 * (2.0 * sigma_sf[i] / z_grid).mapv(|x| x.powi(9))
- 15.0 * (2.0 * sigma_sf[i] / z_grid).mapv(|x| x.powi(3))))
.mapv(|x| x.min(0.0));
bh_tail
+ &(2.0 * PI * mi * epsilon1_k_sf[i] * sigma_sf[i].powi(3) * rho_s / 45.0
* (2.0 * (sigma_sf[i] / z_grid).mapv(|x| x.powi(9))
- 15.0 * (sigma_sf[i] / z_grid).mapv(|x| x.powi(3))))
}
Self::FreeEnergyAveraged {
coordinates,
sigma_ss,
epsilon_k_ss,
pore_center,
system_size,
n_grid,
cutoff_radius,
} => {
// combining rules
let epsilon_k_sf =
(fluid_parameters.epsilon_k_ff()[i] * epsilon_k_ss).mapv(|e| e.sqrt());
let sigma_sf = (fluid_parameters.sigma_ff()[i] + sigma_ss) * 0.5;
calculate_fea_potential(
z_grid,
mi,
coordinates,
sigma_sf,
epsilon_k_sf,
pore_center,
system_size,
n_grid,
temperature,
Geometry::Cartesian,
*cutoff_radius,
)
}
Self::Custom(_) => unreachable!(),
});
}
ext_pot
}
// Evaluate the external potential in cylindrical coordinates for a given grid and fluid parameters.
pub fn calculate_cylindrical_potential<P: FluidParameters>(
&self,
r_grid: &Array1<f64>,
pore_size: f64,
fluid_parameters: &P,
temperature: f64,
) -> Array2<f64> {
if let ExternalPotential::Custom(potential) = self {
return potential.clone();
}
// Allocate external potential
let m = fluid_parameters.m();
let mut ext_pot = Array2::zeros((m.len(), r_grid.len()));
for (i, &mi) in m.iter().enumerate() {
ext_pot.index_axis_mut(Axis_nd(0), i).assign(&match self {
Self::HardWall { sigma_ss } => {
let sigma_sf = (fluid_parameters.sigma_ff()[i] + *sigma_ss) * 0.5;
r_grid.mapv(|r| {
if r > pore_size - sigma_sf {
f64::INFINITY
} else {
0.0
}
})
}
Self::LJ93 {
sigma_ss,
epsilon_k_ss,
rho_s,
} => {
// combining rules
let epsilon_k_sf =
(fluid_parameters.epsilon_k_ff() * *epsilon_k_ss).mapv(|e| e.sqrt());
let sigma_sf = (fluid_parameters.sigma_ff() + *sigma_ss) * 0.5;
(phi(6, &(r_grid / pore_size), sigma_sf[i] / pore_size)
- phi(3, &(r_grid / pore_size), sigma_sf[i] / pore_size))
* 2.0
* PI
* mi
* epsilon_k_sf[i]
* sigma_sf[i].powi(3)
* *rho_s
}
Self::SimpleLJ93 {
sigma_ss: _,
epsilon_k_ss: _,
} => {
unimplemented!()
}
Self::CustomLJ93 {
sigma_sf: _,
epsilon_k_sf: _,
} => {
unimplemented!()
}
Self::Steele {
sigma_ss,
epsilon_k_ss,
rho_s,
xi,
} => {
// combining rules
let epsilon_k_sf =
(fluid_parameters.epsilon_k_ff() * *epsilon_k_ss).mapv(|e| e.sqrt());
let sigma_sf = (fluid_parameters.sigma_ff() + *sigma_ss) * 0.5;
(2.0 * PI * mi * xi.unwrap_or(1.0) * epsilon_k_sf[i])
* (sigma_sf[i].powi(2) * DELTA_STEELE * rho_s)
* (psi(6, &(r_grid / pore_size), sigma_sf[i] / pore_size)
- psi(3, &(r_grid / pore_size), sigma_sf[i] / pore_size)
- sigma_sf[i] / DELTA_STEELE
* phi(
3,
&(r_grid / (pore_size + DELTA_STEELE * 0.61)),
sigma_sf[i] / (pore_size + DELTA_STEELE * 0.61),
))
}
Self::DoubleWell {
sigma_ss,
epsilon1_k_ss,
epsilon2_k_ss,
rho_s,
} => {
// combining rules
let epsilon1_k_sf =
(fluid_parameters.epsilon_k_ff() * *epsilon1_k_ss).mapv(|e| e.sqrt());
let epsilon2_k_sf =
(fluid_parameters.epsilon_k_ff() * *epsilon2_k_ss).mapv(|e| e.sqrt());
let sigma_sf = (fluid_parameters.sigma_ff() + *sigma_ss) * 0.5;
let bh_tail = ((phi(6, &(r_grid / pore_size), 2.0 * sigma_sf[i] / pore_size)
- phi(3, &(r_grid / pore_size), 2.0 * sigma_sf[i] / pore_size))
* 2.0
* PI
* mi
* epsilon2_k_sf[i]
* sigma_sf[i].powi(3)
* *rho_s)
.mapv(|x| x.min(0.0));
bh_tail
+ &((phi(6, &(r_grid / pore_size), sigma_sf[i] / pore_size)
- phi(3, &(r_grid / pore_size), sigma_sf[i] / pore_size))
* 2.0
* PI
* mi
* epsilon1_k_sf[i]
* sigma_sf[i].powi(3)
* *rho_s)
}
Self::FreeEnergyAveraged {
coordinates,
sigma_ss,
epsilon_k_ss,
pore_center,
system_size,
n_grid,
cutoff_radius,
} => {
// combining rules
let epsilon_k_sf =
(fluid_parameters.epsilon_k_ff()[i] * epsilon_k_ss).mapv(|e| e.sqrt());
let sigma_sf = (fluid_parameters.sigma_ff()[i] + sigma_ss) * 0.5;
calculate_fea_potential(
r_grid,
mi,
coordinates,
sigma_sf,
epsilon_k_sf,
pore_center,
system_size,
n_grid,
temperature,
Geometry::Cylindrical,
*cutoff_radius,
)
}
Self::Custom(_) => unreachable!(),
});
}
ext_pot
}
// Evaluate the external potential in spherical coordinates for a given grid and fluid parameters.
pub fn calculate_spherical_potential<P: FluidParameters>(
&self,
r_grid: &Array1<f64>,
pore_size: f64,
fluid_parameters: &P,
temperature: f64,
) -> Array2<f64> {
if let ExternalPotential::Custom(potential) = self {
return potential.clone();
}
// Allocate external potential
let m = fluid_parameters.m();
let mut ext_pot = Array2::zeros((m.len(), r_grid.len()));
for (i, &mi) in m.iter().enumerate() {
ext_pot.index_axis_mut(Axis_nd(0), i).assign(&match self {
Self::HardWall { sigma_ss } => {
let sigma_sf = (fluid_parameters.sigma_ff()[i] + *sigma_ss) * 0.5;
r_grid.mapv(|r| {
if r > pore_size - sigma_sf {
f64::INFINITY
} else {
0.0
}
})
}
Self::LJ93 {
sigma_ss,
epsilon_k_ss,
rho_s,
} => {
// combining rules
let epsilon_k_sf =
(fluid_parameters.epsilon_k_ff() * *epsilon_k_ss).mapv(|e| e.sqrt());
let sigma_sf = (fluid_parameters.sigma_ff() + *sigma_ss) * 0.5;
PI * mi
* epsilon_k_sf[i]
* rho_s
* (sigma_sf[i].powi(12) / 90.
* ((r_grid - 9.0 * pore_size)
/ (r_grid - pore_size).mapv(|x| x.powi(9))
- (r_grid + 9.0 * pore_size)
/ (r_grid + pore_size).mapv(|x| x.powi(9)))
- sigma_sf[i].powi(6) / 3.
* ((r_grid - 3.0 * pore_size)
/ (r_grid - pore_size).mapv(|x| x.powi(3))
- (r_grid + 3.0 * pore_size)
/ (r_grid + pore_size).mapv(|x| x.powi(3))))
/ r_grid
}
Self::SimpleLJ93 {
sigma_ss: _,
epsilon_k_ss: _,
} => {
unimplemented!()
}
Self::CustomLJ93 {
sigma_sf: _,
epsilon_k_sf: _,
} => {
unimplemented!()
}
Self::Steele {
sigma_ss,
epsilon_k_ss,
rho_s,
xi,
} => {
// combining rules
let epsilon_k_sf =
(fluid_parameters.epsilon_k_ff() * *epsilon_k_ss).mapv(|e| e.sqrt());
let sigma_sf = (fluid_parameters.sigma_ff() + *sigma_ss) * 0.5;
(2.0 * PI * mi * xi.unwrap_or(1.0) * epsilon_k_sf[i])
* (sigma_sf[i].powi(2) * DELTA_STEELE * rho_s)
* (2.0 / 5.0 * sum_n(10, r_grid, sigma_sf[i], pore_size)
- sum_n(4, r_grid, sigma_sf[i], pore_size)
- sigma_sf[i] / (3.0 * DELTA_STEELE)
* (sigma_sf[i].powi(3)
/ r_grid
.mapv(|r| (pore_size + 0.61 * DELTA_STEELE - r).powi(3))
+ sigma_sf[i].powi(3)
/ r_grid.mapv(|r| {
(pore_size + 0.61 * DELTA_STEELE + r).powi(3)
})
+ 1.5
* sum_n(
3,
r_grid,
sigma_sf[i],
pore_size + 0.61 * DELTA_STEELE,
)))
}
Self::DoubleWell {
sigma_ss,
epsilon1_k_ss,
epsilon2_k_ss,
rho_s,
} => {
// combining rules
let epsilon1_k_sf =
(fluid_parameters.epsilon_k_ff() * *epsilon1_k_ss).mapv(|e| e.sqrt());
let epsilon2_k_sf =
(fluid_parameters.epsilon_k_ff() * *epsilon2_k_ss).mapv(|e| e.sqrt());
let sigma_sf = (fluid_parameters.sigma_ff() + *sigma_ss) * 0.5;
let bh_tail = (2.0
* PI
* mi
* epsilon2_k_sf[i]
* sigma_sf[i].powi(2)
* rho_s
* (2.0 / 5.0 * sum_n(10, r_grid, 2.0 * sigma_sf[i], pore_size)
- sum_n(4, r_grid, 2.0 * sigma_sf[i], pore_size)))
.mapv(|x| x.min(0.0));
bh_tail
+ &(2.0
* PI
* mi
* epsilon1_k_sf[i]
* sigma_sf[i].powi(2)
* rho_s
* (2.0 / 5.0 * sum_n(10, r_grid, sigma_sf[i], pore_size)
- sum_n(4, r_grid, sigma_sf[i], pore_size)))
}
Self::FreeEnergyAveraged {
coordinates,
sigma_ss,
epsilon_k_ss,
pore_center,
system_size,
n_grid,
cutoff_radius,
} => {
// combining rules
let epsilon_k_sf =
(fluid_parameters.epsilon_k_ff()[i] * epsilon_k_ss).mapv(|e| e.sqrt());
let sigma_sf = (fluid_parameters.sigma_ff()[i] + sigma_ss) * 0.5;
calculate_fea_potential(
r_grid,
mi,
coordinates,
sigma_sf,
epsilon_k_sf,
pore_center,
system_size,
n_grid,
temperature,
Geometry::Spherical,
*cutoff_radius,
)
}
Self::Custom(_) => unreachable!(),
});
}
ext_pot
}
}
fn phi(n: i32, r_r: &Array1<f64>, sigma_r: f64) -> Array1<f64> {
let m3n2 = 3.0 - 2.0 * n as f64;
let n2m3 = 2.0 * n as f64 - 3.0;
(1.0 - &r_r.mapv(|r| r.powi(2))).mapv(|r| r.powf(m3n2)) * 4.0 * PI.sqrt() / n2m3
* sigma_r.powf(n2m3)
* gamma(n as f64 - 0.5)
/ gamma(n as f64)
* taylor_2f1_phi(r_r, n)
}
fn psi(n: i32, r_r: &Array1<f64>, sigma_r: f64) -> Array1<f64> {
(1.0 - &r_r.mapv(|r| r.powi(2))).mapv(|r| r.powf(2.0 - 2.0 * n as f64))
* 4.0
* PI.sqrt()
* gamma(n as f64 - 0.5)
/ gamma(n as f64)
* sigma_r.powf(2.0 * n as f64 - 2.0)
* taylor_2f1_psi(r_r, n)
}
fn sum_n(n: i32, r_grid: &Array1<f64>, sigma: f64, pore_size: f64) -> Array1<f64> {
let mut result = Array1::zeros(r_grid.len());
for i in 0..n {
result = result
+ sigma.powi(n) / (pore_size.powi(i) * r_grid.mapv(|r| (pore_size - r).powi(n - i)))
+ sigma.powi(n) / (pore_size.powi(i) * r_grid.mapv(|r| (pore_size + r).powi(n - i)));
}
result
}
fn taylor_2f1_phi(x: &Array1<f64>, n: i32) -> Array1<f64> {
match n {
3 => {
1.0 + 3.0 / 4.0 * x.mapv(|x| x.powi(2)) + 3.0 / 64.0 * x.mapv(|x| x.powi(4))
- 1.0 / 256.0 * x.mapv(|x| x.powi(6))
- 15.0 / 16384.0 * x.mapv(|x| x.powi(8))
}
6 => {
1.0 + 63.0 / 4.0 * x.mapv(|x| x.powi(2))
+ 2205.0 / 64.0 * x.mapv(|x| x.powi(4))
+ 3675.0 / 256.0 * x.mapv(|x| x.powi(6))
+ 11025.0 / 16384.0 * x.mapv(|x| x.powi(8))
}
_ => unreachable!(),
}
}
fn taylor_2f1_psi(x: &Array1<f64>, n: i32) -> Array1<f64> {
match n {
3 => {
1.0 + 9.0 / 4.0 * x.mapv(|x| x.powi(2))
+ 9.0 / 64.0 * x.mapv(|x| x.powi(4))
+ 1.0 / 256.0 * x.mapv(|x| x.powi(6))
+ 9.0 / 16384.0 * x.mapv(|x| x.powi(8))
}
6 => {
1.0 + 81.0 / 4.0 * x.mapv(|x| x.powi(2))
+ 3969.0 / 64.0 * x.mapv(|x| x.powi(4))
+ 11025.0 / 256.0 * x.mapv(|x| x.powi(6))
+ 99125.0 / 16384.0 * x.mapv(|x| x.powi(8))
}
_ => unreachable!(),
}
}
extern "C" {
fn tgamma(x: c_double) -> c_double;
}
fn gamma(x: f64) -> f64 {
unsafe { tgamma(x) }
}