Model Potentials

These EOS for model potentials are useful for understanding theory, and capture some (but perhaps not all) of the physics of “real” fluids.

The unit systems for all model potentials covered here are reduced quantities defined by (nicely summarized in the appendix to the dissertation of Karsten Meier):

Time: $ t^* = t:nbsphinx-math:`frac{sqrt{epsilon/m}}{sigma}`$

Velocity: \(v^* = v\frac{m}{\epsilon}\)

Temperature: \(T^* = \frac{kT}{\epsilon}\)

Particle density: \(\rho^* = \rho\sigma^3\)

Pressure: \(p^* = \frac{p\sigma^3}{\epsilon}\)

Second density virial coefficient: \(B^* = \frac{B}{\sigma^3}\)

Within the implementation for the model potentials, the value of \(R\) is set to 1, which has the desired cancellation in reduced units. For instance the pressure in reduced units is obtained from:

\[p^* = \rho^*T^*(1+\Lambda^{\rm r}_{01}(T^*,\rho^*))\]

In the case of chains, it is always the segment (or monomer) density that is an input to the routines.

import teqp
teqp.__version__

'0.22.0'

Square-well

The potential is defined by

\[\begin{split}V(r) = \left\lbrace \begin{array}{cc} \infty & r < \sigma \\ -\varepsilon & \sigma < r < \lambda\sigma \\ 0 & r > \lambda \sigma \end{array}\right.\end{split}\]

from which an EOS can be developed by correlating results from molecular simulation. The EOS is from:

Rodolfo Espíndola-Heredia, Fernando del Río and Anatol Malijevsky Optimized equation of the state of the square-well fluid of variable range based on a fourth-order free-energy expansion J. Chem. Phys. 130, 024509 (2009); https://doi.org/10.1063/1.3054361

model = teqp.make_model({
  "kind": "SW_EspindolaHeredia2009",
  "model": {
      "lambda": 1.3
  }
})

EXP-6

model = teqp.make_model({
  "kind": "EXP6_Kataoka1992",
  "model": {
      "alpha": 12
  }
})

Lennard-Jones Fluid

The Lennard-Jones potential is given by

\[V(r) = 4\varepsilon\left((\sigma/r)^{12}-(\sigma/r)^{6}\right)\]

and EOS are available from many authors. teqp includes the EOS from Thol, Kolafa-Nezbeda, and Johnson.

for kind, crit in [
    ["LJ126_TholJPCRD2016", (1.32, 0.31)], # Note the true critical point was not used
    ["LJ126_KolafaNezbeda1994", (1.3396, 0.3108)],
    ["LJ126_Johnson1993", (1.313, 0.310)]]:

    j = { "kind": kind, "model": {} }
    model = teqp.make_model(j)
    print(kind, model.solve_pure_critical(1.3, 0.3), crit)

LJ126_TholJPCRD2016 (1.303512554910004, 0.3103860327864474) (1.32, 0.31)
LJ126_KolafaNezbeda1994 (1.3396478193468155, 0.31080389777229156) (1.3396, 0.3108)
LJ126_Johnson1993 (1.3130000571792173, 0.3099999768607838) (1.313, 0.31)

Mie Fluid

The Mie potential is given by

\[u(r) = C\epsilon \left((\sigma/r)^{\lambda_r}-(\sigma/r)^{\lambda_a}\right)\]

with

\[C = \frac{\lambda_r}{\lambda_r-\lambda_a}\left(\frac{\lambda_r}{\lambda_a}\right)^{\lambda_a/(\lambda_r-\lambda_a)}\]

The SAFT-VR-Mie model can be used for this fluid, or two new models from 2023

for kind, model in [
    ["Mie_Chaparro2023", {"lambda_r": 12, "lambda_a": 6}],
    ["Mie_Pohl2023", {"lambda_r": 12}]]:

    j = { "kind": kind, "model": model }
    model = teqp.make_model(j)
    print(kind, model.solve_pure_critical(1.3, 0.3))

Mie_Chaparro2023 (1.3302552193659323, 0.30398356369036467)
Mie_Pohl2023 (1.3219460984743199, 0.3044747955342046)

Two-Center Lennard-Jones Fluid

model = teqp.make_model({
  'kind': '2CLJF-Dipole',
  'model': {
      "author": "2CLJF_Lisal",
      'L^*': 0.5,
      '(mu^*)^2': 0.1
  }
})
print(model.solve_pure_critical(1.3, 0.3))

model = teqp.make_model({
  'kind': '2CLJF-Quadrupole',
  'model': {
      "author": "2CLJF_Lisal",
      'L^*': 0.5,
      '(Q^*)^2': 0.1
  }
})
print(model.solve_pure_critical(1.3, 0.3))

(2.828297206218807, 0.20050466666340064)
(2.8325743035618367, 0.20031946554633046)