PyDFTlj

An python library for calculations using the classical Density Functional Theory (cDFT) for Lennard-Jones fluids in 1D and 3D geometries.

Dependencies

• NumPy is the fundamental package for scientific computing with Python.
• SciPy is a collection of fundamental algorithms for scientific computing in Python.
• PyFFTW is a pythonic wrapper around FFTW, the speedy FFT library.
• PyTorch is a high-level library for machine learning, with multidimensional tensors that can also be operated on a CUDA-capable NVIDIA GPU.
• Matplotlib is a comprehensive library for creating static, animated, and interactive visualizations in Python.
• Optional: SciencePlots is a Matplotlib styles for scientific figures

Installation

Option 1: Using pip to install directly from the GitHub repo

You can run

pip install pydftlj

and then you will be able to access the pydftlj library.

Option 2: Using setup.py

Clone PyDFTlj repository if you haven't done it yet.

git clone https://github.com/elvissoares/PyDFTlj

Go to PyDFTlj's root folder, there you will find setup.py file, and run the command below:

pip install -e .

The command -e permits to edit the local source code and add these changes to the pydftlj library.

cDFT basics

The cDFT is the extension of the equation of state to treat inhomogeneous fluids. For a fluid with temperature T, total volume V, and chemical potential $\mu$ specified, the grand potential, $\Omega$, is written as

$$\Omega[\rho(\boldsymbol{r})] = F[\rho (\boldsymbol{r})] + \int_{V} [ V^{(\text{ext})}(\boldsymbol{r}) - \mu ]\rho(\boldsymbol{r}) d\boldsymbol{r}$$

where $F[\rho (\boldsymbol{r})] $ is the free-energy functional, $V^{(\text{ext})} $ is the external potential, and $\mu $ is the chemical potential. The free-energy functional can be written as a sum $ F = F^\text{id} + F^\text{exc} $, where $F^\text{id} $ is the ideal gas contribution and $F^\text{exc}$ is the excess contribution.

The ideal-gas contribution $F^\text{id} $ is given by the exact expression

$$ F^{\text{id}}[\rho (\boldsymbol{r})] = k_B T\int_{V} \rho(\boldsymbol{r})[\ln(\rho (\boldsymbol{r})\Lambda^3)-1] d\boldsymbol{r}$$

where $k_B $ is the Boltzmann constant, and $\Lambda $ is the well-known thermal de Broglie wavelength.

The excess Helmholtz free-energy, $F^{\text{exc} }$, is the free-energy functional due to particle-particle interactions and can be splitted in the form

$$ F^{\text{exc}}[\rho (\boldsymbol{r})] = F^{\text{hs}}[\rho (\boldsymbol{r})] + F^{\text{att}}[\rho (\boldsymbol{r})] $$ where $F^{\text{hs}} $ is the hard-sphere repulsive interaction excess contribution and $F^{\text{att}} $ is the attractive interaction excess contribution.

The hard-sphere contribution, $F^{\text{hs}} $, represents the hard-sphere exclusion volume correlation and it can be described using different formulations of the fundamental measure theory (FMT) as

• Rosenfeld Functional (RF) - Rosenfeld, Y., Phys. Rev. Lett. 63, 980–983 (1989)
• White Bear version I (WBI) - Yu, Y.-X. & Wu, J., J. Chem. Phys. 117, 10156–10164 (2002); Roth, R., Evans, R., Lang, A. & Kahl, G., J. Phys. Condens. Matter 14, 12063–12078 (2002)
• White Bear version II (WBII) - Hansen-Goos, H. & Roth, R. J., Phys. Condens. Matter 18, 8413–8425 (2006)

The attractive contribution, $F^\text{att}$, of the Lennard-Jones potential can be described by several formulations as listed below:

• Mean Field Approximation (MFA) -
• Weighted Density Approximation (WDA) - Shen, G., Ji, X., & Lu, X. (2013). The Journal of Chemical Physics, 138(22), 224706.
• Modified Mean-Field Approximation (MMFA) - Soares, E. do A., Barreto, A. G., & Tavares, F. W. (2021). Fluid Phase Equilibria, 542–543, 113095.

where [x] represents the implemented functionals.

The thermodynamic equilibrium is given by the functional derivative of the grand potential in the form

$$ \frac{\delta \Omega}{\delta \rho(\boldsymbol{r})} = k_B T \ln(\rho(\boldsymbol{r}) \Lambda^3) + \frac{\delta F^{\text{exc}}[\rho]}{\delta \rho(\boldsymbol{r})} +V^{(\text{ext})}(\boldsymbol{r})-\mu = 0$$

When necessary, we use the MBWR¹ equation of state for Lennard-Jones Fluids. We also describe the direct correlation function using the double Yukawa potential from the FMSA².

Cite PyDFTlj

If you use PyDFTlj in your work, please consider to cite it using the following reference:

Soares, Elvis do A, Amaro G Barreto, and Frederico W Tavares. 2023. “Classical Density Functional Theory Reveals Structural Information of H2 and CH4 Fluids Adsorbed in MOF-5.” Fluid Phase Equilibria, July, 113887. ArXiv: 2303.11384

Bibtex:

@article{Soares2023, 
author = {Soares, Elvis do A and Barreto, Amaro G and Tavares, Frederico W}, 
doi = {10.1016/j.fluid.2023.113887}, 
issn = {03783812}, 
journal = {Fluid Phase Equilibria}, 
keywords = {Adsorption,Density functional theory,Metal–organic framework,Structure factor}, 
month = {jul}, 
pages = {113887}, 
title = {{Classical density functional theory reveals structural information of H2 and CH4 fluids adsorbed in MOF-5}}, 
url = {https://linkinghub.elsevier.com/retrieve/pii/S037838122300167X}, 
year = {2023} 
} 

Contact

Elvis Soares: elvis@peq.coppe.ufrj.br

Universidade Federal do Rio de janeiro - UFRJ

Programa de Engenharia Química - COPPE

Usage examples

To access the examples folder you will need to clone PyDFTlj repository if you haven't done it yet.

git clone https://github.com/elvissoares/PyDFTlj

The, you can access our examples folder and you can find different applications of the PyDFTlj.

Lennard-Jones equation of State (Example1-Phasediagram-Methane.ipynb)

Fig.1 - The phase diagram of the LJ fluid. The curve represents the MBWR EoS1.	Fig.2 - The saturation pressure as a function of the inverse of the temperature.

Confined LJ fluid (Example2-Hardwall3D.ipynb)

Fig.3 - The density profiles of LJ fluid near a hardwall with reduce temperature T*=1.35 and reduced density of ρ*=0.5. Symbols: MC data. Lines: Different DFT formulations.

Fig.4 - The density profiles of LJ fluid confined in slit-like pores at reduced density of ρ*=0.5925 and reduced temperature of T*=1.2 for pore size of H = 7.5, 4.0, 3.0, 1.8$\sigma$. Symbols: MC data. Lines: Different DFT formulations.

LJ fluid Radial Distribution Function (Example4-RadialDistributionFunction.ipynb)

Fig.7 - The radial distribution function of LJ fluid at reduced density of ρ*=0.84 and reduced temperature of T*=0.71. Symbols: MC data. Lines: Different DFT formulations.

Adsorption of CH4 inside MOF-5 (Example5-Adsorption3D_CH4_on_MOFs.ipynb)

Fig.8 - Excess adsorbed quantity of CH4 inside the MOF-5 at 300 K. Symbols: MC data. Lines: Different DFT formulations.

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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. ↩ ↩²

2. Tang, Y., & Lu, B. C. Y. (2001). On the mean spherical approximation for the Lennard-Jones fluid. Fluid Phase Equilibria, 190(1–2), 149–158. ↩