# Nanothermodynamic Description and Molecular Simulation of a Single-Phase Fluid in a Slit Pore

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Hill’s Nanothermodynamics

#### 2.2. Maxwell Relations for a Slit Pore

#### 2.3. The Disjoining Pressure

#### 2.4. A Mechanical Description of the Slit Pore

## 3. Simulation Details

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Israelachvili, J.N. Intermolecular and Surface Forces; Academic Press: Cambridge, MA, USA, 1985. [Google Scholar]
- McDonald, T.M.; Mason, J.A.; Kong, X.; Bloch, E.D.; Gygi, D.; Dani, A.; Crocella, V.; Giordanino, F.; Odoh, S.O.; Drisdell, W.S.; et al. Cooperative insertion of CO
_{2}in diamine-appended metal-organic frameworks. Nature**2015**, 519, 303–308. [Google Scholar] [CrossRef] [PubMed] - Vlugt, T.; Krishna, R.; Smit, B. Molecular simulations of adsorption isotherms for linear and branched alkanes and their mixtures in silicalite. J. Phys. Chem. B
**1999**, 103, 1102–1118. [Google Scholar] [CrossRef] [Green Version] - Bresme, F.; Oettel, M. Nanoparticles at fluid interfaces. J. Phys. Condens. Matter
**2007**, 19, 413101. [Google Scholar] [CrossRef] [PubMed] - Bresme, F.; Lehle, H.; Oettel, M. Solvent-mediated interactions between nanoparticles at fluid interfaces. J. Chem. Phys.
**2009**, 130, 214711. [Google Scholar] [CrossRef] [Green Version] - Galteland, O.; Bresme, F.; Hafskjold, B. Solvent-Mediated Forces between Ellipsoidal Nanoparticles Adsorbed at Liquid–Vapor Interfaces. Langmuir
**2020**, 36, 48. [Google Scholar] [CrossRef] - Derjaguin, B. Untersuchungen über die Reibung und Adhäsion, IV. Kolloid Z.
**1934**, 69, 155–164. [Google Scholar] [CrossRef] - Moura, M.; Flekkøy, E.G.; Måløy, K.J.; Schäfer, G.; Toussaint, R. Connectivity enhancement due to film flow in porous media. Phys. Rev. Fluids
**2019**, 4, 094102. [Google Scholar] [CrossRef] [Green Version] - Das, D.; Hassanizadeh, S. Upscaling Multiphase Flow in Porous Media; Springer: Berlin, Germany, 2005. [Google Scholar]
- Khanamiri, H.H.; Berg, C.F.; Slotte, P.A.; Schlüter, S.; Torsæter, O. Description of Free Energy for Immiscible Two-Fluid Flow in Porous Media by Integral Geometry and Thermodynamics. Water Resour. Res.
**2018**, 54, 9045–9059. [Google Scholar] [CrossRef] - Armstrong, R.T.; McClure, J.E.; Robins, V.; Liu, Z.; Arns, C.H.; Schlüter, S.; Berg, S. Porous media characterization using minkowski functionals: Theories, applications and future directions. Transp. Porous Med.
**2019**, 130, 305–335. [Google Scholar] [CrossRef] - Slotte, P.A.; Berg, C.F.; Khanamiri, H.H. Predicting Resistivity and Permeability of Porous Media Using Minkowski Functionals. Transp. Porous Med.
**2020**, 131, 705–722. [Google Scholar] [CrossRef] [Green Version] - Kjelstrup, S.; Bedeaux, D.; Hansen, A.; Hafskjold, B.; Galteland, O. Non-isothermal transport of multi-phase fluids in porous media. the entropy production. Front. Phys.
**2018**, 6, 126. [Google Scholar] [CrossRef] - Kjelstrup, S.; Bedeaux, D.; Hansen, A.; Hafskjold, B.; Galteland, O. Non-isothermal transport of multi-phase fluids in porous media. Constitutive equations. Front. Phys.
**2019**, 6, 150. [Google Scholar] [CrossRef] [Green Version] - Balbuena, P.B.; Berry, D.; Gubbins, K.E. Solvation pressures for simple fluids in micropores. J. Phys. Chem.
**1993**, 97, 937–943. [Google Scholar] [CrossRef] - Gubbins, K.E.; Long, Y.; Śliwinska-Bartkowiak, M. Thermodynamics of confined nano-phases. J. Chem. Thermodyn.
**2014**, 74, 169–183. [Google Scholar] [CrossRef] - Bedeaux, D.; Kjelstrup, S. Hill’s nano-thermodynamics is equivalent with Gibbs’ thermodynamics for surfaces of constant curvatures. Chem. Phys. Lett.
**2018**, 707, 40–43. [Google Scholar] [CrossRef] [Green Version] - Gjennestad, M.A.; Wilhelmsen, Ø. Thermodynamic stability of volatile droplets and thin films governed by the disjoining pressure in open and closed containers. Langmuir
**2020**, 36, 27. [Google Scholar] [CrossRef] - Gjennestad, M.A.; Wilhelmsen, Ø. Thermodynamic stability of droplets, bubbles and thick films in open and closed pores. Fluid Phase Equilibr.
**2020**, 505, 112351. [Google Scholar] [CrossRef] - Strøm, B.A.; Simon, J.M.; Schnell, S.K.; Kjelstrup, S.; He, J.; Bedeaux, D. Size and shape effects on the thermodynamic properties of nanoscale volumes of water. Phys. Chem. Chem. Phys.
**2017**, 19, 9016–9027. [Google Scholar] [CrossRef] - Galteland, O.; Bedeaux, D.; Hafskjold, B.; Kjelstrup, S. Pressures inside a nano-porous medium. The case of a single phase fluid. Front. Phys.
**2019**, 7, 60. [Google Scholar] [CrossRef] [Green Version] - Erdős, M.; Galteland, O.; Bedeaux, D.; Kjelstrup, S.; Moultos, O.A.; Vlugt, T.J. Gibbs Ensemble Monte Carlo Simulation of Fluids in Confinement: Relation between the Differential and Integral Pressures. Nanomaterials
**2020**, 10, 293. [Google Scholar] [CrossRef] [Green Version] - Rauter, M.T.; Galteland, O.; Erdős, M.; Moultos, O.A.; Vlugt, T.J.; Schnell, S.K.; Bedeaux, D.; Kjelstrup, S. Two-Phase Equilibrium Conditions in Nanopores. Nanomaterials
**2020**, 10, 608. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Strøm, B.A.; He, J.; Bedeaux, D.; Kjelstrup, S. When Thermodynamic Properties of Adsorbed Films Depend on Size: Fundamental Theory and Case Study. Nanomaterials
**2020**, 10, 1691. [Google Scholar] [CrossRef] [PubMed] - Bedeaux, D.; Kjelstrup, S.; Schnell, S.K. Nanothermodynamics. General Theory; NTNU: Trondheim, Norway, 2020. [Google Scholar]
- Hill, T.L. Thermodynamics of Small Systems - Two Volumes Bound as One; Dover: New York, NY, USA, 1964. [Google Scholar]
- Hill, T.L.; Chamberlin, R.V. Extension of the thermodynamics of small systems to open metastable states: An example. Proc. Natl. Acad. Sci. USA
**1998**, 95, 12779–12782. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hill, T.L.; Chamberlin, R.V. Fluctuations in energy in completely open small systems. Nano Lett.
**2002**, 2, 609–613. [Google Scholar] [CrossRef] - Hansen, J.P.; McDonald, I.R. Theory of Simple Liquids; Elsevier: Amsterdam, The Netherlands, 1990. [Google Scholar]
- Radke, C. Film and membrane-model thermodynamics of free thin liquid films. J. Colloid Interf. Sci.
**2015**, 449, 462–479. [Google Scholar] [CrossRef] [Green Version] - Long, Y.; Palmer, J.C.; Coasne, B.; Śliwinska-Bartkowiak, M.; Gubbins, K.E. Pressure enhancement in carbon nanopores: A major confinement effect. Phys. Chem. Chem. Phys.
**2011**, 13, 17163–17170. [Google Scholar] [CrossRef] - Van Dijk, D. Comment on “Pressure enhancement in carbon nanopores: A major confinement effect” by Y. Long, J. C. Palmer, B. Coasne, M. Sliwinska-Bartkowiak and K. E. Gubbins, Phys. Chem. Chem. Phys., 2011, 13, 17163. Phys. Chem. Chem. Phys.
**2020**, 22, 9824–9825. [Google Scholar] [CrossRef] - Long, Y.; Palmer, J.C.; Coasne, B.; Shi, K.; Śliwińska-Bartkowiak, M.; Gubbins, K.E. Reply to the ‘Comment on “Pressure enhancement in carbon nanopores: A major confinement effect”’by D. van Dijk, Phys. Chem. Chem. Phys., 2020, 22. Phys. Chem. Chem. Phys.
**2020**, 22, 9826–9830. [Google Scholar] [CrossRef] - Schofield, P.; Henderson, J.R. Statistical mechanics of inhomogeneous fluids. Proc. R. Soc. Lon. Ser. A
**1982**, 379, 231–246. [Google Scholar] - Harasima, A. Molecular theory of surface tension. Adv. Chem. Phys.
**1958**, 1, 203–237. [Google Scholar] - Irving, J.; Kirkwood, J.G. The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics. J. Chem. Phys.
**1950**, 18, 817–829. [Google Scholar] [CrossRef] - Hafskjold, B.; Ikeshoji, T. Microscopic pressure tensor for hard-sphere fluids. Phys. Rev. E
**2002**, 66, 011203. [Google Scholar] [CrossRef] [PubMed] - Shi, K.; Shen, Y.; Santiso, E.E.; Gubbins, K.E. Microscopic pressure tensor in cylindrical geometry: Pressure of water in a carbon nanotube. J. Chem. Theory Comput.
**2020**, 16, 5548–5561. [Google Scholar] [CrossRef] [PubMed] - Ikeshoji, T.; Hafskjold, B.; Furuholt, H. Molecular-level calculation scheme for pressure in inhomogeneous systems of flat and spherical layers. Mol. Simulat.
**2003**, 29, 101–109. [Google Scholar] [CrossRef] - Evans, R.; Marini Bettolo Marconi, U. Phase equilibria and solvation forces for fluids confined between parallel walls. J. Chem. Phys.
**1987**, 86, 7138–7148. [Google Scholar] [CrossRef] - Frenkel, D.; Smit, B. Understanding Molecular Simulation: From Algorithms to Applications; Elsevier: Amsterdam, The Netherlands, 2001; Volume 1. [Google Scholar]
- Shinoda, W.; Shiga, M.; Mikami, M. Rapid estimation of elastic constants by molecular dynamics simulation under constant stress. Phys. Rev. B
**2004**, 69, 134103. [Google Scholar] [CrossRef] - Plimpton, S. Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys.
**1995**, 117, 1–19. [Google Scholar] [CrossRef] [Green Version] - Hafskjold, B.; Travis, K.P.; Hass, A.B.; Hammer, M.; Aasen, A.; Wilhelmsen, Ø. Thermodynamic properties of the 3D Lennard-Jones/spline model. Mol. Phys.
**2019**, 117, 3754–3769. [Google Scholar] [CrossRef] - Stukowski, A. Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool. Model. Simul. Mater. Sci.
**2009**, 18, 015012. [Google Scholar] [CrossRef]

**Figure 1.**Visualization of the fluid particles in a slit pore of height ${L}_{x}=4\sigma $, chemical potential ${\mu}^{*}=1$, and temperature ${T}^{*}=2$. The fluid particles are rendered in red, and their diameter is rendered at $\sigma $. The solid is not rendered. The solid lines illustrates the edges of the simulation box. The simulation was rendered with Open Visualization Tool (OVITO) [45].

**Figure 2.**(

**a**) Normal mechanical pressure, (

**b**) tangential mechanical pressure, and (

**c**) fluid number density $\rho $ as a function of th e x-direction for slit pore heights $h=2.04\sigma $, $2.59\sigma $, and $8\sigma $.

**Figure 3.**Fluid number density $\rho =N/V$ as a function of slit pore height h. The bulk fluid number density is shown as a dashed line.

**Figure 4.**Entropy density $s=S/V$ as a function of slit pore height h. See Equation (36). The dashed line shows the entropy density of the bulk ${s}^{\mathrm{b}}$.

**Figure 5.**Internal energy density $u=U/V$ as a function of slit pore height h. See Equation (35). The dashed line shows the internal energy density of the bulk ${u}^{\mathrm{b}}$.

**Figure 8.**Surface tension $\gamma $ as a function of slit pore height h, see Equation (33). The dashed line shows the surface tension at infinite separation ${\gamma}^{\infty}$.

**Figure 10.**The scaling of normal pressure minus integral pressure as a function of the inverse slit pore height h.

**Table 1.**The reduced units are denoted with an asterisk in superscript, for example ${T}^{*}$. The variables are reduced using the molecular diameter $\sigma $, potential well depth $\u03f5$, fluid particle mass m and Boltzmann constant ${k}_{\mathrm{B}}$.

Description | Definition |
---|---|

Energy | ${E}^{*}=E/\u03f5$ |

Entropy | ${S}^{*}=S/{k}_{\mathrm{B}}$ |

Temperature | ${T}^{*}=T{k}_{\mathrm{B}}/\u03f5$ |

Distance | ${x}^{*}=x/\sigma $ |

Pressure | ${p}^{*}=p{\sigma}^{3}/\u03f5$ |

Chemical potential | ${\mu}^{*}=\mu /\u03f5$ |

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**MDPI and ACS Style**

Galteland, O.; Bedeaux, D.; Kjelstrup, S.
Nanothermodynamic Description and Molecular Simulation of a Single-Phase Fluid in a Slit Pore. *Nanomaterials* **2021**, *11*, 165.
https://doi.org/10.3390/nano11010165

**AMA Style**

Galteland O, Bedeaux D, Kjelstrup S.
Nanothermodynamic Description and Molecular Simulation of a Single-Phase Fluid in a Slit Pore. *Nanomaterials*. 2021; 11(1):165.
https://doi.org/10.3390/nano11010165

**Chicago/Turabian Style**

Galteland, Olav, Dick Bedeaux, and Signe Kjelstrup.
2021. "Nanothermodynamic Description and Molecular Simulation of a Single-Phase Fluid in a Slit Pore" *Nanomaterials* 11, no. 1: 165.
https://doi.org/10.3390/nano11010165