# Gibbs Ensemble Monte Carlo Simulation of Fluids in Confinement: Relation between the Differential and Integral Pressures

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## Abstract

**:**

## 1. Introduction

## 2. Simulation Details

## 3. Theory

## 4. Results and Discussion

#### 4.1. Difference between the Differential and Integral Pressure

#### 4.2. Ratio of Driving Forces

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic representation of the Monte Carlo (MC) simulation scheme. Simulation Box 1 represents the bulk fluid with differential pressure P. Simulation Box 2 contains the confined fluid with (average) integral pressure $\langle \widehat{p}\rangle $. (

**a**,

**b**) The two investigated systems are shown where the bulk fluid is in equilibrium with the nanoconfined fluid in a cylinder and in a slit pore, respectively. Due to the particle exchange, the chemical potential of the two boxes are equal, but in general $P\ne \langle \widehat{p}\rangle $.

**Figure 2.**The chemical potential (

**a**) and density (

**b**) of the bulk, Box 1, and confined fluids, Box 2, as a function of the cylinder radius, R, at $P=0.2$ (in Simulation Box 1). The blue and red colors represent Box 1 and Box 2, respectively. The temperature is fixed at $T=2$. All values are presented in reduced units. The error bars are smaller than the symbol sizes.

**Figure 3.**Schematic representation of a bulk fluid in equilibrium with a nanoconfined fluid in a pore. The concept of the ratio of driving forces for mass transport, $\frac{\mathrm{d}\langle \widehat{p}\rangle}{\mathrm{d}P}$, can be introduced based on the definition of ratio of driving forces in the two systems, $\frac{\mathrm{d}\langle \widehat{p}\rangle}{\mathrm{d}L}$ and $\frac{\mathrm{d}P}{\mathrm{d}L}$.

**Figure 4.**The difference between the differential, P, and the ensemble average of integral pressure, $\langle \widehat{p}\rangle $, is shown as a function of the inverse radius ${R}^{-1}$ of Box 2 at $P=$ 0.2, 2.0, and 6.0 for cylindrical and slit pores with fluid-wall interactions. (

**a**,

**c**,

**e**) The pressure difference is shown for cylindrical pores at differential pressure $P=0.2,$ 2.0, and 6.0, respectively. (

**b**,

**d**,

**f**) The pressure difference is shown for slit pores at differential pressure $P=0.2,$ 2.0, and 6.0, respectively. The simulation results are shown with symbols, while the lines are fits to the data points. The equation used for the fitting is $A{R}^{-1}+B$, where A and B are constants. The colors denote the different level of attraction between the wall of Box 2 and the fluid, repulsive wall potential (black), ${\u03f5}_{\mathrm{wf}}=$ 0.3 (blue), 0.5 (red), 0.7 (gray), 1.0 (orange), and 1.5 (cyan). The results for the cylindrical pore are shown with closed circles and for the slit pores with open rectangles. The temperature of both boxes is set to $T=2$. The average densities of Box 1 are $\rho \approx $ 0.10, 0.58, and 0.8 at $P=$ 0.2, 2.0, and 6.0, respectively.

**Figure 5.**The ratio of driving forces, $\frac{\mathrm{d}\langle \widehat{p}\rangle}{\mathrm{d}P}$, is shown as a function of the inverse radius, ${R}^{-1}$, of Box 2 at $P=0.2,2.0$, and 6.0 for the cylindrical and slit pore cases with repulsive and attractive wall potentials. (

**a**,

**c**,

**e**) $\frac{\mathrm{d}\langle \widehat{p}\rangle}{\mathrm{d}P}$ is shown for cylindrical pores at differential pressure $P=0.2$, 2.0, and 6.0, respectively. (

**b**,

**d**,

**f**) $\frac{\mathrm{d}\langle \widehat{p}\rangle}{\mathrm{d}P}$ is shown for slit pores at differential pressures $P=0.2$, 2.0, and 6.0, respectively. The simulation results are shown with symbols, while the lines are fits to the data points. The equation used for the fitting is $A{R}^{-1}+B$, where A and B are constants. The results for the cylindrical pore are shown with closed circles and for the slit pores with open rectangles. The colors denote the different wall potential used in Box 2, repulsive wall potential (black), ${\u03f5}_{\mathrm{wf}}=0.3$ (blue), ${\u03f5}_{\mathrm{wf}}=0.5$ (red), ${\u03f5}_{\mathrm{wf}}=0.7$ (gray), ${\u03f5}_{\mathrm{wf}}=1.0$ (orange), and ${\u03f5}_{\mathrm{wf}}=1.5$ (cyan). The temperature of both boxes is set to $T=2$. The average densities of Box 1 are $\rho \approx $ 0.10, 0.58, and 0.8 at $P=$ 0.2, 2.0, and 6.0, respectively.

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**MDPI and ACS Style**

Erdős, M.; Galteland, O.; Bedeaux, D.; Kjelstrup, S.; Moultos, O.A.; Vlugt, T.J.H.
Gibbs Ensemble Monte Carlo Simulation of Fluids in Confinement: Relation between the Differential and Integral Pressures. *Nanomaterials* **2020**, *10*, 293.
https://doi.org/10.3390/nano10020293

**AMA Style**

Erdős M, Galteland O, Bedeaux D, Kjelstrup S, Moultos OA, Vlugt TJH.
Gibbs Ensemble Monte Carlo Simulation of Fluids in Confinement: Relation between the Differential and Integral Pressures. *Nanomaterials*. 2020; 10(2):293.
https://doi.org/10.3390/nano10020293

**Chicago/Turabian Style**

Erdős, Máté, Olav Galteland, Dick Bedeaux, Signe Kjelstrup, Othonas A. Moultos, and Thijs J. H. Vlugt.
2020. "Gibbs Ensemble Monte Carlo Simulation of Fluids in Confinement: Relation between the Differential and Integral Pressures" *Nanomaterials* 10, no. 2: 293.
https://doi.org/10.3390/nano10020293