# The Effect of Topology on Phase Behavior under Confinement

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Minkowski Functionals

## 3. Methods

#### 3.1. Density Functional Theory

#### 3.2. Geometries

#### 3.3. Simulation Parameters

_{2}in a SiO

_{2}slit pore with a width of $L/{\sigma}_{\mathrm{ff}}=500$. The results confirm that the parameters listed in Table 1 are a reasonable choice. With more advanced models for the interaction between N

_{2}and SiO

_{2}, a better match can be obtained between cDFT simulations and experiments [35,60]. However, the choice of the same potential for particle-particle and wall-particle keeps the system simple and the results easier to interpret. The computations are performed in the grand canonical ($\mu ,V,T$) ensemble and the relation between the chemical potential and pressure is obtained from a bulk cDFT simulation. Since the pore space is in equilibrium with the bulk, both the temperature and chemical potential are constant throughout the system [63].

#### 3.4. Minkowski Coefficients

## 4. Results

## 5. Discussion & Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The various 2D geometries and topologies used in the simulations. Along the vertical axis, the various shapes show the cross-section of pores (i.e., disks) with different radii, ${r}_{p}$. Along the horizontal axis, rods (i.e., small disks) with radius ${r}_{p}=1.0$ (≈$0.36\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}$) are placed inside the pores to modify the Euler characteristic, $\chi $. The open-pore space is colored white and gray. The pores in gray are simulation cases where the distance between walls of either the pores or the rods are smaller than ≈$10{\sigma}_{\mathrm{ff}}$, which is the distance at which Hadwiger’s theorem starts to break down [31].

**Figure 2.**Comparison between experiments [62] and the cDFT simulations for the adsorption isotherm of N

_{2}in a SiO

_{2}slit pore with a width of $L=500{\sigma}_{\mathrm{ff}}$. The chosen wall potential does not fully capture the interactions between N

_{2}and SiO

_{2}, but the results show a good match. This confirms that the used parameters shown in Table 1 are a reasonable choice.

**Figure 3.**Dimensionless 2D grand potential, $\Omega /L$, as a function of the dimensionless chemical potential, $\mu $, for a Lennard-Jones fluid. Only the cDFT simulation results when $\chi =1$ are shown. Inset (

**a**) shows the continuous phase transition which marks the onset of gas adsorption onto the wall. Panel (

**d**) shows the corresponding diverging derivative of the excess adsorption. Inset (

**b**) shows the discrete (first-order) capillary condensation phase transition. Panel (

**c**) shows the matching jump in the dimensionless excess adsorption.

**Figure 4.**Dimensionless pressure difference between the continuous phase transition associated with adsorption of gas onto a wall in the bulk and inside a pore times the surface area Minkowski functional, $\Delta {p}_{w}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}A\left(X\right)$, as function of circumference Minkowski functional, $C\left(X\right)$. The Minkowski functional for topology is zero in all the shown simulations, $K\left(X\right)=0$. The curve fit is equal to: ${f}_{w}\left({C}^{3/4}\right)={\Omega}_{\mathrm{lg}}+{\sigma}_{\mathrm{lg}}^{\prime}{C}^{3/4}$, where the Minkowski functional $C\left(X\right)$ only depends on ${r}_{p}$. The coefficients are: ${\Omega}_{\mathrm{lg}}=-8.3\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-3}\pm \phantom{\rule{3.33333pt}{0ex}}0.1\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-3}$ and ${\sigma}_{\mathrm{lg}}^{\prime}=1.538\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-3}\pm \phantom{\rule{3.33333pt}{0ex}}0.008\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-3}$. In dimensional form, from left to right the pressure differences are $1.8\phantom{\rule{3.33333pt}{0ex}}\mathrm{kPa}$, $1.3\phantom{\rule{3.33333pt}{0ex}}\mathrm{kPa}$, $1.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{kPa}$, $0.69\phantom{\rule{3.33333pt}{0ex}}\mathrm{kPa}$, and $0.40\phantom{\rule{3.33333pt}{0ex}}\mathrm{kPa}$.

**Figure 5.**Dimensionless pressure difference between the continuous phase transition associated with adsorption of gas onto a wall in the bulk and inside a pore times the surface area Minkowski functional minus the function from Figure 4, $\Delta {p}_{w}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}A\left(X\right)-{f}_{w}\left({C}^{3/4}\right)$, as function of the Minkowski functional, $K\left(X\right)$. The graph shows a collapse of the data and a linear fit with: ${g}_{w}\left(K\right)={\kappa}_{\mathrm{lg}}^{\prime}K\left(X\right)$, where ${\kappa}_{\mathrm{lg}}^{\prime}=4.6\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-4}\pm 0.2\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-4}$.

**Figure 6.**Dimensionless pressure difference between the capillary condensation pressure and bulk phase transition pressure times the surface area Minkowski functional, $\Delta p\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}A\left(X\right)$, as a function of pore radius, ${r}_{p}$. The Minkowski functional for topology is zero in all the shown simulations, $K\left(X\right)=0$. The curve fit is equal to: $f({C}^{3/4},{A}^{1/2})={\Omega}_{\mathrm{lg}}+{\sigma}_{\mathrm{lg}}^{\prime}{C}^{3/4}+{p}_{\mathrm{lg}}^{\prime}{A}^{1/2}$, where the Minkowski functionals $C\left(X\right)$ and $A\left(X\right)$ only depend on ${r}_{p}$. The coefficients are ${\Omega}_{\mathrm{lg}}=-0.4\pm 0.2$, ${\sigma}_{\mathrm{lg}}^{\prime}=0.12\pm 0.03$, and ${p}_{\mathrm{lg}}^{\prime}=-0.04\pm 0.02$. In dimensional form, from left to right the pressure differences are $97\phantom{\rule{3.33333pt}{0ex}}\mathrm{kPa}$, $69\phantom{\rule{3.33333pt}{0ex}}\mathrm{kPa}$, $49\phantom{\rule{3.33333pt}{0ex}}\mathrm{kPa}$, $29\phantom{\rule{3.33333pt}{0ex}}\mathrm{kPa}$, and $15\phantom{\rule{3.33333pt}{0ex}}\mathrm{kPa}$.

**Figure 7.**Dimensionless pressure difference between capillary condensation pressure and bulk phase transition pressure times the surface area Minkowski functional minus the function from Figure 6, $\Delta p\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}A\left(X\right)-f({C}^{3/4},{A}^{1/2})$, as a function of the Minkowski functional, $K\left(X\right)$. The graph shows a collapse of the data and a linear fit with: $g\left(K\right)={\kappa}_{\mathrm{lg}}^{\prime}K\left(X\right)$, where ${\kappa}_{\mathrm{lg}}^{\prime}=-0.041\pm 0.002$.

**Figure 8.**Dimensionless 2D grand potential, $\Omega /L$, as a function of the dimensionless chemical potential, $\mu $, for a Lennard-Jones fluid. Only the simulation results when $\chi =1$ are shown. The different lines show the results of the cDFT simulations while the symbols show the grand potential as reconstructed from the Minkowski functionals and one set of Minkowski functional coefficients: pressure, $p\left(\right)open="("\; close=")">\mu ,T$, surface tension, $\sigma \left(\right)open="("\; close=")">\mu ,T$, bending rigidity, $\kappa \left(\right)open="("\; close=")">\mu ,T$, and the pseudo pressure and surface tension terms ${p}^{\prime}\left(\right)open="("\; close=")">\mu ,T$ and ${\sigma}^{\prime}\left(\right)open="("\; close=")">\mu ,T$.

**Figure 9.**(

**a**) Dimensionless Minkowski functional coefficients: pressure, $p\left(\right)open="("\; close=")">\mu ,T$, surface tension, $\sigma \left(\right)open="("\; close=")">\mu ,T$, bending rigidity, $\kappa \left(\right)open="("\; close=")">\mu ,T$, and the pseudo pressure and surface tension terms ${p}^{\prime}\left(\right)open="("\; close=")">\mu ,T$ and ${\sigma}^{\prime}\left(\right)open="("\; close=")">\mu ,T$, as a function of the dimensionless chemical potential, $\mu $. These are the values of the coefficients that are used in Figure 8 to reconstruct the grand potential as a function of the chemical potential. The pressure coefficient is very similar to the bulk pressure, ${p}_{b}$. (

**b**) Contribution of the Minkowski functional coefficients to the grand potential as a function of the dimensionless chemical potential for a pore size of ${r}_{p}=25$ (≈$9\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}$) and an Euler characteristic of $\chi =-5$.

**Figure 10.**Average absolute relative error $<|{\u03f5}_{\Omega}|>$ as a function of the minimal characteristic length scale of the system, ${l}_{\mathrm{min}}$. In the case of a pore without rods, this distance is twice the radius. When rods are present within the pore this is the smallest distance between the pore wall and a rod or between two different rods.

**Figure 11.**Dimensionless 2D excess adsorption, $\Gamma /L$, as a function of the dimensionless chemical potential, $\mu $, for a Lennard-Jones fluid. Only the simulation results from Figure 1 when $\chi =1$ are shown. The different lines show the results of the cDFT simulations while the symbols show the grand potential as reconstructed from the Minkowski functionals and one set of Minkowski functional coefficients: the derivatives of pressure, $\partial (p-{p}_{b})/\partial \mu $, pressure per surface area, ${p}^{\prime}\left(\right)open="("\; close=")">\mu ,T$, surface tension, $\partial \sigma /\partial \mu $, bending rigidity, $\partial \kappa /\partial \mu $, and the pseudo pressure and surface tension terms $\partial {p}^{\prime}/\partial \mu $ and $\partial {\sigma}^{\prime}/\partial \mu $ with respect to the chemical potential.

**Figure 12.**(

**a**) Dimensionless Minkowski functional coefficients: the derivatives of pressure, $\partial (p-{p}_{b})/\partial \mu $, pressure per surface area, ${p}^{\prime}\left(\right)open="("\; close=")">\mu ,T$, surface tension, $\partial \sigma /\partial \mu $, bending rigidity, $\partial \kappa /\partial \mu $, and the pseudo pressure and surface tension terms $\partial {p}^{\prime}/\partial \mu $ and $\partial {\sigma}^{\prime}/\partial \mu $, with respect to the chemical potential. These are the values of the coefficients that are used in Figure 11 to reconstruct the excess adsorption as a function of the chemical potential. (

**b**) Contribution of the Minkowski functional coefficients to the excess adsorption for a pore with a radius of ${r}_{p}=25$ (≈$9\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}$) and an Euler characteristic of $\chi =-5$.

**Figure 13.**Average absolute relative error $<|{\u03f5}_{\Gamma}|>$ as a function of the minimal characteristic length scale of the system, ${l}_{\mathrm{min}}$. In the case of a pore without rods, this distance is twice the radius. When rods are present within the pore this is the smallest distance between the pore wall and a rod or between two different rods. Because the excess adsorption is a derivative of the grand potential, the observed error is larger than Figure 10.

**Table 1.**cDFT parameters of N

_{2}and SiO

_{2}[60]. The number density for SiO

_{2}is ${\rho}_{\mathrm{s}}=66.15\phantom{\rule{3.33333pt}{0ex}}{\mathrm{nm}}^{-3}$ [35]. Fluid-fluid interactions are truncated at $5{\sigma}_{\mathrm{ff}}$. The simulations are performed at $77.3\phantom{\rule{3.33333pt}{0ex}}\mathrm{K}$.

${\mathit{\u03f5}}_{\mathbf{ff}}/{\mathit{k}}_{\mathit{B}}$ | ${\mathit{\sigma}}_{\mathbf{ff}}$ | ${\mathit{d}}_{\mathbf{HS}}$ | ${\mathit{\u03f5}}_{\mathbf{sf}}/{\mathit{k}}_{\mathit{B}}$ | ${\mathit{\sigma}}_{\mathbf{sf}}$ | |
---|---|---|---|---|---|

$\left[\mathbf{K}\right]$ | $\left[\mathbf{nm}\right]$ | $\left[\mathbf{nm}\right]$ | $\left[\mathbf{K}\right]$ | $\left[\mathbf{nm}\right]$ | |

N_{2} | 94.45 | 0.3575 | 0.3575 | 147.3 | 0.317 |

**Table 2.**An overview of the dimensionless variables used in the results section. “$\phantom{\rule{0.277778em}{0ex}}\widehat{}\phantom{\rule{0.277778em}{0ex}}$” denotes the dimensional equivalent of a dimensionless variable. In addition, ${k}_{b}$ is the Boltzmann constant, ${\sigma}_{\mathrm{ff}}=0.3575\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}$, $T=77.3\phantom{\rule{3.33333pt}{0ex}}\mathrm{K}$, and L is one dimensionless unit length.

Description | Symbol | Dimensionless Definition |
---|---|---|

Distance | $\mathrm{length}$ | $\widehat{\mathrm{length}}/{\sigma}_{\mathrm{ff}}$ |

Density | $\rho $ | $\widehat{\rho}{\sigma}_{\mathrm{ff}}^{3}$ |

Grand potential | $\Omega /L$ | $\widehat{\Omega}\phantom{\rule{0.277778em}{0ex}}{\sigma}_{\mathrm{ff}}/\widehat{L}\phantom{\rule{0.277778em}{0ex}}{k}_{b}T$ |

Excess adsorption | $\Gamma /L$ | $\widehat{\Gamma}{\sigma}_{\mathrm{ff}}/\widehat{L}$ |

Chemical potential | $\mu $ | $\widehat{\mu}/{k}_{b}T$ |

Pressure | p | $\widehat{p}{\sigma}_{\mathrm{ff}}^{3}/{k}_{b}T$ |

Surface tension | $\sigma $ | $\widehat{\sigma}{\sigma}_{\mathrm{ff}}^{2}/{k}_{b}T$ |

Bending rigidity | $\kappa $ | $\widehat{\kappa}{\sigma}_{\mathrm{ff}}/{k}_{b}T$ |

Pseudo pressure | ${p}^{\prime}$ | ${\widehat{p}}^{\prime}{\sigma}_{\mathrm{ff}}^{2}/{k}_{b}T$ |

Pseudo surface tension | ${\sigma}^{\prime}$ | ${\widehat{\sigma}}^{\prime}{\sigma}_{\mathrm{ff}}^{7/4}/{k}_{b}T$ |

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Boelens, A.M.P.; Tchelepi, H.A.
The Effect of Topology on Phase Behavior under Confinement. *Processes* **2021**, *9*, 1220.
https://doi.org/10.3390/pr9071220

**AMA Style**

Boelens AMP, Tchelepi HA.
The Effect of Topology on Phase Behavior under Confinement. *Processes*. 2021; 9(7):1220.
https://doi.org/10.3390/pr9071220

**Chicago/Turabian Style**

Boelens, Arnout M. P., and Hamdi A. Tchelepi.
2021. "The Effect of Topology on Phase Behavior under Confinement" *Processes* 9, no. 7: 1220.
https://doi.org/10.3390/pr9071220