# Molecular Simulation of the Adsorption and Diffusion in Cylindrical Nanopores: Effect of Shape and Fluid–Solid Interactions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Simulation Details

_{cylinder,1}−LJ

_{fluid}, the medium interaction is C

_{cylinder,2}−LJ

_{fluid}, and the weak interaction is C

_{cylinder,3}−LJ

_{fluid}, with the interaction parameters detailed in Table 1. An arbitrary choice was made with respect to the absolute values of these size and energy parameters in order to provide measurable quantities. All results may be generalized by taking into account reduced properties; throughout the paper, we have expressed these reduced units alongside the absolute ones. Similarly, an arbitrary mass of 100 g·mol

^{−1}was assigned to each Mie sphere.

_{12}is the harmonic force constant in kJ·mol

^{−1}·m

^{−2}, and r

_{0}is the equilibrium bond length set here to be equal to the value of σ of the fluid. The force constant chosen for this harmonic potential had a value of 6000 kJ·mol

^{−1}·m

^{−2}. This is a relatively large value, essentially a rigid spring maintaining the bead tangent. A harmonic angle potential was employed to moderate the rigidity of linear chains:

_{123}is the force constant in kJ·mol

^{−1}·rad

^{−2}, and θ

_{0}is the reference angle, taken to be 180°. The fully flexible model is defined as a chain with k

_{123}= 0, while the rigid linear is characterized by k

_{123}= 3000 kJ·mol

^{−1}·rad

^{−2}. The ring trimer is defined by an equilateral triangle of side σ formed by the centers of the three molecules.

_{b}T/ε = 1.5), which was below the bulk critical temperature of both the chain and the ring-like configuration by a Nosé–Hoover thermostat, using a relaxation constant of 1 ps. The number of beads used for both the cylindrical pore and the repulsive wall were close to 30,000 (depending on the pore), while the number of beads for the fluid phase was approximately 2000. All the simulations were performed with a 0.008 ps time step and a total production run length (after equilibration) of 70 ns.

_{solid}/2) inward from that defined by the plane of the centers of the particles in the wall (c.f. Figure 3). In order to obtain the density of the adsorption region, the number of particles inside the cylindrical pore was counted over the time, and the average number of particles was taken when the loading in the pore was equilibrated. In the bulk region, the average number of particles was taken in the cubic zone showed in Figure 4. For the adsorption isotherms, the results are presented as the number of particles adsorbed as a function of the pressure in the system. In the simulation cell, the adsorbed region inside the cylinder was in equilibrium with the bulk phase, so the normal pressure inside the pores corresponded to the bulk (isotropic) pressure. Separate NVT isotropic single-phase simulations employing 500 particles were carried out at the density obtained from the bulk region to calculate the pressure of the system employing the classical virial (mechanical) route [38].

_{z}, in the direction of the pore axis (z coordinate in Figure 4) we employed the Einstein relation, i.e., we followed the limiting value of the time dependence of the mean-square displacement (MSD) [17]:

## 3. Results

#### 3.1. Adsorption of Trimers in Cylindrical Pores

_{b}value of 13.8 K, which is around 70% of the medium energy and 55% of the strong energy interaction. In this case, the energy was relatively low, so it is not possible to observe the inversion in the adsorption hierarchy. Here, the rigid linear morphology showed the maximum density value for the smallest pore, but the ring-like and the rigid linear converged to similar values. A more important observation is that, with the lowest energy value, we were able to produce type V isotherms (Figure 5h,i), which is characteristic of macroporous solids with weak fluid–solid interactions. This weakness caused the uptake of fluid at low pressures to be quite small, but once the particles became adsorbed, the fluid–fluid interactions added to the fluid–solid interaction in order to adsorb further molecules. In the isotherms, this led to a first-order transition from vapor phase to liquid phase, which is reflected as an abrupt discontinuity in the curve.

#### 3.2. Nematic Order Parameter

#### 3.3. Diffusion Coefficients

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Sample Availability: Not available. |

**Figure 1.**Temperature–density vapor–liquid coexistence curve for a 3-bead Lennard-Jones fluid. The solid blue line corresponds to the SAFT-VR-Mie equation of state (EoS) for chains, the solid black line corresponds to the SAFT-VR Mie EoS for rings [24], the filled black circles are obtained by molecular dynamic (MD) simulations for the ring trimer, the filled blue triangles are the simulation results for a fully flexible trimer, and the empty red squares are the simulation results for a rigid (stiff) linear trimer.

**Figure 2.**Fluid–solid intermolecular potentials for the cylindrical pore of radius 1.5 nm using three different values of fluid-wall energy, normalized with respect to the well depth of the weakest (ε

_{p}= 13.8 k

_{b}). Distances are normalized with respect to the soft diameter of a single fluid bead (σ = 0.3 nm).

**Figure 3.**Representation of the pores and molecules employed, drawn to scale. Diagram shows the convention employed to define the inner pore diameter.

**Figure 4.**Schematic representation of the simulation box used to obtain the adsorption isotherms and the bulk density. Periodic boundary conditions are applied in all Cartesian directions.

**Figure 5.**Adsorption isotherms at 150 K for trimers in a cylindrical pore. The top row corresponds to a strong fluid–solid energy interaction of εp/k

_{b}= 25.0 K per particle: (

**a**) rp = 0.5 nm, (

**b**) rp = 1.0 nm, and (

**c**) rp = 1.5 nm. The middle row corresponds to the medium fluid–solid energy interaction of εp/k

_{b}= 19.3 K per particle: (

**d**) rp = 0.5 nm, (

**e**) rp = 1.0 nm, and (

**f**) rp = 1.5 nm. The lower row corresponds to the weak fluid–solid energy interaction of εp/k

_{b}= 13.8 K per particle: (

**g**) rp = 0.5 nm, (

**h**) rp = 1.0 nm, and (

**i**) rp = 1.5 nm. In all cases, the symbols in the isotherms correspond to rings (filled black circles), rigid linear chains (filled red squares), and fully flexible chains (filled blue triangles). Solid lines are empirical nonlinear fits. Error bars are of the size of the symbols.

**Figure 6.**Nematic order parameter, S, as a function of the system pressure for the rigid linear morphology. From left to right, the energy of the fluid–solid interaction is increasing. Error bars are of the size of the symbols.

**Figure 7.**Self-diffusion coefficient in the z direction, Dz, for the three morphologies at 150 K in a cylindrical nanopore. (

**a**) The image on the left shows the diffusion coefficient as a function of the pore radius: 0.5 nm, 1.0 nm, 1.5 nm, and bulk phase at the high-density limit (15,167 mol/m

^{3}). (

**b**)The image on the right shows diffusion coefficient as a function of the pore density for rp = 1.0 nm.

**Table 1.**Summary of the molecular parameters for the solid–fluid and fluid–fluid potentials. k

_{b}is the Boltzmann constant.

σ [nm] | ε/k_{b} [K] | λ_{r} | λ_{a} | |
---|---|---|---|---|

C_{cylinder,1}-LJ_{fluid} | 0.25 | 25.00 | 11.0 | 4.0 |

C_{cylinder,2}-LJ_{fluid} | 0.25 | 19.30 | 11.0 | 4.0 |

C_{cylinder,3}-LJ_{fluid} | 0.25 | 13.80 | 11.0 | 4.0 |

LJ_{fluid} | 0.30 | 100.00 | 12.0 | 6.0 |

**Table 2.**Nematic order parameter, S, for the adsorbed liquid phase for different pore sizes (r

_{p}) and different values of fluid–solid energy interaction (ε

_{fluid-wall}/k

_{b}). Values are averages taken at high densities/pressures.

ε_{fluid-wall}/k_{b}[K] | 13.80 | 19.30 | 25.00 | ||||||
---|---|---|---|---|---|---|---|---|---|

r_{p} [nm] | 0.5 | 1.0 | 1.5 | 0.5 | 1.0 | 1.5 | 0.5 | 1.0 | 1.5 |

S | 0.69 | 0.477 | 0.45 | 0.815 | 0.506 | 0.47 | 0.86 | 0.53 | 0.48 |

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**MDPI and ACS Style**

Cárdenas, H.; Müller, E.A. Molecular Simulation of the Adsorption and Diffusion in Cylindrical Nanopores: Effect of Shape and Fluid–Solid Interactions. *Molecules* **2019**, *24*, 608.
https://doi.org/10.3390/molecules24030608

**AMA Style**

Cárdenas H, Müller EA. Molecular Simulation of the Adsorption and Diffusion in Cylindrical Nanopores: Effect of Shape and Fluid–Solid Interactions. *Molecules*. 2019; 24(3):608.
https://doi.org/10.3390/molecules24030608

**Chicago/Turabian Style**

Cárdenas, Harry, and Erich A. Müller. 2019. "Molecular Simulation of the Adsorption and Diffusion in Cylindrical Nanopores: Effect of Shape and Fluid–Solid Interactions" *Molecules* 24, no. 3: 608.
https://doi.org/10.3390/molecules24030608