On the hysteresis loop and equilibrium transition in slit-shaped ink-bottle pores

Abstract

Bin grand canonical Monte Carlo simulations have been carried out to study adsorption–desorption of argon at 87.3 K in a model ink-bottle mesoporous solid in order to investigate the interplay between the pore blocking process, controlled by the evaporation through the pore mouth via the meniscus separating the adsorbate and the bulk gas surroundings, and the cavitation process, governed by the instability of the stretched fluid (with a decrease in pressure) in the cavity. The evaporation mechanism switches from pore blocking to cavitation when the size of the pore neck is decreased, and is relatively insensitive to the neck length under conditions where cavitation is the controlling mechanism. We have applied the recently-developed Mid-Density scheme to determine the equilibrium branch of the hysteresis loop, and have found that, unlike ideal simple pores of constant size and infinite length, where the equilibrium transition is vertical, the equilibrium branch of an ink-bottle pore has three distinct sub-branches within the hysteresis loop. The first sub-branch is steep but continuous and is close to the desorption branch (which is typical for a pore with two open ends); this is associated with the equilibrium state in the neck. The third sub-branch is much steeper and is close to the adsorption branch (which is typical for either a pore with one end closed or a closed pore), and is associated with the equilibrium state in the cavity. The second sub-branch, connecting the other two sub-branches, has a more gradual slope. The domains of these three sub-branches depend on the relative sizes of the cavity and the neck, and their respective lengths. Our investigation of the effects of changing neck length clearly demonstrates that cavitation depends, not only on fluid properties, as frequently stated, but also on pore geometry.

Appendix

The interaction of a particle and a strip which is finite in one direction (y-direction) and infinite in the other (x-direction) (shown in Fig. 18) is calculated from the Bojan–Steele equation (Bojan and Steele 1988, 1989, 1993):

$$U_{i,s} = \sum\limits_{a = 1}^{M} {2\pi \varepsilon_{i}^{a,s} \rho_{s} (\sigma_{i}^{a,s} )^{2} \left\{ {\left[ {U_{rep} (y_{i}^{a, + } ,z_{i}^{a} ) - U_{rep} (y_{i}^{a, - } ,z_{i}^{a} )} \right]} \right.} \;\; - \left. {\left[ {U_{att} (y_{i}^{a, + } ,z_{i}^{a} ) - U_{att} (y_{i}^{a, - } ,z_{i}^{a} )} \right]} \right\}$$

(6)

where

$$y_{i}^{a, + } = \frac{L}{2} - y_{i}^{a} ;\,\quad y_{i}^{a, - } = - \frac{L}{2} - y_{i}^{a}$$

(7)

The repulsive and attractive terms on the right hand side of Eq. (6) are given by:

$$\begin{aligned} U_{\text{rep}} (y,z) = & & \frac{y}{{\sqrt {y^{2} + z^{2} } }}\left\{ {\frac{1}{5}\left( {\frac{{\sigma_{i}^{a,s} }}{z}} \right)} \right.^{10} + \frac{1}{10}\frac{{(\sigma_{i}^{a,s} )^{10} }}{{z^{8} (y^{2} + z^{2} )}} \\ + \frac{3}{40}\frac{{(\sigma_{i}^{a,s} )^{10} }}{{z^{6} (y^{2} + z^{2} )^{2} }} + \frac{1}{16}\frac{{(\sigma_{i}^{a,s} )^{10} }}{{z^{4} (y^{2} + z^{2} )^{3} }} \\ \left. { + \frac{7}{128}\frac{{(\sigma_{i}^{a,s} )^{10} }}{{z^{2} (y^{2} + z^{2} )^{4} }}} \right\} \\ \end{aligned}$$

(8)

$$U_{\text{att}} (y,z) = \frac{y}{{\sqrt {y^{2} + z^{2} } }}\left\{ {\frac{1}{2}\left( {\frac{{\sigma_{i}^{a,s} }}{z}} \right)^{4} + \frac{1}{4}\frac{{(\sigma_{i}^{a,s} )^{4} }}{{z^{2} (y^{2} + z^{2} )}}} \right\}$$

(9)

where $z_{i}^{a}$ is the distance of site a of molecule i from the graphite surface, $\varepsilon_{i}^{a,s}$ and $\sigma_{i}^{a,s}$are the adsorptive-graphite interaction potential well-depth and intermolecular collision diameter respectively, and ρ _s is the surface density (taken as 38.2 nm⁻² in this study). The cross parameters, $\sigma_{i}^{a,s}$and $\varepsilon_{i}^{a,s}$can be determined by the Lorentz–Berthelot mixing rules: $\sigma_{i}^{a,s} = (\sigma_{i}^{a,a} + \sigma_{{}}^{s} )/2$ and $\varepsilon_{i}^{a,s} = (\varepsilon_{i}^{a,a} \varepsilon_{{}}^{s} )^{1/2}$ where the parameters for a carbon atom in a graphene layer are σ ^s = 0.34 nm and ε ^s /k _B  = 28.0 K.The potential equation energy in Eqs. (6)–(9) is valid for any positions around the strip, except z ^a _i  → 0, and in such cases we use the following Taylor series expansion:

$$\begin{gathered} \frac{{U_{i,s} }}{{2\pi K\rho_{s} }} = \sum\limits_{a = 1}^{M} {\varepsilon_{i}^{a,s} (\sigma_{i}^{a,s} )^{2} \left( {\,\,\,\left\{ {\frac{63}{1280}\left[ {\left( {\frac{{\sigma_{i}^{a,s} }}{{y_{i}^{a, - } }}} \right)^{10} - \left( {\frac{{\sigma_{i}^{a,s} }}{{y_{i}^{a, + } }}} \right)^{10} } \right]} \right.} \right.} \\ - \left. {\frac{3}{16}\left[ {\left( {\frac{{\sigma_{i}^{a,s} }}{{y_{i}^{a, - } }}} \right)^{4} - \left( {\frac{{\sigma_{i}^{a,s} }}{{y_{i}^{a, + } }}} \right)^{4} } \right]} \right\} \\ - \left( {\frac{{z_{i}^{a} }}{{\sigma_{i}^{a,s} }}} \right)^{2} \left\{ {\frac{231}{1024}\left[ {\left( {\frac{{\sigma_{i}^{a,s} }}{{y_{i}^{a, - } }}} \right)^{12} - \left( {\frac{{\sigma_{i}^{a,s} }}{{y_{i}^{a, + } }}} \right)^{12} } \right]} \right. \\ - \left. {\frac{5}{16}\left[ {\left( {\frac{{\sigma_{i}^{a,s} }}{{y_{i}^{a, - } }}} \right)^{6} - \left( {\frac{{\sigma_{i}^{a,s} }}{{y_{i}^{a, + } }}} \right)^{6} } \right]} \right\} \\ + \left( {\frac{{z_{i}^{a} }}{{\sigma_{i}^{a,s} }}} \right)^{4} \left\{ {\frac{1287}{2048}\left[ {\left( {\frac{{\sigma_{i}^{a,s} }}{{y_{i}^{a, - } }}} \right)^{14} - \left( {\frac{{\sigma_{i}^{a,s} }}{{y_{i}^{a, + } }}} \right)^{14} } \right]} \right. \\ - \left. {\frac{105}{256}\left[ {\left( {\frac{{\sigma_{i}^{a,s} }}{{y_{i}^{a, - } }}} \right)^{8} - \left( {\frac{{\sigma_{i}^{a,s} }}{{y_{i}^{a, + } }}} \right)^{8} } \right]} \right\} \\ - \left( {\frac{{z_{i}^{a} }}{{\sigma_{i}^{a,s} }}} \right)^{6} \left\{ {\frac{45045}{32768}\left[ {\left( {\frac{{\sigma_{i}^{a,s} }}{{y_{i}^{a, - } }}} \right)^{16} - \left( {\frac{{\sigma_{i}^{a,s} }}{{y_{i}^{a, + } }}} \right)^{16} } \right]} \right. \\ - \left. {\frac{63}{128}\left[ {\left( {\frac{{\sigma_{i}^{a,s} }}{{y_{i}^{a, - } }}} \right)^{10} - \left( {\frac{{\sigma_{i}^{a,s} }}{{y_{i}^{a, + } }}} \right)^{10} } \right]} \right\} \\ + \left( {\frac{{z_{i}^{a} }}{{\sigma_{i}^{a,s} }}} \right)^{8} \left\{ {\frac{85085}{32768}\left[ {\left( {\frac{{\sigma_{i}^{a,s} }}{{y_{i}^{a, - } }}} \right)^{18} - \left( {\frac{{\sigma_{i}^{a,s} }}{{y_{i}^{a, + } }}} \right)^{18} } \right]} \right. \\ \left. { - \left. {\frac{1155}{2048}\left[ {\left( {\frac{{\sigma_{i}^{a,s} }}{{y_{i}^{a, - } }}} \right)^{12} - \left( {\frac{{\sigma_{i}^{a,s} }}{{y_{i}^{a, + } }}} \right)^{12} } \right]} \right\} + O(z/\sigma_{i}^{a,s} )^{10} } \right) \\ \end{gathered}$$

(10)

where K = 1 for positive values of $y_{i}^{a, + }$and $y_{i}^{a, - }$, K = −1 for negative values of $y_{i}^{a, + }$ and $y_{i}^{a, - }$.

Fig. 18

Configuration of a finite strip