Figure 1

Diagrams showing 2D sections of linear pores of $N=5$ cells unoccupied by matrix, labeled $i=1,...,5$ (white squares), and different configurations of matrix cells (gray squares) in the 3D system. (a) Pore of type I completely surrounded by matrix. (b) Pore of type II with open ends. (c) Pore of type III with one open and one closed end consisting of two sections with statistically different matrix-fluid interaction represented by light- and dark-gray cells. The interaction of the fluid in cells $1\le i\le N_1=3$ and in cells $4<i\le N$ with the matrix represented by light-gray and dark-gray squares is described by $\langle w_1^{\text{mf}}\rangle$, and $\langle w_2^{\text{mf}}\rangle$, respectively. For clarity, two relevant matrix cells per unoccupied cell, on top of and underneath the white cells, are not shown in this 2D diagram.Reuse & Permissions

Figure 2

Condensation in pores of types I and II [cf. Figs. 1a and 1b]. The mean $\langle V\rangle$ (upper panel) and variance $\text{Var}\left[V\right]$ (lower panel) of occupied volume of pores of length $N=100$ are plotted vs $\mu$. The bold arrows show the direction in which $\mu$ changes for adsorption and desorption curves. The solid (dashed) curves correspond to pores of type I (II). Different columns refer to different degrees of normal disorder in $w_{ij}^{\text{mf}}$ with the same mean value $\langle w^{\text{mf}}\rangle =1.0$ and the same $w^{\text{ff}}=1.0$: (a) $\rho \left(w^{\text{mf}}\right)$ has a width $\Delta =0.05$ and (b) a width $\Delta =0.25$. Symbols refer to the numerical results.Reuse & Permissions

Figure 3

Condensation in pores of types I and II. As in Fig. 2, mean $\langle V\rangle$ (upper panel) and variance $\text{Var}\left[V\right]$ (lower panel) of occupied volume of pores of length $N=100$ are plotted vs $\mu$. Columns show different values of normal disorder $\Delta$ with constant, $\langle w^{\text{mf}}\rangle =1.0$ and $w^{\text{ff}}=4.0$. The degree of disorder is $\Delta =0.1$ in column (a) and $\Delta =0.4$ in column (b).Reuse & Permissions

Figure 4

Characteristic values of $\mu$ relevant to adsorption for a range of strengths $\Delta$ of normal disorder. Shown are the values of $\langle {\mu }_{\text{min}}^{\text{c}}\rangle$ (solid line), $\langle {\mu }_{\text{min}}^{\text{o}}\rangle =\langle {\mu }_{\text{min}}^{\text{inner}}\rangle$ (dashed line), and ${\mu }_{\text{max}}^{\text{pin}}$ (dot-dashed line). In both panels $\langle w^{\text{mf}}\rangle =1.0$ and $N=100$. The fluid-fluid interaction is $w^{\text{ff}}=1.0$ in panel (a) and $w^{\text{ff}}=4.0$ in panel (b). The vertical lines with upward arrows indicate values of $\Delta$ corresponding to the four different regimes, (1)–(4), for adsorption described in the text.Reuse & Permissions

Figure 5

Four regimes for (a) adsorption and (b) desorption are shown in the parameter space of $\langle w^{\text{mf}}\rangle$ and $w^{\text{ff}}$ for pores of type I of length $N=100$ and $\Delta =0.1$. The characteristic points A$({\langle w^{\text{mf}}\rangle }_{-},w^{\text{ff}})$ and D$({\langle w^{\text{mf}}\rangle }_{+},w^{\text{ff}})$ are located at ${\langle w^{\text{mf}}\rangle }_{\pm }=\Delta [\sqrt{8}{\text{erfc}}^{-1}(2/N)\pm \sqrt{5/\pi }]$ and $w^{\text{ff}}=2\sqrt{8}\Delta {\text{erfc}}^{-1}(2/N)$. Adsorption in pores of type II exhibits two regimes, (2) and (4). Regime (2) corresponds to the region (2) in panel (a) and regime (4) spans over the regions (1), (3), and (4) in (a). Similarly, desorption in pores of type II occurs only in two regimes, (1) and (2). Regime (2) corresponds to the region (2) in panel (b) and regime (1) spans over the regions (1), (3), and (4) in (b).Reuse & Permissions

Figure 6

Characteristic values of $\mu$ for (a) adsorption and (b) desorption for exponential disorder as functions of the degree of disorder, $\Delta$, for a pore of length $N=100$ with $\langle w^{\text{mf}}\rangle =1.0$ and $w^{\text{ff}}=1.25$. In panel (a), solid, dashed, and dot-dashed lines indicate $\langle {\mu }_{\text{min}}^{\text{c}}\rangle$, $\langle {\mu }_{\text{min}}^{\text{o}}\rangle =\langle {\mu }_{\text{min}}^{\text{inner}}\rangle$ and ${\mu }_{\text{max}}^{\text{pin}}$, respectively. The vertical lines with upward arrows give examples of adsorption in regimes (1), (2), and (3). Panel (b) shows $\langle {\mu }_{\text{max}}^{\text{c}}\rangle$ (solid line), $\langle {\mu }_{\text{max}}^{\mathrm{o}}\rangle$ (dashed line), ${\mu }_{\text{min}}^{\text{pin}}$ (dot-dashed line), and $\langle {\mu }_{\max }^{\mathrm{inner}}\rangle$ (dotted line). The dashed curve in panel (b) merges with the dotted and solid curves for large $\Delta$ (off the right-hand edge of the graph). Examples of desorption occurring in regimes (1), (2), and (3) are indicated by vertical lines with downward arrows in panel (b).Reuse & Permissions

Figure 7

Condensation for linear pores of length $N=100$ with exponential disorder of strength $\Delta =0.02$ [panel (a)] and $\Delta =0.2$ [panel (b)]. Same line styles are used as in Figs. 2 and 3. The interaction parameters are $\langle w^{\text{mf}}\rangle =1.0$ and $w^{\text{ff}}=1.25$. The locations of the cusps are marked by arrows and given by ${\mu }_{\text{max}}^{\text{c}}=-5.25$, ${\mu }_{\text{max}}^{\text{o}}=M^{\text{pin}}=-4.45$, and $M^{\text{inner}}=-5.7$.Reuse & Permissions

Figure 8

Condensation in pores of type III [cf. Fig. 1c]. $\langle V\rangle$ (upper panel) and $\text{Var}\left[V\right]$ (lower panel) plotted vs $\mu$ for pores of length $N=100$. The curves of different styles refer to different geometries, i.e., ink-bottle (solid lines), with $N_1=50$, $w_1=1.2$, $w_2=1.0$, and funnel (dashed lines), with $N_1=50$, $w_1=1.0$, $w_2=1.2$. The light dotted lines in the upper panels of (a)–(c) correspond to the mean occupied volume of two separate open-ended pores [shape (b) in Fig. 1] of length $N=50$, one of which has $\langle w_{ij}^{\text{mf}}\rangle =1.0$ for all cells and the other $\langle w_{ij}^{\text{mf}}\rangle =1.2$ for all cells. In (c), the solid, dashed, and dotted lines coincide on the scale of the graph. In all cases, $w^{\text{ff}}=1.0$. Each column represents a different degree or form of disorder: (a) normally distributed, $\Delta =0.1$; (b) correlated exponentially distributed, $\Delta =0.1$; and (c) correlated exponentially distributed, $\Delta =0.25$. Arrows show the direction of change of $\mu$ for adsorption and desorption and symbols refer to numerical data.Reuse & Permissions

Figure 9

Adsorption and desorption in pores of type III at nonzero temperature. Upper panel: Volume of adsorbed fluid vs $\mu$ for Metropolis dynamics simulations. In panel (a), the system consists of $N=100$ cells with an ink-bottle shape (solid and dot-dashed lines) and funnel shape (dashed lines) with $\langle w_1^{\text{mf}}\rangle =w^{\text{ff}}=1.0$, $\langle w_2^{\text{mf}}\rangle =1.2$ and Gaussian disorder of strength $\Delta =0.1$. Two values of temperature are considered: $\beta =50.0$ (solid and dashed lines) and $\beta =5$ (dot-dashed lines). In panel (b), the system has an ink-bottle shape with $\langle w_1^{\text{mf}}\rangle =w^{\text{ff}}=1.0$, $\langle w_2^{\text{mf}}\rangle =1.2$ and correlated exponential distribution of $\rho \left(w^{\text{mf}}\right)$ with $\Delta =0.25$ and three values of temperature: $\beta =50$ (solid curves), $\beta =15$ (double-dot dashed curves), and $\beta =5$ (dot-dashed curves). Middle and lower panels: The variance $\text{Var}\left[V\right]$ vs $\mu$ for the same systems as in the upper panels (with the same line styles). For clarity, the curves for $\beta =50$ have been presented on the middle panel while curves corresponding to $\beta =15$ and $\beta =5$ are on the lower panel. In all of the simulations, the rate of change of $\mu$ was $r=0.004{\text{MCSS}}^{-1}$.Reuse & Permissions