Effect of disorder on condensation in the lattice gas model on a random graph

The lattice gas model of condensation in a heterogeneous pore system, represented by a random graph of cells, is studied using an exact analytical solution. A binary mixture of pore cells with different coordination numbers is shown to exhibit two phase transitions as a function of chemical potential in a certain temperature range. Heterogeneity in interaction strengths is demonstrated to reduce the critical temperature and, for large-enough degreeS of disorder, divides the cells into ones which are either on average occupied or unoccupied. Despite treating the pore space loops in a simplified manner, the random-graph model provides a good description of condensation in porous structures containing loops. This is illustrated by considering capillary condensation in a structural model of mesoporous silica SBA-15.

Figure 1

Panel (a): Random graph of fixed degree distribution with several loops. The cells drawn with dashed circles are assumed to be a long enough distance from cell $i$ that they do not significantly affect it and can be removed. Panel (b): The same random graph with distant cells removed to make it a tree. The gray dashed cells marked $k_1,k_2,...$ are on the edge of the tree and their effect on the cells marked $l_1,l_2,l_3$ neighboring cell $i$ can be accounted for in terms of effective fields, $w_{l_1{k}_1}^{\mathrm{eff}},w_{l_1{k}_2}^{\mathrm{eff}},...$, given by Eq. (7).

Figure 2

Equilibrium phase diagrams for homogeneous $q$-regular graphs with interaction parameters $w_0^{\mathrm{ff}}=1$ and $w_0^{\mathrm{mf}}=0$. (a) $\mu$-$T$ phase diagram for $q$-regular graphs with coordination numbers $q_1=4$ (dashed line) and $q_2=10$ (solid line). For both values of $q$, first-order phase transitions occur when crossing the vertical lines below the critical points, C1 for $q_1=4$ at ${\mu }_1=-2$ and C2 for $q_2=10$ at ${\mu }_2=-5$. The two additional figures in panel (a) show the dependence of the mean density, $\overline{\rho }$, on chemical potential, $\mu$, for a 4-regular graph above ($T=1.0$, upper figure) and below ($T=0.6$, lower figure) the critical point. (b) Liquid-gas coexistence curves (binodals) in the $(\overline{\rho },T)$ plane for fluids in 4- (dashed line) and 10-regular graphs (continuous line). The dot-dashed lines and double dot-dashed lines in both panels correspond to the temperatures for which the density profiles are shown in the right-hand figures of panel (a).

Figure 3

Panel (a): Distribution, $R\left(\rho \right)$, of cell fluid densities, $\rho$, for binary system with a representative set of parameters, $\mu =-2.8$, $T=0.1$, $w_0^{\mathrm{ff}}=1$, and $w_0^{\mathrm{mf}}=0$, for disorder in coordination number with $f=0.77$. The mean value of the density for this distribution is $\overline{\rho }\simeq 0.441$ [see the star in Fig. 6]. Panel (b): Cumulative distribution $C\left(\rho \right)={\int }_0^{\rho }R\left(\rho \right)d\rho$ of the cell fluid densities. In both panels, the lines are calculated by solving Eqs. (17) and (18) self-consistently and using Eq. (19). The circles in panel (b) are calculated using Metropolis dynamics simulations described in Sec. 8. The system size is $N={10}^6$ and the mean occupation, ${\rho }_i$, was measured over a period of ${10}^5$ Monte Carlo steps per spin (MCSS).

Figure 4

Phase diagram in the $(\mu ,T)$ plane (a) and binodal in the $(\overline{\rho },T)$ plane (b) of a random graph without disorder in interaction strengths and with bimodal disorder in coordination number consisting of a fraction $f=0.77$ of $q_1=4$-coordinated cells and a fraction $1-f$ of $q_2=10$-coordinated cells. The locations of critical points (C1 and C2) and triple point (t) are marked by circles. Solid lines correspond to the coexistence curves of two phases where the chemical potential takes values ${\mu }_1\left(T\right)$ between C1 and t, ${\mu }_2\left(T\right)$ between C2 and t, and ${\mu }_3\left(T\right)$ below t. The dashed lines correspond to the spinodals, i.e., the boundaries within which metastable states exist, and are marked by ${\mu }_1^{\prime }$, ${\mu }_1^{\prime \prime }$, ${\mu }_2^{\prime }$ and ${\mu }_2^{\prime \prime }$ in panel (a). The horizontal dot-dashed, double-dot dashed, and double-dash dotted lines correspond to the values of $T$ and $\mu$ for which the sorption curves are shown in Figs. 5, 5, and 5, respectively. The temperature ranges corresponding to the four regimes I, II, III, and IV are indicated by the numbers and arrows on the right.

Figure 5

Mean fluid density (left panels) and free energy per cell (right panels) as a function of chemical potential for a range of temperatures, $T$. The system in all panels has no disorder in interaction strengths and coordination number disorder with $q_1=4$ and $q_2=10$ for $f=0.77$ as in Fig. 4. Curves represent the substitution of solutions of Eqs. (17) and (18) into Eqs. (19) and (20). Panel (a) shows $\overline{\rho }$ and $\overline{F}/N$ for several temperatures in regime I (i.e., for $T>T_{\mathrm{C}2}$): $T=1.2$ (solid), $1.5$ (dashed), $1.8$ (dot-dashed), and 2 (dot double-dashed). Panels (b), (c), and (d) show examples of regime II ($T=0.5$), regime III ($T=0.2$), and regime IV ($T=0.1$), respectively [see dot-dashed, double-dot dashed, and double-dash dotted lines in Figs. 4 and 4]. In panels (b)–(d), the solid lines represent stable states while dashed lines represent metastable states. The insets in the right-hand sides of panels (c) and (d) show a magnification of the free energies of the low-, intermediate-, and high-density states around the point where they are equal with the stable state marked black and the metastable states marked with dotted lines. Symbols represent the results of Metropolis dynamics simulations (see Sec. 8). The rates of change of $\mu$ are $\dot{\mu }=\pm {10}^{-3}{\mathrm{MCSS}}^{-1}$ [panel (a)] and $\dot{\mu }=\pm 2\times {10}^{-5}{\mathrm{MCSS}}^{-1}$ [panels (b), (c), and (d)] with crosses and circles corresponding to positive, $\dot{\mu }>0$, and negative, $\dot{\mu }<0$, rates of change, respectively. Magnifications of the left-hand sides of panels (c) and (d) are shown in Fig. 6.

Figure 6

Detail of sorption curves of $\overline{\rho }$ vs. $\mu$ for the same system as in Fig. 5 and $T=0.2$ [panel (a)] and $T=0.1$ [panel (b)]. Solid and dashed lines represent stable and metastable states as in Figs. 5 and 5. The values of $\overline{\rho }$ for the intermediate-density states averaged across only $q_1$- and $q_2$-coordinated cells are shown by double-dot dashed and dot-dashed lines, respectively. Symbols represent Metropolis dynamics simulations for rates of change of $\mu$ given by $\dot{\mu }=2\times {10}^{-5}$ [panel (a)] and $\dot{\mu }={10}^{-6}$ [panel (b)] (same symbol styles as in Fig. 5). Black arrows indicate the variation of $\mu$ in numerical simulations of desorption from the stable state with $\rho \sim 1$ (density shown by circles). White arrows show the variation of the system from a metastable state with $\rho \sim 1$ (density shown by crosses). The star refers to the mean density of the p.d.f. presented in Fig. 3.

Figure 7

Phase diagrams for a system consisting of a mixture of $q_1=4$ and $q_2=10$ coordinated nodes and constant interaction strengths. The fraction of $q_1$-coordinated nodes is (a) $f=0.7$, (b) $f=0.77$, (c) $f=0.95$, and (d) $f=0.957$. Locations of lines of first-order phase transitions (solid lines) and boundaries of the regions where metastable states exist (dashed lines) are shown in the space of chemical potential and temperature. The locations of critical points (C1 and C2), critical end points (CEP), and triple points (t) are marked with circles for the values of $f$ where they exist. The critical end point is unique and is found at $T\simeq 0.586$, $\mu \simeq -2.1481$, and $f\simeq 0.957$. The first-order transition lines are found by using Eqs. (17, 18, 19, 20).

Figure 8

Values of tricritical temperature $T_{\mathrm{t}}$ (dashed line) and critical temperatures $T_{\mathrm{C}1}$ (solid line) and $T_{\mathrm{C}2}$ (dot-dashed line) as a function of fraction $f$ of $q_1=4$-coordinated nodes. The value of $f_2$ corresponds to the location of the critical end point (marked CEP).

Figure 9

Sorption curves for systems with different interaction strengths for cells of different coordination number. Panel (a): $q_1=3$, $q_2=4$, $f=0.666$, $w_1^{\mathrm{mf}}=5$ ${\tilde{w}}_0^{\mathrm{mf}}=0$ and $T=0.05$. The inset magnifies the region of the low-$\mu$ first-order phase transition at $\mu \simeq -6.08$. Panel (b): Binomial distribution of coordination numbers with $q_{\max }=20$, $p=0.9$, $w_1^{\mathrm{mf}}=20$, ${\tilde{w}}_0^{\mathrm{mf}}=0$, and $T=0.05$. The inset shows the detail of the region where one of the first-order phase transitions occurs at $\mu \simeq -33.6$. The crosses (adsorption) and circles (desorption) refer to the results of Metropolis dynamics with $\dot{\mu }={10}^{-5}{\mathrm{MCSS}}^{-1}$ [panel (a)] and $\dot{\mu }={10}^{-4}{\mathrm{MCSS}}^{-1}$ [panel (b)].

Figure 10

Critical temperatures as a function of degree of disorder for a 4-regular graph with normal (solid curve) and exponential (dot-dashed) disorder in $w^{\mathrm{mf}}$ (i.e., ${\Delta }^{\mathrm{mf}}=\Delta$, ${\Delta }^{\mathrm{ff}}=0$) and cut-normal distribution in $w^{\mathrm{ff}}$ (${\Delta }^{\mathrm{ff}}=\Delta$, ${\Delta }^{\mathrm{mf}}=0$; dashed curve).

Figure 11

Distributions of the mean occupancies, ${\rho }_i$, of the cells in a $q=4$-regular graph with $\mu =-2$, constant $w^{\mathrm{ff}}$ for all cells and normal disorder in $w^{\mathrm{mf}}$. Panel (a): Temperature, $T=1.0$, and width of disorder ${\Delta }^{\mathrm{mf}}=0.3$ (solid curve), $0.6$ (dashed), $1.2$ (dot-dashed), $1.8$ (double-dot dashed), and $2.4$ (dot double-dashed). The circles represent a normal distribution of width given by Eq. (B11) and the crosses were found by Metropolis dynamics simulations in which the values of ${\rho }_i$ for each of $N={10}^6$ cells were averaged over a duration of ${10}^3$ MCSS. Panel (b): $T=0.4$ with ${\Delta }^{\mathrm{mf}}=0.75$ (solid and dashed curves representing low- and high-density states, respectively) and $0.8$ (dot-dashed curve).

Figure 12

Phase diagram for binary system of $q_1=4$ and $q_2=10$-coordinated cells with a fraction $f=0.77$ of $q_1$-coordinated cells. The matrix-fluid interactions are normally distributed with coordination number–dependent standard deviations, ${\Delta }^{\mathrm{mf}}(q=4)=0.24$ and ${\Delta }^{\mathrm{mf}}(q=10)=0$, while disorder in fluid-fluid interactions is a cut-normal distribution with ${\Delta }^{\mathrm{ff}}=0.04$.

Figure 13

The pore cell network for the Euclidean model of SBA-15. The light gray (blue) hexagons represent the pore cells vertically connected to each other by thin solid lines (vertical fluid-fluid interactions, $w_{\mathrm{vert}}^{\mathrm{ff}}$). The thick horizontal lines account for horizontal fluid-fluid interactions, $w_{\mathrm{horiz}}^{\mathrm{ff}}$, between pore cells in different vertical mesopores through the micropores. The cylindrical walls of the mesopores are split into matrix cells shown by dark gray hexagon sides. The dashed horizontal lines represent matrix-fluid interactions, $w^{\mathrm{mf}}$.

Figure 14

Sorption isotherms for structural models of SBA-15 with $w_{\mathrm{horiz}}^{\mathrm{ff}}/w_{\mathrm{vert}}^{\mathrm{ff}}=0.1$, $w_0^{\mathrm{mf}}=0.2$, and ${\Delta }^{\mathrm{mf}}=0.04$. The lines correspond to the results for the random-graph model. The symbols represent the results of Metropolis dynamics simulations for $50\times 50\times 50$ pore cells arranged on a lattice (see Fig. 13) with periodic horizontal and open vertical boundary conditions. (a) $p_t=0.05$, $T^{-1}=2$ (dashed line and squares), and $T^{-1}=15$ (solid line and circles); (b) $p_t=0.2$, $T^{-1}=2$ (double dot-dashed line and stars), $T^{-1}=5$ (dashed line and squares), and $T^{-1}=20$ (solid line, solid and open circles for adsorption and desorption, respectively, with $\dot{\mu }={10}^{-8}$ MCSS${}^{-1}$). The inset shows the binodal (phase coexistence curve) for the random-graph model with critical temperature $T_c\simeq 0.118\pm 0.002$.

Figure 15

Sorption isotherms for structural models of SBA-15 with $w_{\mathrm{horiz}}^{\mathrm{ff}}=w_{\mathrm{vert}}^{\mathrm{ff}}=1$, $w_0^{\mathrm{mf}}=1$, and ${\Delta }^{\mathrm{mf}}=0.2$. The lines and symbols are for the random-graph and Euclidean models, respectively. (a) $p_t=0.05$, $T^{-1}=2$ (dashed line and circles), and $T^{-1}=10$ (solid line and squares); (b) $p_t=0.2$, $T^{-1}=1$ (double dot-dashed line and stars), $T^{-1}=2$ (dot-dashed line and squares), and $T^{-1}=10$ (solid line, solid and open circles for adsorption and desorption, respectively, with $\dot{\mu }={10}^{-7}$ MCSS${}^{-1}$). The inset shows the binodal (phase coexistence curve) for the random-graph model with critical temperature $T_c\simeq 0.358\pm 0.001$.

Figure 16

Phase diagram for fluid condensation on a $q=4$-regular graph with homogeneous interaction strengths, $w_0^{\mathrm{ff}}=1$, $w_0^{\mathrm{mf}}=0$. The solid line represents the phase boundary and dashed lines represent the limits of metastability (spinodal lines).