Phase behavior of fluids in undulated nanopores

The geometry of walls forming a narrow pore may qualitatively affect the phase behavior of the confined fluid. Specifically, the nature of condensation in nanopores formed of sinusoidally shaped walls (with amplitude $A$ and period $P$) is governed by the wall mean separation $L$ as follows. For $L>L_t$, where $L_t$ increases with $A$, the pores exhibit standard capillary condensation similar to planar slits. In contrast, for $L<L_t$, the condensation occurs in two steps, such that the fluid first condenses locally via bridging transition connecting adjacent crests of the walls, before it condenses globally. For the marginal value of $L=L_t$, all the three phases (gaslike, bridge, and liquidlike) may coexist. We show that the locations of the phase transitions can be described using geometric arguments leading to modified Kelvin equations. However, for completely wet walls, to which we focus on, the phase boundaries are shifted significantly due to the presence of wetting layers. In order to take this into account, mesoscopic corrections to the macroscopic theory are proposed. The resulting predictions are shown to be in a very good agreement with a density-functional theory even for molecularly narrow pores. The limits of stability of the bridge phase, controlled by the pore geometry, is also discussed in some detail.

Figure 1

A sketch illustrating three possible phases in a nanopore of a mean width $L$ formed by sinusoidally shaped walls with an amplitude $A$ and period $P$: (a) gas phase, (b) bridge phase, and (c) liquid phase.

Figure 2

A scheme of a bridge phase inside a nanopore formed by two walls of the local height $z_w\left(x\right)$ and $-z_w\left(x\right)$ relative to the horizontal axis. The macroscopic picture assumes that the menisci demarcating the liquid bridge are parts of a circle of the Laplace radius $R$ which meets tangentially the walls at the points $[\pm x_0,\pm z_0]$.

Figure 3

Illustration of the macroscopic estimation of the lower (a) and upper (b) spinodals of bridging transition in sinusoidal pores.

Figure 4

Illustration of the RP geometric construction of the bridge phase in a completely wet sinusoidal nanopore by taking into account the wetting layers [43]. (a) The walls are first coated by wetting layers, whose normal width is ${\ell }_{\pi }$ corresponding to a thickness of the liquid film adsorbed on a planar wall at the given chemical potential. In the second step, the circular menisci of the Laplace radius $R=\gamma /\left(\delta \mu \mathrm{\Delta }\rho \right)$ are drawn, such that they meet the wetting layers tangentially; (b) the construction of the shape of the interface $\tilde{\psi }\left(x\right)$ corresponding to the wetting layers. The unit vectors $\mathbf{n}$ and $\mathbf{t}$ are normal and tangent to the wall at a given point $x^{\prime }$, respectively, using which the height of the wetting layer can be determined at the point $x$, shifted from $x^{\prime }$ according to Eq. (43).

Figure 5

Adsorption isotherms obtained from DFT for nanopores formed by walls with $A=2\sigma$ and $P=50\sigma$. The mean distance between the walls is (a) $L=8\sigma$, (b) $L=9\sigma$, and (c) $L=10\sigma$.

Figure 6

Equilibrium 2D density profiles corresponding to (a) capillary gas, (b) bridge, and (c) capillary liquid phases in the nanopore with $A=2\sigma$, $P=50\sigma$, and $L=8\sigma$ [cf. Fig. 5].

Figure 7

DFT results (symbols) showing a dependence of $\delta {\mu }_{\mathrm{cc}}$ on the aspect ratio $a=A/P$ for nanopores with $P=50\sigma$ and $L=50\sigma$. The solid line represents the solution of the Kelvin equation (25) and the dashed line shows the value of $\delta {\mu }_{\mathrm{cc}}^{\parallel }$ for capillary condensation in the planar slit obtained from 1D DFT. The inset shows the log-log plot of the DFT results and the straight line with the slope of 2 confirms the prediction (25).

Figure 8

DFT results for a dependence of $\delta {\mu }_{\mathrm{cc}}$ on $A$ and $P$, such that $a=A/P=0.1$. The horizontal dotted line indicates the prediction given by Kelvin's equation (24).

Figure 9

A comparison between DFT results (symbols) and the prediction given by Kelvin's equation (24) (line) for a dependence of $\delta {\mu }_{\mathrm{cc}}$ on $L$ for walls with amplitudes $A=2\sigma$ (a) and $A=5\sigma$ (b) and period $P=50\sigma$.

Figure 10

A dependence of $x_0$, specifying the location where the bridging menisci meet the walls, on $\delta \mu$, for the slits with $A=2\sigma$ and $L=8\sigma$ (a) and $A=5\sigma$ and $L=14\sigma$ (b). The period of the walls is $P=50\sigma$ in both cases. A comparison is made between DFT results (symbols), the prediction given by the solution of the quartic equation, (26), (full line), and its simple approximative solution, (32), based on the perturbative scheme (dotted line). The DFT results include states where the bridges are stable (full circles), as well as the states where the bridges are metastable (open circles).

Figure 11

Comparison of the location of G-B transition, $\delta {\mu }_{\mathrm{gb}}$, as a function of $L$ obtained from DFT (symbols) and the macroscopic prediction given by Eq. (17) (solid line) for nanopores formed by sinusoidally shaped walls with the amplitude $A=2\sigma$ (a) and $A=5\sigma$ (b) and period $P=50\sigma$. Also shown are the estimated lower (red dotted line) and upper (red dashed line) spinodals of B phase, as obtained from Eqs. (38) and (39), respectively. The DFT results include states where the bridges are stable (full circles), as well as the states where the bridges are metastable (open circles).

Figure 12

Comparison of the location of B-L transition, $\delta {\mu }_{\mathrm{bl}}$, as a function of $L$ obtained from DFT (symbols) and the macroscopic prediction given by Eq. (18) (line) for nanopores formed by sinusoidally shaped walls with the amplitude $A=2\sigma$ (a) and $A=5\sigma$ (b) and the period $P=50\sigma$. The DFT results include states where the bridges are stable (full circles), as well as the states where the bridges are metastable (open circles).

Figure 13

Comparison of the threshold mean width $L_t$, allowing for a three-phase coexistence, as a function of the wall amplitude $A$, obtained from DFT (symbols) and from the macroscopic prediction given by Eq. (67) (line).

Figure 14

Phase diagrams showing the phase behavior of fluids in nanopores with the walls of amplitudes $A=2\sigma$ (a) and $A=5\sigma$ (b) and the period $P=50\sigma$, in the $\delta \mu \text{−}L$ plane. The phase boundaries between G, B and L phases correspond to the DFT results (black solid line) and the macroscopic theory (black dashed line). Also shown are the spinodals demarcating the limits of stability of B phase, as determined by DFT (solid red lines) and the macroscopic theory (dashed red lines). All the three phase boundaries meet at the triple point T, for which $L=L_t$ (cf. Fig. 13). The DFT results also include the critical point ${\mathrm{C}}_{\mathrm{gb}}$, whose presence allows for a continuous formation of bridges, cf. Fig. 15. The vertical dotted lines depicted in the upper panel correspond to adsorption isotherms shown in Fig. 5 (blue) and in Fig. 15 (green).

Figure 15

Adsorption isotherm corresponding to the nanopore with $A=2\sigma$, $P=50\sigma$, and $L=7.4\sigma$ illustrating continuous formation of B phase. The thermodynamic path corresponds to the green line in the phase diagram shown in Fig. 14.

Figure 16

DFT results showing the thickness ${\ell }_{\pi }$ of the liquid film adsorbed on a planar Lennard-Jones wall as a function of $\delta \mu$. For small values of $\delta \mu$, the results are consistent with the expected asymptotic power law, as is verified by the log-log plot shown in the inset, where the straight line has a slope of $-1/3$. The line in the figure corresponds to the fit of the power law to the DFT data, which gives ${\ell }_{\pi }=1.363\delta {\mu }^{-1/3}$.

Figure 17

Comparison of the dependence of $\delta {\mu }_{\mathrm{cc}}$ on the aspect ratio $a=A/P$ between DFT (symbols), the macroscopic theory, Eq. (24), (solid line) and the mesoscopic theory, Eq. (41), (dashed line) for nanopores with $P=50\sigma$ and $L=50\sigma$. The dotted line indicates the value of $\delta {\mu }_{\mathrm{cc}}^{\parallel }$ for capillary condensation in the planar slit obtained from 1D DFT. The inset shows the log-log plot of the DFT results and the straight line with the slope of 2 confirms the prediction (25).

Figure 18

Comparison of the dependence of $\delta {\mu }_{\mathrm{cc}}$ on $L$ between DFT results (symbols), the prediction given by the fully macroscopic Kelvin equation (24) (dotted line) and its mesoscopic correction given by Eq. (41) (solid line). The nanopores are formed of sinusoidally shaped walls with the amplitudes of $A=2\sigma$ (a) and $A=5\sigma$ (b), and period $P=50\sigma$. The DFT results include states which are stable (full circles) and also metastable (open circles).

Figure 19

Comparison of the location of G-B transition, $\delta {\mu }_{\mathrm{gb}}$, as a function of $L$, obtained from DFT (symbols), the macroscopic prediction given by Eq. (17) (dotted line) and its mesoscopic correction based on the RP construction (full line), for nanopores formed of sinusoidal walls with the amplitude $A=2\sigma$ (a) and $A=5\sigma$ (b) and period $P=50\sigma$. The macroscopic results terminate at the (macroscopically predicted) upper limit of B stability (denoted by the cross), when the radius of the bridge menisci becomes $R=R_s^{+}$. The DFT results include states which are stable (full circles) and also metastable (open circles).

Figure 20

Comparison of the location of B-L transition, $\delta {\mu }_{\mathrm{bl}}$, as a function of $L$ obtained from DFT (symbols), the macroscopic prediction given by Eq. (18) (dotted line) and its mesoscopic correction based on the RP construction (full line), for nanopores formed of sinusoidal walls with the amplitude $A=2\sigma$ (a) and $A=5\sigma$ (b) and period $P=50\sigma$. The macroscopic results terminate at the (macroscopically predicted) lower limit of B stability (denoted by the cross), when the radius of the bridge menisci becomes $R=R_s^{-}$. The DFT results include states which stable (full circles) and also metastable (open circles).