Adsorption-induced shape transitions in bistable nanopores with atomically thin walls

Atomically thin cylindrical nanopores can change shape in response to physically adsorbed gas inside. Coupled to a gas reservoir, an initially collapsed pore can expand to allow the adsorbed gas to form concentric shells on the inner part of the pore, driven by adsorption energetics, not gas pressure. A lattice gas model describes the evolution of the nanotube pore shape and absorbed gas as a function of gas chemical potential at zero temperature. We found that narrow-enough tubes are always expanded and gas inside adsorbs in sequences of concentric shells as the gas chemical potential increases. Wider tubes, which are collapsed without gas, can expand with one or more concentric shells adsorbed on the inner surface of the expanded region.

Figure 1

Equilibrium shapes of a partially collapsed tube of length $L$ with gas inside. Left and right ends are held open and closed by boundary conditions. Characterizing the tube shape by the length $\ell$ of the expanded region, the following configurations are considered: collapsed with $\ell =0$ (C, top), mixed collapsed-expanded with $0\le \ell \le L$ (CE, middle), and expanded with $\ell =L$ (E, bottom).

Figure 2

Allowed locations for gas atoms within the lattice gas model for both expanded and collapsed shapes. Red lines show shells.

Figure 3

The difference in energy per unit length $\mathrm{\Delta }U\left(R\right)=U_{\circ }-U_{-}$ between expanded ($U_{\circ }$) and collapsed ($U_{-}$) configurations for an empty tube. The expanded shape minimizes the energy for tubes with radii below the vertical dashed line at $R_t$; the collapsed shape minimizes for radii above. The two shapes provide different accessible surface areas for atoms adsorbed onto the inner surface. Note that both shapes are metastable at large-enough radius.

Figure 4

Schematic showing two different scenarios for the evolution in gas coverage as $\mu$ increases. Narrow tubes simply add successive layers to the inner surface of the expanded cross section, following the sequence $\mathbf{E}0\to \mathbf{E}1 \to \mathbf{E}2 \to \cdots \to \mathbf{E}m$. Wider tubes adsorb first in the bulbs; further gas adsorption then forces the pore open along the sequence $\mathbf{C}0\to \mathbf{C}1\to \mathbf{E}1\to \mathbf{E}2 \to \cdots \to \mathbf{E}m$.

Figure 5

The shape of tube of radius 26.7 Å as a function of the gas chemical potential $\mu$ and the gas-gas interaction strength $\epsilon$. As $\mu$ increases, the bulbs of the initially empty collapsed tube fill with gas at the lower blue line. The tube then expands into a state with one or two shells of adsorbed gas. Further increase of $\mu$ yields additional concentric shells. $\epsilon$ and $\mu$ are given in units of ${\epsilon }_C$.

Figure 6

Chemical potential corresponding to a given number of adsorbed shells in the expanded tube for various adsorbed gases and tube radii. The shell thicknesses on the negative slope to the left of minimal ${\mu }_m$ are omitted, and those systems expanded directly into a state with multiple shells. $\mu$ is given in units of ${\epsilon }_C$.

Figure 7

The number $m(\epsilon ,D)$ of shells formed in the expanded region of carbon and boron-nitride tubes during the initial expansion C$1\to$ E$m$. Bold font indicates a transition directly into a fully filled pore: Capillary condensation simultaneous with shape transition. The bottom of the range of radii shown corresponds to $R_t$, the size at which collapse becomes energetically favorable. $\epsilon$ is given in units of ${\epsilon }_C$.

Figure 8

Periodic (in axial direction) unit cells of (40,40) collapsed and expanded tube configurations. Similar cells of different diameter tubes were used for computations of per unit length energies $U_{-}$, $U_{\circ }$, and difference $\mathrm{\Delta }U=U_{\circ }-U_{-}$ shown in Fig. 3.

Figure 9

Shell structures of helium (red) absorbed inside a (44,44) carbon nanotube (gray) at $T=10\mathrm{K}$. The portion of the system cloest to the viewer is removed for clarity to expose the interior. (a) Longitudinal view showing the interior of the tube and the transition between collapse (top) and expansion (bottom). Figure 1 shows a side view of a similar system. (b) Close-up of the collapsed region, showing a close-packed single shell of helium absorbed in the bulb. The gray area is the region where the opposing tube walls are in direct contact. (c) Side view of single shell in the bulb. (d) Close-up of the expanded region with close-packed gas shells. (e) Gas is absorbed into two shells with the second one being incomplete.

Figure 10

Schematic showing geometric parameters of the collapsed tube cross section. $2a$ is the width of the collapsed region; $b$ is the length of the noncontacted wall part; $r$ is the radius of the bulb region.

Figure 11

The difference in energy per unit length $\mathrm{\Delta }U\left(R\right)=U_{\circ }-U_{-}$ between the expanded ($U_{\circ }$) and collapsed ($U_{-}$) configurations for an empty tube. The blue (dotted) line shows results of molecular dynamics simulations; the black (solid) and orange (dashed) lines show $U\left(R\right)$ results from the simplified continuum model of Eq. (B4). The collapsed shape minimizes the energy for tubes with radius $R$ for which $\mathrm{\Delta }U\left(R\right)>0$.