Influence of Pore Size on the van der Waals Interaction in Two-Dimensional Molecules and Materials

Despite the importance of porous two-dimensional (2D) molecules and materials in advanced technological applications, the question of how the void space in these systems affects the van der Waals (vdW) scaling landscape has been largely unanswered. Analytical and numerical models presented herein demonstrate that the mere presence of a pore leads to markedly different vdW scaling across nonasymptotic distances, with certain relative pore sizes yielding effective power laws ranging from simple monotonic decay to the formation of minima, extended plateaus, and even maxima. These models are in remarkable agreement with first-principles approaches for the 2D building blocks of covalent organic frameworks (COFs), and reveal that COF macrocycle dimers and periodic bilayers exhibit unique vdW scaling behavior that is quite distinct from their nonporous analogs. These findings extend across a range of distances relevant to the nanoscale, and represent a hitherto unexplored avenue towards governing the self-assembly of complex nanostructures from porous 2D molecules and materials.

Figure 1

Effective power law exponents $P_{\mathrm{vdW}}^{A\text{-ann}}$ for the pairwise (lines) and many-body (circles) vdW interaction between a point particle and an annulus with $R=10$ as a function of $r$ and $D$ (all in Å). The mere presence of a pore fundamentally alters the vdW scaling landscape and leads to a markedly slower $E_{\mathrm{vdW}}^{A\text{-ann}}$ decay rate across all nonasymptotic distances.

Figure 2

Top: Effective power law exponents $P_{\mathrm{vdW}}^{\mathrm{ann}\text{−}\mathrm{ann}}$ for the pairwise vdW interaction between two stacked annuli with $R=10$ as a function of $r$ and $D$ (all in Å). Depending on the relative pore size, $P_{\mathrm{vdW}}$ exhibits widely varying behavior at nonasymptotic distances, ranging from simple monotonic decay (closed disks) to extended plateaus and maxima (rings). Bottom: Extended plateaus (shown here for $R=10–40\AA$) occur for annuli with $r/R=0.75$ at $P_{\mathrm{vdW}}=-4.85$ (dotted line), and correspond to an enhanced vdW interaction across a wide range of intermediate distances. Maxima of $P_{\mathrm{vdW}}=-4.88$ (dashed line) correspond to minima in the $E_{\mathrm{vdW}}$ decay rate and occur for rings at a relative distance of $D/R=1.33$.

Figure 3

Effective power law exponents $P_{\mathrm{vdW}}^{\mathrm{layer}\text{−}\mathrm{layer}}$ for the pairwise vdW interaction between stacked periodic hexagonal bilayers with $\overline{R}=10$ as a function of $\overline{r}$ and $D$ (all in Å). When $\overline{r}\ne 0$, such porous 2D materials exhibit fundamentally different behavior (including the formation of minima in $P_{\mathrm{vdW}}$) than the limiting case of two infinite nonporous plates ($\overline{r}=0$), which has $P_{\mathrm{vdW}}=-4$ for all interlayer separations.

Figure 4

Top: 2D building blocks for COF-5 (red), TP-COF (blue), and HHTP-DPB COF (orange). In this work, COF macrocycles have capped terminal hydroxyl ($\text{-OH}$) groups. Bottom: Effective power law exponents for stacked COF macrocycle dimers (MC) and periodic bilayers (BL) obtained using the analytical ($\mathrm{ann}[r,R]$) and numerical ($\text{layer}[\overline{r},\overline{R}]$) models introduced herein (solid lines), and the effective pairwise PBE+TS-vdW approach (TS, circles and squares). $P_{\mathrm{vdW}}^{\mathrm{ann}\text{−}\mathrm{ann}}$ ($P_{\mathrm{vdW}}^{\mathrm{layer}\text{−}\mathrm{layer}}$) are provided for a range of $r$ and $R$ ($\overline{r}$ and $\overline{R}$) based on simple estimates of the COF radii [56], and are in high fidelity with the first-principles based PBE+TS-vdW approach. Quite interestingly, we find that these COF systems have relative pore sizes that lead to characteristic features such as extended plateaus (MC) and minima (BL) in $P_{\mathrm{vdW}}$.

Figure 5

Effective power law exponents for COF macrocycle dimers (MC) when accounting for pairwise (TS, open circles) and many-body (MBD, closed circles) vdW interactions.