Relations between the diffusion anomaly and cooperative rearranging regions in a hydrophobically nanoconfined water monolayer

We simulate liquid water between hydrophobic walls, separated by 0.5 nm, to study how the diffusion constant $D_{\parallel }$ parallel to the walls depends on the microscopic structure of water. At low temperature $T$, water diffusion can be associated with the number of defects in the hydrogen bond network. However, the number of defects solely does not account for the peculiar diffusion of water, with maxima and minima along isotherms. Here, we calculate a relation that quantitatively reproduces the behavior of $D_{\parallel }$, focusing on the high-$T$ regime. We clarify how the interplay between breaking of hydrogen bonds and cooperative rearranging regions of 1-nm size gives rise to the diffusion extrema in nanoconfined water.

Figure 1

Diffusion coefficient $D_{\parallel }$ from MC simulations (symbols) as a function of pressure along isotherms for (from top to bottom) $T$ between 972 and 483 K. For $T<972$ K, $D_{\parallel }$ has maxima $D_{\parallel }^{\max }$ at $P^{\max }$ (dotted-dashed line) and minima $D_{\parallel }^{\min }$ at $P^{\min }$ (dashed line). Solid lines are from Eq. (6), as described in the text.

Figure 2

Microscopic configuration changes along isotherms for temperatures 1006 K (squares) and 693 K (circles), corresponding to the onset of anomalous $D_{\parallel }$ and to the fully developed anomaly of $D_{\parallel }$, respectively. Top: Probability $p_B\left[i\right]$ for a molecule to participate in $i=0,\cdots ,4$ HBs. Bottom: Probability $p_F\left[i\right]$ for a molecule to have $i=0,\cdots ,4$ free n.n. cells available for diffusion. The discontinuity in $p_F\left[i\right]$ corresponds to the liquid-gas first-order phase transition. Among different panels, probability values change up to two orders of magnitude.

Figure 3

Upper inset: Average numbers of free n.n. cells around a molecule $\langle n_F\rangle$ (circles) and of HBs formed by a molecule $\langle n_{\mathrm{HB}}\rangle$ (squares) as a function of $P$ at $T=693$ K. The discontinuity in $\langle n_F\rangle$ at low $P$ corresponds to the gas-liquid first-order phase transition. At higher $P$, both quantities are monotonic. Main panel: The plot of $D_{\parallel }$ vs $W_{\nu ,\mu }$ along the six isotherms represented in Fig. 1 shows that $D_{\parallel }$ is a linear function of $W_{\nu ,\mu }$ within the numerical noise. The parameters $\nu$ and $\mu$ change with $T$, as described in the text. Lower inset: Example of a cooperative (shaded) region of $\sim$1 nm size, with $\nu =12$ molecules with n.n. free cells (not all represented in the scheme) and $\mu {\mathcal{P}}_b=5$ HBs.