# Transport of thin water films: from thermally activated random walks to hydrodynamics

Simon Gravelle, Christian Holm, Alexander Schlaich

► To cite this version:

Simon Gravelle, Christian Holm, Alexander Schlaich. Transport of thin water films: from thermally activated random walks to hydrodynamics. The Journal of Chemical Physics, 2022, 157 (10), 10.1063/5.0099646. [hal-05002825](https://hal.science/hal-05002825v1)

HAL Id: hal-05002825

https://hal.science/hal-05002825v1

Submitted on 24 Mar 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Transport of thin water films: from thermally activated random walks to hydrodynamics

Simon Gravelle,<fnref n="1" /> Christian Holm,<fnref n="1" /> and Alexander Schlaich<fnref n="1" />,<fnref n="2" />

Under ambient atmospheric conditions, a thin film of water wets many solid surfaces, including insulators, ice, and salt. The film thickness as well as its transport behavior sensitively depend on the surrounding humidity. Understanding this intricate interplay is of highest relevance for water transport through porous media, particularly in the context of soil salinization induced by evaporation. Here, we use molecular simulations to evaluate the transport properties of thin water films on prototypical salt and soil interfaces, namely NaCl and silica solid surfaces. Our results show two distinct regimes for water transport: at low water coverage, the film permeance scales linearly with the adsorbed amount, in agreement with the activated random walk model. For thicker water films, the permeance scales as the adsorbed amount to the power of 3, in line with the Stokes equation. By comparing results obtained for silica and NaCl surfaces, we find that, at low water coverage, water permeance at the silica surface is considerably lower than at the NaCl surface, which we attribute to difference in hydrogen bonding. We also investigate the effect of atomic surface defects on the transport properties. Finally, in the context of water transport through porous material, we determine the humidity-dependent crossover between a vapor dominated and a thin film dominated transport regimes depending on the pore size.

## I. INTRODUCTION

Under ambient conditions, a thin film of water covers the surfaces of insulators, metals, semiconductors, and even ice [1–4]. Depending on the nature of the substrate and on the ambient relative humidity and temperature, the water coverage can vary from a fraction of a monolayer to many molecular layers. The thermodynamic and dynamic properties of such a thin film of water are controlled by the two interfaces formed with the vapor and the solid, respectively [4]. These properties are crucial to phenomena such as dynamic wetting [5] or the caking of salt particles [3], as well as to understand flow through porous media. In particular, the evaporation of water from a porous medium can be accompanied by the formation of an efflorescent layer of crystallized salt [6, 7], which can either enhance [8, 9] or hinder [10] the evaporation rate of water. A better understanding of the water transport properties of salt crust is particularly important for arid regions where soil salinization due to irrigation is a critical problem [11]. Here, we use molecular simulations to evaluate the transport properties of a thin film of water on the prototypical salt surface NaCl, which we compare with results obtained for silica, i.e. a prototypical soil surface.

Let us consider a thin water film of thickness $h$ on an atomically flat crystal surface. For NaCl(100) surface, water typically forms a monolayer for relative humidity of 40%, with $h \approx 0.4$ nm [2, 12, 13]. The thin film has its normal in the $e_z$ direction, and is exposed to a gradient of chemical potential $\nabla\mu = -f$ along the $e_x$ direction, where $f$ [N/m$^3$] is the corresponding force per unit of volume acting on the water. Assuming the validity of bulk hydrodynamics, which is typically correct down to system size of ~1 nm [14, 15], one can use the Stokes equation, $\eta \partial_z^2 u_x = -f$, where $\eta$ is the bulk fluid viscosity, $u_x$ the component velocity field along $e_x$, and $\partial_z$ the partial derivative over the $z$ coordinate. Assuming the no-slip boundary condition ($u|_{z=0} = 0$) at the surface in $z = 0$, which holds reasonably well for the systems considered here (Fig. S4),

and the free flow boundary condition ($\partial u/\partial z|_{z=h} = 0$) at the air-vapor interface in $z = h$, the velocity field inside the thin film reads

$$u(z) = -\frac{f}{\eta} \left( \frac{z^2}{2} - hz \right). \qquad (1)$$

Integrating $\rho_w u(z)$ over the thickness $h$ of the thin film, where $\rho_w$ is the bulk water density (Fig. S2), and multiplying by the film depth $w$ (i.e. the extension of the system over the $e_y$ direction), leads to the expression for the total water flux of the thin film in the direction of the forcing:

$$Q_{\text{Stokes}} = \frac{\rho_w w h^3}{3\eta} f, \qquad (2)$$

where $Q_{\text{Stokes}}$ has the units of s$^{-1}$. By dividing $Q_{\text{Stokes}}$ by $f$ and $w$, we define the permeance per unit length for the thin film,

$$L_{\text{Stokes}} = \frac{\rho_w}{3\eta} h^3, \qquad (3)$$

where $L_{\text{Stokes}}$ has the units of Pa$^{-1}$ s$^{-1}$.

Equations (2-3) are expected to hold for a thin film with thickness $h \ge 1$ nm. However, in the limit of extremely low water coverage, $\theta \to 0$, where $\theta$ is the number of water molecule per surface adsorption site, the mobility of the fluid is expected to involve the hopping of the adsorbed molecules from one site to another [16]. In the limit $\theta \to 0$, for a random walk on a lattice, the resulting water flux is

$$Q_{\text{RW}} = \frac{w D_s N_s \theta}{k_B T \rho_w} f, \qquad (4)$$

where $D_s$ is the diffusion coefficient, $N_s$ the surface site density, $k_B$ is the Boltzmann constant, and $T$ the temperature. The product $N_s \theta$ is the water amount per surface area, and $N_s \theta / \rho_w$

<footnote n="1">¹) Institute for Computational Physics, University of Stuttgart, D-70569 Stuttgart, Germany</footnote>

<footnote n="2">²) Stuttgart Center for Simulation Science (SC SimTech), University of Stuttgart, 70569 Stuttgart, Germany</footnote>
the effective thickness of the water layer h. Again, dividing $Q_{RW}$ by f and w, we obtain a permeance per unit length:

$$L_{RW} = \frac{N_s D_s}{k_B T \rho_w} \theta. \qquad (5)$$

Equation (5) is similar to the expression obtained by Reed and Ehrlich [17] for non-interacting atoms in a Langmuir film, in the limit of low occupancy $\theta \to 0$.

An important difference between Eq. (3) and (5) is the scaling with the adsorbed water amount. Rewriting Eq. (3) in terms of the water coverage $\theta$ using $h = \theta N_s / \rho_w$ [4], one gets

$$L_{\text{Stokes}} = \frac{N_s^3}{3\eta\rho_w^2}\theta^3. \qquad (6)$$

Therefore one expects the permeance of the water layer to scale as $L \propto \theta^3$ for $h \ge 1$ nm, and as $L \propto \theta$ for $\theta \to 0$. However, Eqs. (4-5) are, a priori, valid only in the limit of a low water content $\theta \to 0$. Here the transition between activated random walk transport and hydrodynamics is explored using molecular simulations.

We consider the case of a thin film of water on a NaCl(100) surface (Fig. 1). The NaCl(100) surface is of particular interest due to its structural simplicity and because its interaction with water has been extensively characterized [2, 12, 13, 18]. The NaCl(100) surface consists of $Na^+$ and $Cl^-$ ions disposed on a cubic lattice, and can be produced by cleaving a NaCl crystal [12, 13]. Early TEM measurements have revealed the modification of NaCl surface after exposure to ambient conditions [19], and AFM measurements have showed the migration of atomic steps at the NaCl surface when exposed to air of relative humidity larger than a certain threshold [20]. Infrared spectroscopy measurement have given hints on the hydrogen bonding network (gas, liquid, and ice water have different signatures) and film thickness, and have allowed Ewing and coworkers to extract adsorption isotherms. Their results show that for a relative humidity of 40%, water forms a monolayer on NaCl(100) [2, 12, 13]. For a relative humidity of 80%, water forms up to 4 layers, which is consistent with contact force spectroscopy experiments [21]. In parallel to experiments, molecular simulations of water at the NaCl(100) interface have provided information on the molecular structure of the thin film [18], and have revealed for instance that water molecules favor adsorption at the $Na^+$ over $Cl^-$ ions, and that, in the limit of low water content $\theta$, water molecules form clusters due to formation of hydrogen bonds.

In this article, we use molecular simulations to study the thermodynamic and dynamic properties of a thin film of water at the NaCl(100) surface. First, grand canonical Monte Carlo simulations are used to evaluate the adsorption of water at the interface for different values of the relative humidity, and adsorption isotherms are compared with experimental data. Then, non-equilibrium molecular dynamics is used to measure the transport properties of the water at the solid surface for varying water content. Our results are compared with the predictions of Eqs. (3) and (5), as well as with results obtained for a charged silica surface. Finally, the effect of defects at the NaCl surface are evaluated. We end with a discussion on the

<figure><img src="image_2.png" /><figcaption>Figure 1. a) Lateral views ($e_x$, $e_z$) of the molecular simulations, showing the NaCl(100) slit nanopore with water content $\theta = 1$. Periodic boundary conditions were applied at the dashed line. b, c, d) Top views ($e_x$, $e_y$) of the NaCl(100) surface for water content $\theta = 1$ (b), $\theta = 0.3$ (c), and $\theta = 0.1$ (d). Sodium ($Na^+$) atoms are in purple, chloride ($Cl^-$) in green, oxygen (O) in red, and hydrogen (H) in white.</figcaption></figure>

relative contribution of the thin film and of vapor to the overall water flux in porous material such as salt crusts at various relative humidities.

## II. RESULTS

### A. Thermodynamics of the NaCl(100)-water interface

The water content at the NaCl(100) surface was extracted for varying water relative humidity (RH) using the hybrid Grand Canonical Monte Carlo - Molecular Dynamics (GCMC-MD) simulation method implemented using LAMMPS [22]. The system is made of a planar block of NaCl that is made effectively infinite in the $e_x$ and $e_y$ directions thanks to periodic boundary conditions (Fig. 1). In the $e_z$ direction, the distance between the two faces of the block is about 7 nm. A temperature equal to $T = 297.15$ K was chosen to allow for a direct comparison with experimental data of Refs. 12 and 13. The transferable force field developed by Loche et al. [23] was used for $Na^+$ and $Cl^-$, and the TIP4P/$\epsilon$ force field [24] was used for water. More details about the numerical procedure are given in the supplementary material, and input files are openly available in DaRUS repository [25].

The adsorption isotherm we obtained using the GCMC-MD method shows three separate regions (blue symbols in Fig. 2). For RH lower than 0.3, the water coverage is almost null;
<figure><img src="image_3.png" /><figcaption>Figure 2. Amount of water $\theta$ at the NaCl(100) surface as a function of the relative humidity RH, with $\theta = 1$ corresponding to one water molecule per surface atom. Full disks are simulation data obtained using the GCMC/MD method, open squares are experimental data from Ref. 13, and open triangles are experimental data from Ref. 12. The two dashed vertical lines highlight the two transitions at RH = 0.3 and 0.65, respectively. These results have been obtained for $T = 297.15^{\circ}$C.</figcaption></figure>

$\theta \ll 1$. For RH in between 0.3 and 0.6, there is a sharp increase of $\theta$ with RH. Finally, for RH larger than 0.6, the slit nanochannel was completely filled with liquid water due to capillary condensation [26]. Our results are in good qualitative agreement with the experimental measurements performed by Ewing et al. for RH < 0.65 [12, 13] (open symbols in Fig. 2). For RH > 0.65, experimental isotherms show an abrupt increase in coverage, which has been linked to the spontaneous dissolution of NaCl [4]. Note that such deliquescence in water of the NaCl(100) surface was never observed during our simulations, which is consistent with previous molecular simulations [27].

After the system has reached equilibrium, the water isosteric heat of adsorption $Q_{st}$ was measured in the grand canonical ensemble using [28]

$$Q_{st} = RT - \frac{\langle UN \rangle - \langle U \rangle \langle N \rangle}{\langle N^2 \rangle - \langle N \rangle^2}, \qquad (7)$$

where R is the gas constant, U the energy of the system, N the adsorbed amount, and the angular brackets denote the ensemble average. Our results yield $Q_{st} \approx 45-48$ kJ/mol, in reasonable agreement with experimental data (Fig. S1). As a comparison, the heat of vaporisation of the TIP4P/ε water model is 46 kJ mol⁻¹ [24], suggesting that the strength of the interaction of the water molecules with the NaCl(100) is similar to the interaction between water molecules in the liquid phase.

<figure><img src="image_4.png" /><figcaption>Figure 3. Permeance L of a single water layer as a function of the amount of water $\theta$. Symbols are results from NEMD simulations for NaCl(100) surface (full disks) and charged silica surface (open squares), respectively. The dashed line is obtained from Eq. (3), the dot-dashed line is Eq. (5) with $D_s = 7 \cdot 10^{-10}$ m² s⁻¹. The dotted line corresponds to Eq. (5) with $D_s = 1 \cdot 10^{-11}$ m² s⁻¹, and the full lines denote the overall transport according to Eq. (11). Insert: Same results as in the main panel, but as a function of the relative humidity RH.</figcaption></figure>

The overall good agreement between our molecular simulation results and experimental data shows the validity of the force fields used in this study to describe the thermodynamics of the water-NaCl(100) interface.

## B. Transport properties of the water layer

The transport properties of a thin film of water on the NaCl(100) surface was evaluated using Non-Equilibrium Molecular Dynamics (NEMD). The initial water content was varied from $\theta = 0.03$ to 4, the temperature was set to 300 K, and a driving force $f_0$ was applied to the water molecules in the $e_x$ direction parallel to the NaCl(100) surface. After an equilibration time of 1 ns, the total water flux $Q$ in the direction of the force was recorded and the permeance $L$ was calculated as $L = Q/fw$. More details are given in supplementary material, and input files are openly available in DaRUS repository [25].

Our results show that for $\theta > 1$, the permeance $L \propto \theta^3$, in agreement with the Stokes solution given by Eq. (6) (full symbols in Fig. 3). Using the viscosity and the density of the TIP4P/ε water model, respectively, $\eta = 0.855$ mPa s and $\rho_w = 55.3$ mol/L [24], together with a surface site density estimated as $N_s = \lambda^{-2}$ where $\lambda = 2.8$ Å is the equilibrium Na⁺-Cl⁻ distance, Eq. (6) gives a quite good agreement with MD simulation for $\theta > 1$ (dashed line in Fig. 3), only overestimating $L$ by a factor of about 2.
The difference between our molecular simulation results and the Stokes prediction is likely to originate in the interfacial properties of water, which is known to differ from its bulk properties [15, 29–31]. For instance, the density profile shows some oscillations near the solid wall (Fig. S2 and Fig. S3), which is a consequence of fluid structuring. Accounting for those oscillations in the calculation of $Q_{\text{Stokes}}$ (Eq. (2)) by substituting $\rho_w$ by $\rho(z)$ only leads to a correction of 1% on the value of $Q_{\text{Stokes}}$ for $\theta = 4$, suggesting that density oscillations can be reasonably ignored from the model.

In addition to the density, the viscosity is also known to differ from the bulk at interfaces [30]. In order to account for modified water properties within the first water layer, one can assume the finite slip boundary conditions $u|_{z=0} = b \partial u/\partial z|_{z=0}$, where b is the slip length [14]. In case of slip, Eq. (1) reads

$$u(z) = -\frac{f}{\eta} \left( \frac{z^2}{2} - hz - hb \right). \qquad (8)$$

A reasonably good agreement between Eq. (8) and MD results for $\theta = 4$ is found for $b = -0.16$ nm and $\eta = 0.886$ mPa.s (instead of the 0.855 mPa.s expected for the TIP4P/ε water model [24]) (Fig. S4). In the presence of a finite slip length b, Eq. (3) reads

$$L_{\text{Stokes}} = \frac{\rho_w}{3\eta} h^2 (h+b). \qquad (9)$$

While Eq. (9) with $b = -0.16$ nm and $\eta = 0.886$ mPa.s fits the MD results better than Eq. (3) for large water content $\theta > 2$ (Fig. S5), it also predicts negative values for $L_{\text{Stokes}}$ for $\theta \le 1$, which suggest a breakdown of the Stokes formulation for such extreme water content limit, as expected for sub-nanometric water films [14].

Our results also show that, for $\theta < 1$, $L \propto \theta^\alpha$ with $\alpha \approx 1$, in agreement with a random walk on a lattice as given by Eq. (5) (full symbols in Fig. 3). Such good scaling agreement suggests that, for $\theta \to 0$, the mobility of water at the NaCl(100) surface involves the hopping of the adsorbed molecules from one adsorption site to another. Jumps of the water molecules from site to site can be seen from the molecular dynamics trajectory (see the movies for $\theta = 0.1$ [32]). In addition, the diffusivity of the water molecules for $\theta = 0.1$ varies with temperature according to an equation of the Arrhenius form, as expected for activated surface diffusion [33] (Fig. S8). Here the adsorption sites correspond to both $Na^+$ and $Cl^-$ atoms of the NaCl(100) surface, with the adsorption at the $Na^+$ sites being slightly stronger than that at the $Cl^-$ sites [18]. In order to compare Eq. (5) with molecular dynamics results, we measured the self-diffusion coefficient as $D_s = 7 \cdot 10^{-10} \text{ m}^2 \text{ s}^{-1}$ using Equilibrium Molecular Dynamics (EMD) simulations for $\theta = 0.1$ (details of the EMD simulations are given in the supplementary material). Equation (5) slightly underestimates the permeance by a factor $\approx 2$ (full symbols in Fig. 3). This underestimation could be related to the assumption behind Eq. (5) that the fluid molecules are non-interacting. However, in the case of water molecules on the NaCl(100) surface, small clusters of molecules are always present in our simulations (Fig 1cd), as indicated by the

<figure><img src="image_5.png" /><figcaption>Figure 4. a) Oxygen-oxygen coordination number $C_{O-O}$ from Eq. (10), as a function of the water content $\theta$. The results have been obtained with the NaCl(100) surface (full disks) and charged silica surface (open squares), respectively. b) Average number of hydrogen bonds per water molecule $N_{\text{HB}}$ between the water molecules and the solid surfaces.</figcaption></figure>

coordination number defined as

$$C_{O-O} = 4\pi \int_{0}^{3.4\AA} r^2 g_{O-O}(r) \rho_w dr, \qquad (10)$$

where $g_{O-O}(r)$ is the oxygen-oxygen radial distribution function (Fig 4a), and $r \le 3.4 \AA$ the first coordination shell. Even for $\theta$ as low as 0.03, $C_{O-O} \approx 0.6$ (see also the movies in Ref. [32]). As $\theta$ increases, $C_{O-O}$ increases until it reaches a value of 4 for the largest water content considered here (the bulk value we measured for water is about 4.6).

Assuming that the two contributions, Eqs. (3) and (5) act in parallel, as done in the dusty gas model [34], the total permeance of the thin film reads

$$L_{\text{Film}} = L_{\text{Stokes}} + L_{\text{RW}}. \qquad (11)$$

Eq. (11) compares well with the NEMD results for the full range of water content considered here (full line in Fig. 3).

### C. Comparison between NaCl(100) and charged silica

In order to evaluate whether the results obtained with NaCl(100) are specific to this surface, we perform similar simulations with a charged silica surface with surface charge $\Sigma \approx 0.1 \text{ C m}^{-2}$ corresponding to pH $\approx 7$ [35] (see the supplementary material for details). In the case of the silica, we define the surface coverage such that a water content $\theta = 1$ corresponds to 1 water molecule per NaCl atom on a NaCl(100) surface of equal dimension. For $\theta > 0.3$, values of the permeance $L$ are similar to the ones obtained with NaCl(100) (open symbols in Fig. 3). However, for $\theta \le 0.3$, values of $L$ are much lower for silica than with NaCl(100): for $\theta = 0.03$, $L$ is two orders of magnitude lower than in the case of NaCl(100). The comparatively lower permeance of the silica surface for $\theta \le 0.3$ can be explained, at least in part, by the difference in hydrogen bonding between water and the two surfaces. Our results indicate that there is about 100 times more hydrogen bonds per water molecule between water and the silica surface than between water and the NaCl(100) surface (Fig 4b). Here, a
hydrogen bond corresponds to an inter-oxygen distance < 3 Å, and a minimum angle of 150° between the donor oxygen, the hydrogen, and the acceptor.

On the silica surface, the density of sites with which wa-
ter can form hydrogen bonding is ≈ 3.8 nm⁻², where we ac-
counted for both deprotonated hydroxyl groups (in this case the
water molecule supplies the hydrogen atom) and protonated
hydroxyl groups (in this case the hydroxyl group supplies the
hydrogen atom). Therefore, for water contents up to θ ≈ 0.3,
every water molecule can form, on average, one hydrogen bond
with the surface. However, for θ > 0.3, an unbounded layer of
water molecules appears, as indicated by coordination number
and hydrogen bond measurements (Fig. 4). For the NaCl(100)
surface, there are ≈ 6 chloride atoms per square nanometer
with which a water molecule can potentially form a hydrogen
bond. However, as visible in Fig. 4b, even when there are more
available bounding sites than there are water molecules (i.e. for
θ <approx~ 0.6), only about 0.01 hydrogen bond per water molecule
is recorded, compared to almost 1 in the case of the silica.

The self-diffusion coefficient of water on the silica surface was found to be D<sub>s</sub> = 1 · 10<sup>-11</sup> m<sup>2</sup> s<sup>-1</sup> using EMD simulations for θ = 0.1, i.e. more than one order of magnitude lower as for NaCl(100). Therefore it is not possible to conclude from our data whether the random walk regime L ~ θ is present in the case of the silica surface (note that our results are not in contradiction with the presence of L ~ θ scaling for low θ, see the dotted line in Fig. 3). Our results also indicate that for a water content θ > 0.3, NaCl(100) and charged silicate surfaces have a similar permeance L for a given water content. It is, however, instructive to compare the permeance of both surfaces for a given RH instead of a given water content. Our results show that, for a given RH, the permeance of the NaCl(100) surface is always larger than the permeance of the silica surface by at least one order of magnitude (at least for the range of relative humidity considered here) (Insert in Fig. 3).

D. The effect of roughness

In order to explore the impact of surface defects on the prop-
erties of the water layer, NaCl surfaces with different degrees
of roughness were generated by randomly removing a fraction
1 – *r* of atoms from the top-most and bottom-most layer of the
NaCl(100) block (see surface samples for *r* = 0.1, 0.5, and 0.9
in Fig S6, and movies [36]). We made sure that the generated
systems remained charge neutral. For each value of *r*, three dif-
ferent initial configurations were created, and each simulation
was reproduced three times. To avoid the dissolution of the
incomplete surface layer, the ions were prevented from moving
along the *e*<sub>*x*</sub> direction, but free to move along the *e*<sub>*x*</sub> and *e*<sub>*y*</sub> directions. First, adsorption isotherms for RH ≈ 30, 40, and
50%, where extracted using the GCMC-MD method (Fig. 5a).
All three isotherms show an increase of the adsorbed water
amount for 0 < *r* < 1 with respect to the pristine NaCl(100)
(*r* = 0 or 1) case. Such increase of θ for rough surfaces was
expected, as the roughness effectively increases the effective
surface area of the solid.

The results obtained for RH = 0.3 indicate that, although

a rough surface accommodates up to four times more water
molecules than the pristine NaCl(100) surface (Fig. 5a), its
permeance remains essentially unchanged (Fig. 5b), suggest-
ing that most of the additional water molecules are effectively
trapped by the solid surface, and that the instantaneous num-
ber of water molecules participating in the transport remains
practically unaffected by the surface roughness. Our results
also show that, for RH = 0.3, the number of hydrogen bonds
between the water and the surface only slightly increase for
0 < *r* < 1 (Fig. 5c), and that the coordination number C<sub>O-O</sub>
is quasi-independent of *r* (Fig. 5d), which confirms that the
additional water molecules that adsorb because of the surface
roughness are as ‘attached’ to the solid surface as the water
molecules which are bound to the pristine NaCl(100).

However, even for the highest value of RH considered here (i.e. 0.5), the amount of water increases by ≈ 60% at most for *r* ≈ 0.1–0.2, whereas the permeance is found to increase up to a factor of 4 for *r* = 0.1 as compared to *r* = 0 (Fig. 5b). The coordination number *C*<sub>O–O</sub> is also found to increase significantly for *r* = 0.1 (Fig. 5d), suggesting that for RH = 0.5, the additional water molecules that adsorb due to the roughness of the solid surface are in average less bound to a rough surface than they are to pristine NaCl(100), and are thus more free to contribute to the water transport [37]. Results obtained with RH = 0.4, are in between the 0.3 and the 0.5 cases.

In the simulations presented in Fig. 5, the ions of the incom-
plete layer were constrained along *e*<sub>*z*</sub>, but free to move along *e*<sub>*x*</sub> and *e*<sub>*y*</sub>. As a consequence, and due to the net flux of the
water molecules along *e*<sub>*x*</sub>, a net flux of ions was recorded as
well in the *e*<sub>*x*</sub> direction. Regardless of the respective values
of *r* and RH, the average velocity of the ions was measured to
be about two orders of magnitude lower than the velocity of
water (Fig. S7 ab), with the ion velocity increasing non-linearly
with the water velocity (Fig. S7 c), but linearly with the wa-
ter content θ (Fig. S7 d). In order to evaluate the impact of
constraining the ions along the *e*<sub>*z*</sub> direction on the water ve-
locity, simulations in absence of such a constraint were also
performed. Our results showed that, when the ions of the
incomplete layer are not constrained, their velocities can be
increased by up to one order of magnitude with respect to
the case were they are constrained along *e*<sub>*z*</sub>, with the highest ve-
locities being recorded in the limit of small *r* (compare full and
open symbols in Fig. S7 b). However, the absence (or presence)
of constraint on the ion does not impact significantly the water
velocity (compare full and open symbols Fig. S7 a).

III. DISCUSSION

From our results, we can estimate the contribution from the
thin water film to the overall transport properties of a porous
material such as salt crust. Let us consider a slit pore at equi-
librium with RH = 0.5 and exposed to a gradient of chemical
potential parallel to the solid walls. Let us also assume that
the pore is large enough not to be filled with liquid water by
capillary condensation. The two main contributions to the wa-
ter transport are the thin films (one per surface) and the water
vapor phase. The contribution to the permeance from the vapor
<figure><img src="image_6.png" /><figcaption>Figure 5. Amount of water θ at the rough NaCl surface (a), permeance L of a single water layer (b), average number of hydrogen bonds per water molecule N<sub>HB</sub> between the water molecules and the solid surfaces (c), and coordination number C<sub>O-O</sub> within a cutoff of 3.4 Å (d) as a function of the surface roughness parameter r. Results have been obtained for RH = 0.3 (triangles), RH = 0.4 (squares), and RH = 0.5 (disks), respectively.</figcaption></figure>

layer can be estimated as

$$L_{\text{Vapor}} = (D_{\text{K}}^{-1} + D_{0}^{-1})^{-1} \frac{H_{v}}{RT}, \quad (12)$$

where D<sub>K</sub> = H<sub>v</sub>u<sub>th</sub>/3 is the Knudsen diffusion coefficient, D<sub>0</sub> = 2.5 · 10<sup>-5</sup> m<sup>2</sup>/s is the bulk diffusion coefficient of water vapor [38, 39], H<sub>v</sub> = H - 2h is the thickness of the vapor layer, and u<sub>th</sub> = √8RT/πM<sub>w</sub> is a thermal velocity with M<sub>w</sub> the molar mass of water. The water flux within the vapor layer follows as Q<sub>vapor</sub> = L<sub>vapor</sub>wf, where the force per unit volume f is proportional to the vapor density ρ<sub>vapor</sub> = RHρ<sub>sat</sub>, where ρ<sub>sat</sub> is the saturated vapor pressure. The fraction of total water flux occurring within the thin film is given by

$$\phi = \frac{2Q_{\text{Film}}}{2Q_{\text{Film}} + Q_{\text{Vapor}}}, \quad (13)$$

where Q<sub>Film</sub> = Q<sub>Stokes</sub> + Q<sub>RW</sub>. Our results show that for RH = 0.3, 0.4, and 0.5, the transport is dominated by surface flow contribution for H up to a critical value H* ≈ 70, 220, and 900 nm, respectively (Fig. 6a). Here, in order to predict φ using Eq. (13), a numerical interpolation of the isotherm of Fig. 2 was used to link θ and RH. H* is found to increase quasi-exponentially with RH (Fig. 6b). In the limit of Stokes dominated transport in the film (i.e. Q<sub>Stokes</sub> > Q<sub>Vapor</sub>), and assuming that bulk diffusion dominates over Knudsen diffusion (i.e. H ≫ 3D<sub>0</sub>/u<sub>th</sub> = 126 nm), one gets that H* ∝ h<sup>3</sup>/ρ<sub>vapor</sub>, where ρ<sub>vapor</sub> ∝ RH. Our results indicate that thin film water must be taken into account when predicting transport properties of water through salt crusts, even for pore size in the micrometer range, particularly if RH is large.

<figure><img src="image_7.png" /><figcaption>Figure 6. a) Fraction of the total water flux that is due to the surface layer in a NaCl(100) slit of thickness H as given by Eq. (13). Results are given for three values of the relative humidity: RH = 0.3 (full line), 0.4 (dashed line), and 0.5 (dotted line). The horizontal dashed line marks the transition between thin film dominated water transport (φ > 0.5) and vapor dominated transport (φ < 0.5). b) Critical pore size H* below which the transport is surface dominated, as given by Eq. (13) (blue symbol). The calculation assumes the absence of capillary filling in the pore. Gray dashed line is ∝ exp(RH).</figcaption></figure>

When comparing the properties of thin film water on NaCl(100) and charged silica surfaces, our results reveal a large difference between the two surfaces in terms of conductance in the limit of small water amount, with the silica surface having the lowest permeance for a given water amount (Fig. 3). This difference is mostly due to the much stronger hydrogen bonding between water and silica, as compared with water and NaCl (Fig. 4). The difference in permeance between NaCl(100) and silica could also be due, in part, to the very different roughness of the two surfaces: NaCl(100) is atomically smooth, and silica is decorated with hydroxyl groups, and also bears Na⁺ counter ions. However the results obtained with NaCl surfaces with different degrees of roughness suggest that roughness does not impact significantly the permeance in the limit of low water content (Fig. 5). Note, however, that it is difficult to settle on the impact of roughness on the permeance when comparing NaCl(100) and silica, since different roughness also induces differences in the adsorbed amount of water for a given RH.

## IV. CONCLUSION

The transport properties of a thin film water at the NaCl(100) surface were investigated using molecular simulations. Our results show the transition between lattice-like and a fluid-like transport for water content θ ≈ 0.5, where θ is the number of water molecule per surface atom. In the fluid-like regime, the permeance was found to be given by the resolution of the Stokes equation, with a overestimation of the transport properties by a factor of 2 that we attributed to the modification of the water properties near the surface. In the lattice-like regime, the permeance was found to scale as ∝ θ, in agreement with random walk prediction for atom on a lattice. Our results also show that, at low relative humidity RH = 0.3, the transport properties of the thin film are barely affected by surface defects, but at a higher value of RH = 0.5, the permeance increases in
the presence of defects due to the additional water molecules that adsorb to the surface and that contribute to the transport. Comparisons between charged silica surface and NaCl(100) highlight the strong differences in transport properties between the two surfaces in the limit of a low water content, which we attributed to the strong hydrogen bonding between water and silica. Our results allowed us to estimate the crossover between surface dominated and vapor dominated transport in a slit nanochannel.

## ACKNOWLEDGMENTS

We thank the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) for funding this project Number 327154368 - SFB 1313. A.S. and C.H. acknowledge funding from the DFG under Germany's Excellence Strategy-EXC 2075-390740016 and support by the Stuttgart Center for Simulation Science (SimTech).

## SUPPLEMENTARY MATERIAL

See the supplementary material for molecular simulation details and Fig. S1, S2, S3, S4, S5, S6, S7, and S8.

## DATA AVAILABILITY STATEMENT

LAMMPS and GROMACS input files are openly available in DaRUS repository at doi:10.18419/darus-2726, reference number [25].

## CONFLICT OF INTEREST

The authors have no conflicts to disclose.

## REFERENCES

<footnote n="1">¹I. Langmuir, "The adsorption of gases on plane surfaces of glass, mica and platinum." Journal of the American Chemical society **40**, 1361–1403 (1918).</footnote>

<footnote n="2">²G. E. Ewing, "Thin film water," *The Journal of Physical Chemistry B* **108**, 15953–15961 (2004).</footnote>

<footnote n="3">³G. E. Ewing, "H2o on nacl: From single molecule, to clusters, to monolayer, to thin film, to deliquescence," Intermolecular forces and clusters II, 1–25 (2005).</footnote>

<footnote n="4">⁴G. E. Ewing, "Ambient thin film water on insulator surfaces," *Chemical Reviews* **106**, 1511–1526 (2006).</footnote>

<footnote n="5">⁵P. G. De Gennes, "Wetting: Statics and dynamics," *Review of Modern Physics* **57**, 827 (1985).</footnote>

<footnote n="6">⁶C. Rodriguez-Navarro and E. Doehne, "Salt weathering: influence of evaporation rate, supersaturation and crystallization pattern," *Earth Surface Processes and Landforms: The Journal of the British Geomorphological Research Group* **24**, 191–209 (1999).</footnote>

<footnote n="7">⁷U. Nachshon, N. Weisbrod, R. Katzir, and A. Nasser, "Nacl crust architecture and its impact on evaporation: Three-dimensional insights," *Geophysical Research Letters* **45**, 6100–6108 (2018).</footnote>

<footnote n="8">⁸N. Sghaier and M. Prat, "Effect of efflorescence formation on drying kinetics of porous media," *Transport in Porous Media* **80**, 441–454 (2009).</footnote>

<footnote n="9">⁹H. Eloukabi, N. Sghaier, S. B. Nasrallah, and M. Prat, "Experimental study of the effect of sodium chloride on drying of porous media: The crusty-patchy efflorescence transition," *International Journal of Heat and Mass Transfer* **56**, 80–93 (2013).</footnote>

<footnote n="10">¹⁰U. Nachshon and N. Weisbrod, "Beyond the salt crust: On combined evaporation and subflorescent salt precipitation in porous media," *Transport in Porous Media* **110**, 295–310 (2015).</footnote>

<footnote n="11">¹¹A. Singh, "Soil salinization management for sustainable development: A review," *Journal of environmental management* **277**, 111383 (2021).</footnote>

<footnote n="12">¹²S. J. Peters and G. E. Ewing, "Water on salt: An infrared study of adsorbed H2O on NaCl(100) under ambient conditions," *Journal of Physical Chemistry B* **101**, 10880–10886 (1997).</footnote>

<footnote n="13">¹³M. C. Foster and G. E. Ewing, "Adsorption of water on the NaCl(001) surface. II. an infrared study at ambient temperatures," *Journal of Chemical Physics* **112**, 6817–6826 (2000).</footnote>

<footnote n="14">¹⁴L. Bocquet and E. Charlaix, "Nanofluidics, from bulk to interfaces," *Chemical Society Reviews* **39**, 1073–1095 (2010).</footnote>

<footnote n="15">¹⁵A. Schlaich, J. Kappler, and R. R. Netz, "Hydration friction in nanoconfinement: From bulk via interfacial to dry friction," *Nano Letters* **17**, 5969–5976 (2017).</footnote>

<footnote n="16">¹⁶Y. Choi, L. A. Baker, H. Hillebrenner, C. R. Martin, and S. Matter, "Biosensing with conically shaped nanopores and nanotubes," *Physical Chemistry Chemical Physics* **8**, 4976–4988 (2006).</footnote>

<footnote n="17">¹⁷D. A. Reed and G. Ehrlich, "Surface diffusion, atomic jump rates and thermodynamics," *Surface Science* **102**, 588–609 (1981).</footnote>

<footnote n="18">¹⁸O. Engkvist and A. J. Stone, "Adsorption of water on the nacl(001) surface. iii. monte carlo simulations at ambient temperatures," *Journal of Chemical Physics* **112**, 6827–6833 (2000).</footnote>

<footnote n="19">¹⁹M. Hucher, A. Oberlin Mathieu-Sicaud, and R. Hocart, "Adsorption de vapeur d'eau sur les faces de clivage de quelques halogénures alcalins," *Bulletin de Minéralogie* **90**, 320–332 (1967).</footnote>

<footnote n="20">²⁰H. Shindo, M. Ohashi, O. Tateishi, and A. Seo, "Atomic force microscopic observation of step movements on NaCl (001) and NaF (001) with the help of adsorbed water," *Journal of the Chemical Society, Faraday Transactions* **93**, 1169–1174 (1997).</footnote>

<footnote n="21">²¹V. Vlasov, A. Muslimov, and V. Kanevsky, "Water adsorption on the (001) surface of NaCl," *Journal of Surface Investigation: X-ray, Synchrotron and Neutron Techniques* **14**, 1040–1043 (2020).</footnote>

<footnote n="22">²²A. P. Thompson, H. M. Aktulga, R. Berger, D. S. Bolintineanu, W. M. Brown, P. S. Crozier, P. J. in't Veld, A. Kohlmeyer, S. G. Moore, T. D. Nguyen, et al., "LAMMPS - a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales," *Computer Physics Communications* **271**, 108171 (2022).</footnote>

<footnote n="23">²³P. Loche, P. Steinbrunner, S. Friedowitz, R. R. Netz, and D. J. Bonthuis, "Transferable ion force fields in water from a simultaneous optimization of ion solvation and ion-ion interaction," *The Journal of Physical Chemistry B* **125**, 8581–8587 (2021).</footnote>

<footnote n="24">²⁴R. Fuentes-Azcatl and J. Alejandre, "Non-polarizable force field of water based on the dielectric constant: Tip4p/ε," *Journal of Physical Chemistry B* **118**, 1263–1272 (2014).</footnote>

<footnote n="25">²⁵S. Gravelle, "Molecular simulation scripts for thin film water on NaCl surface," (2022).</footnote>

<footnote n="26">²⁶B. Coasne, A. Galarneau, R. J. Pellenq, and F. Di Renzo, "Adsorption, intrusion and freezing in porous silica: the view from the nanoscale," *Chemical Society Reviews* **42**, 4141–4171 (2013).</footnote>

<footnote n="27">²⁷H. Shinto, T. Sakakibara, and K. Higashitani, "Molecular dynamics simulations of water at NaCl(001) and NaCl(011) surfaces," *Journal of Physical Chemistry B* **102**, 1974–1981 (1998).</footnote>

<footnote n="28">²⁸D. Nicholson and N. G. Parsonage, *Computer simulation and the statistical mechanics of adsorption* (Academic Press, 1982).</footnote>

<footnote n="29">²⁹S. Nakaoka, D. Surblys, Y. Yamaguchi, K. Kuroda, T. Nakajima, and H. Fujimura, "Local viscosity change in water near a solid-liquid interface and its extraction by means of molecular rotational diffusion-a molecular dynamics study," *Chemical Physics Letters* **591**, 306–311 (2014).</footnote>

<footnote n="30">³⁰M. Rezaei, A. R. Azimian, A. R. Pishevar, and D. J. Bonthuis, "Viscous interfacial layer formation causes electroosmotic mobility reversal in monovalent electrolytes," *Physical Chemistry Chemical Physics* **20**, 22517–22524 (2018).</footnote>

<footnote n="31">³¹A. Wolde-Kidan and R. R. Netz, "Interplay of interfacial viscosity, specification, and impurity adsorption determines zeta potentials of phospholipid</footnote>
membranes,” Langmuir **37**, 8463–8473 (2021).

<sup>32</sup>S. Gravelle, “Movies of thin film water on NaCl(100) surface,” (2022).

<sup>33</sup>J.-G. Choi, D. Do, and H. Do, “Surface diffusion of adsorbed molecules in porous media: Monolayer, multilayer, and capillary condensation regimes,” *Industrial & engineering chemistry research* **40**, 4005–4031 (2001).

<sup>34</sup>E. Mason and A. Malinauskas, *Gas Transport in Porous Media: The Dusty-gas Model*, Chemical engineering monographs (Elsevier, 1983).

<sup>35</sup>F. S. Emami, V. Puddu, R. J. Berry, V. Varshney, S. V. Patwardhan, C. C. Perry, and H. Heinz, “Force field and a surface model database for silica to simulate interfacial properties in atomic resolution,” *Chemistry of Materials* **26**, 2647–2658 (2014).

<sup>36</sup>S. Gravelle, “Movies of thin film water on rough NaCl surface,” (2022).

<sup>37</sup>B. Abeles, L. F. Chen, J. W. Johnson, and J. Drake, “Capillary condensation and surface flow in microporous vycor glass,” *Israel journal of chemistry* **31**, 99–106 (1991).

<sup>38</sup>P. S. Nobel, *Physicochemical and environmental plant physiology* (Academic press, 1999).

<sup>39</sup>S. Gravelle and J. Dumais, “A multi-scale model for fluid transport through a bio-inspired passive valve,” *The Journal of chemical physics* **152**, 014502 (2020).