Origins of fast diffusion of water dimers on surfaces

The diffusion of water molecules and clusters across the surfaces of materials is important to a wide range of processes. Interestingly, experiments have shown that on certain substrates, water dimers can diffuse more rapidly than water monomers. Whilst explanations for anomalously fast diffusion have been presented for specific systems, the general underlying physical principles are not yet established. We investigate this through a systematic ab initio study of water monomer and dimer diffusion on a range of surfaces. Calculations reveal different mechanisms for fast water dimer diffusion, which is found to be more widespread than previously anticipated. The key factors affecting diffusion are the balance of water-water versus water-surface bonding and the ease with which hydrogen-bond exchange can occur (either through a classical over-the-barrier process or through quantum-mechanical tunnelling). We anticipate that the insights gained will be useful for understanding future experiments on the diffusion and clustering of hydrogen-bonded adsorbates.

meV compared to the very large 9 × 9 cell. For the other metal (111) surfaces, the smallest lattice constant is only 8% smaller than Pd, hence we expect the convergence with respect to the cell size is similar across all the metal surfaces studied. On NaCl(100), MgO(100), and ZnO(1010), the shortest O-O distances are 8.5Å, 6.1Å, and 6.9Å respectively. Due to that the dimer diffusion barriers on non-metal surfaces are much higher than on metals, the relative impact caused by the interaction between periodic water images should be less significant.
We also tested the convergence of the diffusion barrier with respect to the number of layers on two examples. On Pd(111), the monomer diffusion barrier is only 4 meV different between a 4 layer (6.8Å thick) and a 9 layer slab (13.1Å thick), suggesting that a 4 layer slab is converged. On ZnO(1010), the dimer translation barriers on a 3 layer slab (6.6Å thick) and a 4 layer slab (9.5Å thick) differ by only 3 meV.
The sensitivity of the water monomer and dimer diffusion barriers to DFT exchangecorrelation functional has also been tested on Pd(111). Three other van der Waals inclusive functionals (optB86b-vdW, 1 revPBE-D3, 2,3 and TPSS-D3 3,4 ) are tested, and the results are compared with the optB88-vdW 5 functional used in the main text. These functionals were chosen because they predict accurate lattice parameters for Pd within 2% error with the experimental value of 3.88Å, and have shown excellent performances for gas phase water clusters. [6][7][8] We first relaxed the lattice and surface, and then performed climbing image nudged elastic band (cNEB) calculations for each functional to obtain the diffusion barriers.
The results are shown in Supplementary this step, the water dimer is displaced by one lattice, but the H-bond direction is the opposite as the initial state. Hence, the second step, where the H-bond donor and acceptor exchange roles is needed to complete the diffusion process.

ADSORPTION ENERGY
The adsorption energies (E ad ) of the water monomer and dimer are defined as: where E nH 2 O/surf is the total energy of the n H 2 O adsorbed surface system, E surf is the total energy of the relaxed bare surface slab and E H 2 O is the total energy of the relaxed water monomer in gas phase. Under this definition, E ad is positive for all the systems studied. We decompose the interaction energy of the water dimer on surfaces into three parts: donor water-surface (E D-surf ), acceptor water-surface (E A-surf ), and water-water interactions (E water-water ) which predominately characterises the H-bond interaction at the adsorbate state. They are defined as the following: where E ad (dimer) is the adsorption energy of a water dimer on the surface. E ad (donor/acceptor) is the total energy of a water molecule on the surface fixed at the donor/acceptor geometry of the adsorbed water dimer.) We note that decompositions of adsorption energies are all to a certain extent arbitrary.
Supplementary Table 3 shows the decomposition results for the water dimer adsorption state (initial), and the diffusion transition states (TSs). Decomposition shows that on surfaces where the water-water interactions are smaller than or similar to that in the gas phase Initial Dimer translation TS Dimer DA exchange TS water dimer, both the H-bond donor and acceptor water bind to the surface equally strong.
If the water-water interaction in the dimer is stronger than or similar to E D-surf , the acceptor water interacts much weaker with the surface.

SUPPLEMENTARY NOTE 4. THE IMPACT OF SURFACE FLEXIBILITY ON WATER DIFFUSION ON NON-METAL SURFACES
We also computed the adsorption and diffusion of water monomers and dimers on flexible non-metal surfaces (Supplementary Table 4). We see that compared to fixed surfaces, using flexible surfaces quantitatively increase the water monomer and dimer diffusion barriers by 10%-30%. Yet the adsorption energies of water monomers and dimers are also increased on flexible surfaces, inline with the qualitative idea that the stronger the adsorbate sticks to the surface, the slower it diffuses. The only exception we see is water dimer diffusion on NaCl, where using a flexible surface stabilises the TS more than the initial state.  One can also estimate the water-water interactions for water dimers on flexible surfaces via a more complicated decomposition. One viable treatment is to consider the surface deformation energy as part of the water-surface interactions in the decomposition in Eq. 2: in which E deform = E * surf −E surf (E * surf is the energy of the substrate at the optimised geometry with the adsorbate). E D-surf and E A-surf are defined as same as in Eq. 2. We compared the water-water interactions for a water dimer on flexible and fixed surfaces, shown in Supplementary Table 4. One can see that using flexible surface also does not qualitatively change the water-water interaction in the decomposition. And finally, for the rotation step of the non-transitional water dimer diffusion on NaCl(100), using flexible surface changes the barrier by less than 0.01 eV (to 0.089 eV). This TS geometry is 50 meV higher in energy on Pt(111) compared to the TS discussed in the main text, and on Rh(111) the path in Supplementary Figure 3 We have tested the correlation between the translational diffusion barrier and the adsorption energy, with two other definitions of the dimer adsorption energy: where E nH 2 O is the energy of a optimised gas phase water dimer. This means E ad2 is defined with respect to a gas phase water dimer.
where E * nH 2 O is the energy of a gas phase water dimer fixed at the geometry on the surface. The results are shown in Supplementary Figure 4. The R 2 value of the linear regression for monomer and dimer with E ad2 is 0.68. Using E ad3 gives the same R 2 value. Despite that the correlation becomes better than with E ad , none of the correlations are very good, and clearly worse than the correlation found in the main text.

SUPPLEMENTARY NOTE 6. ADDITIONAL DISCUSSIONS ON THE DIMER DA EXCHANGE DIFFUSION
The process of the two water exchanging their roles is the key step in the water dimer DA exchange diffusion mechanism proposed in ref., 16  parallel to the surface is referred to as a "twist". The rotation of a water around one of its two O-H bonds is referred to as a "flip". In the twist-twist mechanism, the two water molecules rotate in the plane parallel to the surface, while the two O atoms switch height. energy. We used the definition of adsorption energy that gave the best correlation, which is We considered a stepwise water dimer diffusion mechanism and tested it on Cu(111).
The separated water dimer geometry (Supplementary Figure 8) is 0.07 eV higher in energy than the DA exchange TS on Cu(111), an amount close to the E water-water in Supplementary Table 3. This suggests that the stepwise diffusion mechanism is unfavourable compared to the DA exchange mechanism on Cu(111) and other transition metal (111) surfaces, as the DA exchange TS is further stabilised by the water-water interaction.
Relative energy to the intact water dimer (eV) Next we discuss whether water dissociation plays an important role for water diffusion on surfaces. It has been shown that water monomers have high dissociation barriers (∼1 eV) on the surfaces we studied. 18,19 For the water dimer, we computed the energies of different rates are extrapolated to the 9 × 9 cell using the barrier difference between the 4 × 4 and 9 × 9 cell given in Supplementary Figure 1, the rates increase by 1-2 orders of magnitude.
Yet since both the monomer and dimer barriers increase by a similar amount (∼10 meV), the temperature at which the dimer diffusion becomes faster is barely changed by this extrapolation. Furthermore, we considered the impact of a general 10 meV uncertainty on the diffusion rates, as shown by the coloured areas in Supplementary Figure 11. We still see the trend that the dimer diffusion tends to become faster than monomer diffusion as the temperature decreases. However, if the uncertainties make the dimer DA exchange barrier larger than monomer diffusion barrier, i.e. more than 30 meV barrier reduction from tunnelling at 25 K, then dimer diffusion will likely become faster than monomer diffusion at below 25 K instead. The contribution of the anharmonic effects of the water dimer rotation around the axis perpendicular to the surface (see Supplementary Figure 1(c) in the maintext) has also been considered. The rotation barrier, for example, on Pd(111) is ∼9 meV (calculated with cNEB), which is larger than k B T in the temperature range considered (25 − 40 K). We calculated the hindered rotor partition function 22 in this temperature range, and found that it is only twice as large as the harmonic approximation (the harmonic frequency for this mode is ∼20 cm −1 on Pd (111)). Even in the free rotor limit, the rotational partition function will only contribute no more than a factor of 5 to the reduction of the water dimer diffusion rate prefactor. The water monomer also has a rotation mode around the axis perpendicular to the surface (∼40 cm −1 on Pd(111)), in which the free rotor partition function is within a factor of 2 with the harmonic approximation at 25 − 40 K. Hence, the anharmonic rotational partition functions do not have a significant impact on the diffusion rate here, which is dominated by the exponential term.