Supercooled liquid-like dynamics in water near a fully hydrated titania surface: Decoupling of rotational and translational diffusion

We report an ab initio molecular dynamics (MD) simulation investigating the effect of a fully hydrated surface of TiO 2 on the water dynamics. It is found that the universal relation between the rotational and translational diffusion characteristics of bulk water is broken in the water layers near the surface with the rotational diffusion demonstrating progressive retardation relative to the translational diffusion when approaching the surface. This kind of rotation–translation decoupling has so far only been observed in the supercooled liquids approaching glass transition, and its observation in water at a normal liquid temperature is of conceptual interest. This finding is also of interest for the application-significant studies of the water interaction with fully hydrated nanoparticles. We note that this is the first observation of rotation–translation decoupling in an ab initio MD simulation of water.


I. INTRODUCTION
This study represents two extensive and traditionally unrelated research fields: water-solid interfaces and the dynamical anomalies commonly observed in supercooled liquids. We investigate water dynamics at a fully hydrated titanium oxide (TiO 2 , titania) surface. Metal oxide-water interfaces have been investigated with a large number of experimental and computational techniques in the literature in order to address, for example, their catalytic power and adsorption properties. 1,2 Clean metal oxide surfaces expose undercoordinated metal atoms where water molecules adsorb in order to re-establish the natural coordination of the solid bulk phase. 2,3 The adsorbed water molecules then affect the structure and dynamics of the overlying water, resulting in complex and densely packed layered structures. 4 In this way, an adsorbed water film regulates, for example, the adsorption of biological adsorbants [5][6][7][8] with application to nanotoxicology 9 and nanomedicine. 10 Recent simulations of water films on titania 11,12 showed that the water self-diffusion gradually decreased toward the surface, while it was found to reach the bulk water value at about 1 nm-2 nm from the surface (depending on the titania phase and facet). This is consistent with results presented in a review on water self-diffusion in confined water by Tsimpanogiannis et al., 4 where examples of constrained interfacial water with reduced mobility extending up to several nanometers from the oxide surfaces were discussed.
Confined water has also been suggested as a possible model for the well-known anomalies of supercooled liquid water including the super-Arrhenius slowing down of the diffusion and relaxation dynamics upon cooling. 13,14 This was discussed for porous materials and microemulsions in Ref. 15. Enhanced slowing down of water on cooling has also been reported for metal oxide water interfaces. Thus, it was found 16 from quasi-elastic neutron scattering (QENS) and molecular dynamics (MD) simulations that the hydration water dynamics at TiO 2 nano-surfaces exhibits a super-Arrhenius behavior characteristic of supercooled water. This suggests that the dynamical slowing down of water at the TiO 2 surface might also be accompanied by other characteristic anomalies observed in the bulk supercooled liquid state. In this paper, we explore this possibility by investigating a typical dynamical anomaly observed in supercooled water: the decoupling between rotational and translational diffusion where the latter is enhanced with respect to the former.
It is particularly challenging to investigate such an analogy between surface-induced dynamical perturbations and the supercooled liquid state using experimental techniques because of their inability to probe the solid-liquid interface in terms of singlemolecule motions. Here we will use molecular dynamics (MD) simulations as they can provide unique insights into the molecular mechanisms of translational and rotational water dynamics at the interface.
One of the most puzzling anomalies of supercooled water is the decoupling between the translational and rotational diffusion defined as the deviation from the universal relation between these two kinds of diffusion. [17][18][19][20][21] This universality follows from the hydrodynamic arguments describing the diffusion of a macroscopic particle in a solvent, and it is expressed by the Stokes-Einstein and the Stokes-Einstein-Debye relations, namely, where Dtr is the translational diffusion coefficient, Drot is the rotational diffusion coefficient, η is the ambient liquid viscosity, and R is the sphere radius. These relations imply that Dtr/Drot = 3R 2 /2 is a constant independent of the temperature and viscosity. Surprisingly, this purely hydrodynamic model happens to universally hold when describing the molecular diffusion in the normal liquid state. However, this law was found to be violated in the supercooled liquids approaching liquid-glass transition where the rotational diffusion slows down more drastically than the translational diffusion. 18,22,23 The nature and the origin of this anomaly still elude comprehensive understanding despite the intensive research efforts including both real experiments and computer simulations.
In this article, we report an ab initio molecular dynamics simulation study investigating the relationship between rotational and translation diffusion in water at a fully hydrated surface of TiO 2 .
It is found that the universal relation between D rot and D tr that is predicted by the Stokes-Einstein and Stokes-Einstein-Debye relations, and is the characteristics of the normal bulk water diffusion, is broken in the water layers close to the TiO 2 surface. In particular, we find that the translational diffusion demonstrates progressive enhancement relative to the rotational diffusion when approaching the surface. The water molecules in the layers close to the surface translate further for every rotation than in the bulk water. This effect is confined to the first three water layers from the surface, while the dynamics in the fourth layer is indistinguishable from that in the bulk water. The analysis of the diffusion using space-time correlation functions demonstrates that the decoupling is a result of jumps induced in the lateral translational diffusion by the proximity of the TiO 2 surface.
The conceptual significance of this finding is that it is the first demonstration that the enhancement of the translational diffusion relative to the rotational diffusion characteristic of supercooled water can be observed at normal liquid temperature when induced by the proximity of the fully hydrated adsorbing surface. These conditions provide new insight into the mechanism of the decoupling anomaly. We note that translation-rotation decoupling was observed in water under confinement 24,25 and at a protein surface. 26 In both cases, it was characterized by the enhancement of the rotational diffusion relative to the translational diffusion, in contrast to the present result where we observe the opposite effect characteristic of supercooled water. We also note that this study is the first ab initio simulation of water demonstrating supercooled water anomalies.

II. SYSTEM AND AIMD METHOD
We have analyzed the dynamic behavior of water at a TiO 2 anatase (101) slab under full hydration, exploiting MD trajectories available from earlier studies. 11,27 Simulation data for liquid bulk water were analyzed for reference. We explain the significant details of the model here for completeness and convenience of the reader.
A pristine surface was built from the repetition of 4 × 3 × 3 unit cells of anatase TiO 2 . The simulated periodic box (10.37 × 11.42 × 40.0 Å 3 ) was filled with water molecules reproducing the density of liquid water under ambient conditions (cf. Fig. 1). The simulation was performed in the context of Born-Oppenheimer ab initio MD (AIMD) as implemented in CP2K software. 28 Density functional theory (DFT) with the Becke-Lee-Yang-Parr (BLYP) functional 29,30 in combination with D3 dispersion correction of Grimme 31 was used in order to reproduce the liquid properties FIG. 1. Fully hydrated TiO 2 anatase (101) surface. Blue, orange, and white colors represent titanium, oxygen, and hydrogen atoms, respectively. Water layers L 2 , L 3 , and L 4 along the perpendicular direction to the surface are highlighted in red, green, and blue. Water molecules belonging to L 1 are strongly bonded to the five coordinated Ti [5] atoms. They are essentially immobile and are thus considered as part of the surface in this study (see the text). of water. 32 Norm-conserving pseudopotentials of Goedecker, Teter, and Hutter 33,34 and a double-ζ Gaussian basis set with polarization functions (DZVP) 35 were used in order to describe the core and valence electrons, respectively. The simulation was run in an NVT ensemble for 50 ps integrating the equations of motion with a time step of 0.5 fs. The temperature was kept at 310 K by the Bussi thermostat with a time constant of 0.1 ps. 36 As referred to above, we also simulated bulk water using a system of 128 water molecules confined to a periodic cubic box with the same settings. 37 The simulated water slab exhibits a transverse density profile that reveals the formation of four domains of layered water parallel to the TiO 2 surface. 11,12 We define each layer as the space domain confined between two minima in the density profile in the dimension perpendicular to the surface. The layers are denoted as Li, where L 1 is closest to the surface ( Fig. 1) and contains the water molecules that chemically adsorb to five-coordinated Ti atoms (Ti [5] ). They are strongly bound to the surface and do not diffuse; thus, they are considered as part of the surface itself in our analyses of water diffusional dynamics. The second layer (L 2 ) consists of water molecules that adsorb on the bridging oxygen sites (O br ) and the first layer of water molecules by strong hydrogen bond, 12,38,39 and the third layer (L 3 ) is composed of water molecules H-bonded to the second layer. Finally, the fourth layer (L 4 ) shows a homogeneous density profile characteristic of bulk water, apparently losing the correlation with the TiO 2 surface.

A. Lateral translational diffusion
We measure the lateral translational diffusion within a layer Li in terms of the mean-square displacement (MSD) of a water molecule ⟨Δr 2 ⟩ i within time t in the plane parallel to the layer. 40 We identify Ni(t) trajectories of the molecules that remain in the layer Li for time t (the trajectories were allowed to leave the layer within a margin of 0.2 Å). We note that Ni(t) may exceed the total number of the molecules in the system since a molecule can repeatedly re-enter the layer, Then, the lateral diffusion coefficient within the layer can be estimated from the asymptotically linear behavior of the MSD,

B. Rotational diffusion
In a similar way, the rotational mean-square displacement (MSR) of the water molecules confined to the layer Li within the time interval ti is defined as where the rotation angle of a particle j within time t is calculated by integrating the instantaneous angular velocity vector ωj(t), The rotational diffusion constant can be estimated from the asymptotically linear time behavior of the MSR as Alternatively, the rate of the rotational diffusion can be estimated as 1/τ l where τ l is the relaxation time of the rotational correlation function, Here, P l is the Legendre polynomial of order l and uj is the normalized vector bisecting theĤOH angle of the jth water molecule, defining its orientation. The Debye approximation assumes that the reorientation of molecules proceeds in small steps. It predicts that these correlation functions decay exponentially, and the relaxation time is supposed to be related to the rotational diffusion constant as τ −1 l = l(l + 1)Drot. Deviations from this rotational diffusion mechanism were reported for water. 21

C. Description of diffusion in terms of Van Hove correlation functions
The Van Hove correlation function is defined as the probability of finding a particle at position r at time t, given that there was a particle at the origin at time t = 0. The diffusion dynamics in liquids can be conveniently described in terms of the self-part of the time-dependent Van Hove distribution function. For translational diffusion, this function is defined as where ⟨⟩ indicates the ensemble averaging. In the case of twodimensional lateral translational diffusion within a layer, the probability of finding a molecule after time t at the distance r from its initial position at time t = 0 is 2πrGs(r, t).
Its counterpart for rotation, the Van Hove distribution function describing rotational diffusion, can be then defined as Accordingly, the probability of having a molecule that performed angular displacement ϕ after time t is 4πϕ 2 Gs(ϕ, t).
In analogy, the distribution of angular displacement also converges to a Gaussian form at long times, The deviations of these Van Hove correlation functions from the Gaussian approximations at the intermediate time-scale reveal the diffusion mechanism.

IV. RESULTS AND DISCUSSION
A. Layer-resolved translational and rotational diffusion Figure 2 shows MSD and MSR [Eqs. (2) and (4)] as a function of time for the water layers L 2 , L 3 , and L 4 above the TiO 2 surface as well as for bulk water. The diffusion in L 1 is blocked by adsorption and was not considered. The results clearly demonstrate that both the lateral translational diffusion and the rotational diffusion in L 4 behave as bulk water in the long-time limit and that both types of diffusion dynamics progressively slow down closer to the surface. Figure 3 shows C 1 (t) and C 2 (t) [Eq. (8)] for the layers and for the bulk water, respectively. These correlation functions exhibit the same tendency that was observed for the MSR results in Fig. 2: the rotational dynamics slows down when approaching the TiO 2 surface.
Both rotational correlation functions shown in Fig. 3 demonstrate an apparently mono-exponential decay consistent with Eq. (8). The equation predicts that τ 1 /τ 2 = 3, and verifying this prediction represents a stringent test for the Debye model. Smaller values for this ratio in water have been observed in earlier studies, 21,[41][42][43] and it was interpreted as evidence of the presence of large-amplitude angular jumps in the reorientational dynamics of water molecules. [44][45][46] This ratio can be estimated from the asymptotically exponential behavior of the rotational correlation functions as ln C 2 (t)/ln C 1 (t). All the curves plotted in Fig. 3(c) for the layers and bulk water converge in the long-time limit into one and the same value, namely, τ 1 /τ 2 ≈ 2 (see also Table I). This numerical result is in a good agreement with the earlier simulations of normal liquid water using classical 42,46 or ab initio 43 models.
In summary, we have established so far that (i) the water dynamics demonstrates slowing down as induced by the proximity of the TiO 2 surface and that this is true both for the lateral translation and the rotational motion; (ii) in the fourth layer, which corresponds to the distance of about 8 Å from the surface, both lateral translational and rotational dynamics become indistinguishable from those in bulk water in the long-time limit; (iii) the rotational  23 The different libration amplitudes in the layers and the bulk water can be related to the difference in the periodic boundaries. 37 diffusion proceeds by jumps, but the dynamics of the reorientational jumps does not depend on the layer. Next we investigate the layer-dependence of another dynamical feature observed in supercooled water, namely, rotation-translation decoupling. This phenomenon and its conceptual implications will be discussed in the next sub-section.

B. Rotation-translation decoupling
Rotation-translation decoupling is one of the established anomaly characteristics of supercooled bulk water, as mentioned in the Introduction. This phenomenon is commonly discussed in terms of the ratio Drot/Dtr [the diffusion constants were defined in Eqs. (6) and (3)]. However, it has been concluded in the literature 46  has been found to be inconsistent with (τ l Dtr) −1 in reported studies of rotation-translation decoupling in supercooled bulk water simulated using classical models. 17 A qualitatively similar result was observed in the simulation of ortho-terphenyl molecular liquid. 22 Therefore, we avoid here using Drot/Dtr as a measure of rotationtranslation decoupling. Instead, we directly compare rotational diffusion with translational diffusion by plotting MSR and C l for each layer as well as for bulk water as a function of MSD. 47,48 Inspection of Fig. 4 shows that the universal relation between the translational and rotational diffusion characteristics of normal bulk water is broken by the proximity of the TiO 2 surface. The translational diffusion is seen to become progressively enhanced relative to the rotational diffusion for layers closer to the surface. This effect is demonstrated by both MSR and C l for the layers L 2 and L 3 , whereas the long-time behavior of layer L 4 is indistinguishable from that of bulk water. Thus, the molecules within L 2 translate further for each rotation than the molecules in the bulk water (and in L 4 ). This is equivalent to demonstrating that the dynamical slowing down in the layers close to the TiO 2 surface is accompanied by rotation-translation decoupling. L 3 exhibits less pronounced decoupling.
It is also significant that the same decoupling effect is observed using different probes of the rotational diffusion (Drot/Dtr and τ l ). This is in contrast to earlier (classical) simulations of supercooled liquids where some probe-dependence was observed. 17,22 Finally, we remark that the opposite effect, the enhancement of rotational diffusion with respect to translational diffusion, was observed in water under hard confinement. 24,25 C. Analysis of the mechanism of the diffusion anomalies It is well established that the rotational dynamics of bulk water is governed by large jumps, 46 which is also qualitatively confirmed in our system for all the layers by Fig. 3(c). At the same time, the contribution from the long-distance jumps to the translational diffusion in water has been found to increase under supercooling. 49 In the same paper, it was demonstrated that it is exactly the long-jump component of the translational diffusion that is responsible for the Stokes-Einstein breaking in supercooled water.
We will show that the origin of the observed rotationtranslation decoupling of the water diffusion can conveniently be investigated by analyzing the diffusion mechanism of water molecules in each layer. The mechanism of the diffusion process in liquids is commonly analyzed in the literature in terms of the deviations of the actual Van Hove correlation functions of the liquid from Gaussian approximations at the intermediate time-scale. Here, for our water/titania system, we investigate whether, and how, such deviations are affected by the proximity of the water molecules to the TiO 2 surface by analyzing the non-Gaussianity of of Chemical Physics the diffusion dynamics layer-by-layer. Referring to Eqs. (9)-(12), the deviation of Gs(ϕ, t) from GG(ϕ, t) and that of Gs(r, t) from GG(r, t) can be quantified in terms of non-Gaussian parameters. These are α 2 (t) = ⟨r 4 (t)⟩/2⟨r 2 (t)⟩ − 1 for the 2D translational diffusion 50 and α 2 (t) = ⟨ϕ 4 (t)⟩/2⟨ϕ 2 (t)⟩ + ⟨ϕ 2 (t)⟩/6 − 1 for the rotational diffusion. 51 Let us start by examining the mechanism for the rotational diffusion in different layers. Figure 5 shows Gs(ϕ, t) for L 2 , L 3 , and L 4 and for bulk water as functions of ϕ for the times corresponding to the MSR value of 0.75 rad 2 and the corresponding GG(ϕ, t). All Gs(ϕ, t) curves agree reasonably with each other while significantly deviating from the Gaussian. The respective non-Gaussian parameters shown in Fig. 5(b) also display no significant layer dependence. The observation that the non-Gaussianity in the rotational diffusion is not measurably affected by the proximity to the surface indicates that it cannot be regarded as a plausible reason for the rotation-translation decoupling.
The systematic deviation of Gs(ϕ, t) from the Gaussian form can be described as follows. The excess contribution (actual-Gaussian) is mainly observed in two regions: in the main peak, which is slightly shifted to smaller angles, and as the large tail featured beyond ϕ ≈ 1 rad, which demonstrates that the Gaussian approximation underestimates the actual extent of molecular rotation. A larger share of molecules rotate beyond the indicated angle value than what can be expected from the Fickian small-step diffusion model represented by GG(ϕ, t). The described deviations of Gs(ϕ, t) from the Gaussian approximation can be accounted for by the large angular jumps, which were earlier reported in bulk water. 41,44,45 It was also found that the molecules spend some time performing smallamplitude librations within time-limited cages of neighbors before proceeding with angular jumps of about 1 rad. The librations are observed as the excess contribution to the main peak in Gs(ϕ, t) in Fig. 5(a).
Further evidence supporting the angular jump model of rotational diffusion is the existence of a sharp peak at about t ≈ 10 −1 ps in the non-Gaussian parameters, which can be seen in Fig. 5(b). This feature has been interpreted 17 as an indication that the molecules perform short-time small-amplitude librations 23 while being temporarily confined within cages formed by their neighbors. The peak is less pronounced for the bulk water in consistence with the smaller deviation of the first peak of its Gs(ϕ, t) from the Gaussian in Fig. 5(a). This corresponds to the smaller libration amplitude in bulk water as compared with the layers, as discussed above in Sec. IV A [see the caption of Fig. 2(b)].
We conclude that the modest variance of the deviation of the rotational diffusion from Gaussian behavior for different water layers infers that this deviation cannot be the cause of the layerdependent rotation-translation decoupling.
Next, we move on to discussing the translational diffusion in a similar fashion. Figure 6 shows Gs(r, t) in the layers and in the bulk water calculated for ⟨r 2 (t)⟩ = 3 Å 2 compared with the corresponding Gaussian approximation. It demonstrates that the deviation of the distribution from the Gaussian form in a water layer strongly depends on the layer's distance from the adsorbing surface. The distribution curves for layers L 3 and L 4 are close to the one for bulk water, and they all agree reasonably with the Gaussian one whereas the distribution for L 2 is significantly different. Its main peak is considerably higher and more narrow and features a significant shift to the larger r, which indicates that many molecules advanced well beyond the distance suggested by the Gaussian distribution. The narrow width of the L 2 peak indicates that a large share of the molecules in this layer diffuse by jumps to the distance of the peak maximum. This can be regarded as evidence that the proximity to the TiO 2 surface qualitatively changes the mechanism of the translational diffusion in the layer, inducing a considerable degree of large-amplitude jumps. The connection between the onset of jumping diffusion in L 2 and rotation-translation breaking is illustrated in Fig. 7, which compares the displacements of individual molecules in L 2 and L 4 plotted as a function of the MSR for these layers. It is clear that the preferential jumping distance in layer L 2 is consistent with the peak position in Fig. 6(a). The jump event in L 2 translates the molecules well beyond the translation distance in the layer L 4 for the same value of rotation. We can rationalize this effect using the following arguments. The entropy reduction in water induced either by supercooling or by proximity to an adsorbing surface reduces the number of available degrees of freedom per molecule. Therefore, the local structural relaxations that are coupled with the diffusion become more collective. The number of molecules participating in an elementary local structural rearrangement event driving the translational diffusion increases. 52 This impedes the ability of an individual molecule to independently change its orientation while performing a diffusive step as a part of the rearrangement.
These results compel us to conclude that the decoupling between lateral translational diffusion and rotational diffusion that we observed in the water layers close to the TiO 2 surface is a result of the change in the translational diffusion mechanism. The change is characterized by large jumps that are induced by the interaction of the water molecules with the surface. This is consistent with the conclusion by Dueby et al. 49 that the translational jump mechanism is responsible for the Stokes-Einstein breaking in supercooled bulk water. The existence of a translational diffusion with different characteristic length-scales caused by the jumps 23,47 was suggested as a reason of the translation-rotation decoupling in supercooled liquids. 23 Thus, it can be concluded that the proximity of the TiO 2 surface has the same impact on the diffusion dynamics in water as supercooling.

V. CONCLUDING REMARKS
In summary, we presented an ab initio MD simulation investigating how water dynamics is affected at the interface with a TiO 2 surface. We observed a strong enhancement of the translational diffusion relative to rotational diffusion in water layers close to the surface. This decoupling effect has so far only been observed in supercooled water and its observation at normal liquid water temperature, as presented here, is conceptually significant and provides new insight into the molecular mechanism of this phenomenon. The decoupling of rotational and translational diffusion that we discovered is in contrast to the opposite decoupling effect observed in water dynamics under confinement where the rotational diffusion was found to be enhanced relative to the translational diffusion. [24][25][26]53 The analysis we presented makes it possible to conclude that the decoupling can be accounted for by the presence of jumps in the translational diffusion induced by the molecular interaction with the surface. The translational and rotational bulk properties are completely restored in the fourth water layer at the distance of ≈8 Å from the surface.

SUPPLEMENTARY MATERIAL
See supplementary material plot of C 1 and C 2 as a function of time highlighting the short time behavior of these correlation functions.