1. Introduction

Among a lot of important materials, titanium dioxide (TiO₂) takes a particular attention in the recent years in several industrial applications especially in the elimination of organic contamination, sensor, splitting of water, self-cleaning and photocatalytic[¹-⁵]. Furthermore, TiO₂ is abundance [³], inexpensive, nontoxic [⁴], inert and chemical stability [⁵, ⁶]. Rutile TiO₂ (110) surface is the lowest surface energy, thus is very important in studying, especially in catalysis[⁷]. Furthermore, TiO₂ has a high refractive indexranges between 2.4-2.6. Unfortunately, one of the drawbacks of TiO₂ is the wide band gap of 3-3.2 eV, reacts only in the (UV) light under (380 nm), that is available only around 5% in solar energy [⁸-¹¹]. Many researchers treated this deficiency by generating electronic states inside the forbidden gap via inducing defects (oxygen vacancy, Ti interstitial, dislocation,...) and doping with impurities (metal transition, noble metal,..) in the structures of TiO₂. This leads to extend its response in the visible light and near infrared (NIR) region [¹²-¹⁷]. One of the most important applications of TiO₂ surface is water splitting. The H₂O can be adsorbed in two different mechanisms either molecular or dissociative onto rutile TiO₂ (110) surface [¹⁸, ¹⁹]. But this is an intriguing phenomenon, because the adsorption process is affected by many factors (phase components, temperature, vacancies, water coverage...) [²⁰]. Several theoretical methods (Langmuir, Freundlich, Langmuir-Hinshelwood...) are widely used to study the kinetics of adsorptions [²¹-²⁵]. These phenomena have been studied by using various ab inition quantum mechanical approximations highlighting on the important factors that affect on the adsorption and the dissociation of H₂O such as (DFT and DFT-PW and HF...) calculations and molecular dynamics([²¹] and references therein) which have provide numerous groundbreaking insight into the water -TiO₂ interface. However, it is not possible to get adsorption information taking in consideration all the surface sites at the same time. Experimentally, the infrared spectroscopy, especially in Situ ATR-FTIR, is one of the most important and powerful technique used to investigate the water molecules adsorption at solid surfaces [²⁶]. But in experimental, it is not clear and easy to get details at the atomic scale level of adsorption therefore; the theory becomes a useful tool. In light of the above information, we are interested in this context, to study the effect of oxygen vacancies concentration at rutile TiO₂ (110) surface and water flux on the behavior of H₂O molecules adsorption using a system of nonlinear differential equations, carried out taking into account the different cases of adsorption and all surface available sites. This indicates a better understanding of structural and dynamical properties changes of the behavior of H₂O on rutile TiO₂ (110) surface. Thereby, these factors greatly enhancing chemical/catalytic reactivity (among other properties) as well as the splitting of water and self-cleaning. At the end, a summary will be given together with a brief outlook towards challenges and prospects for future theoretical studies [²⁷, ²⁸].

2. The surface structure of rutile TiO₂ (110)

Naturally, TiO₂ exists in three different crystallography forms rutile, anatase and brookite. The structure of each form is described as a chain of octahedron (one atom of Ti surrounded by six oxygen atoms) as shown in Fig. 1. Among the three structures anatase (101) and rutile (110) are the most stable faces and more reactive especially in photocatalytic applications[⁷].

Figure 1 Structure of rutile TiO₂ (110) surface in the z-direction, black balls represent O atoms of the bridge-bonded O species (O _2c ) and three coordinated (O _3c ), red balls are fivefold coordinated surface Ti (Ti _5c ) and six fold coordinated Ti atoms (Ti _6c ), white ball is single oxygen vacancies (O _V ). 

The different surface sites illustrated in this figure play a crucial role in the adsorption and the behavior of water molecules. It is still not clear to distinguish which kind of site is primordial in controlling the adsorption phenomenon. In the following part, we give the possible mechanisms of adsorption used to formulate the proposed theoretical model.

3. Kinetics model description of H₂O - rutile TiO₂ (110) surface interactions

A kinetics model of H₂O on rutile TiO₂ (110) surface, in bellow figures, was developed to describe the behavior of H₂O adsorption, assumed to adsorb in reversible states; different forms model, two types of features found on rutile TiO₂ (1x1) (110) surface as indicated in Fig. 2 and reaction (1). H₂O molecules adsorb in molecular form on defect-free sites (Ti _5c ) at low temperature around 160 K [²⁹] is due a Ti -H₂O bond length (2.16 - 2.29) Å [³⁰], with adsorption energy around (0.5 - 0.7 eV) [¹, ²⁸, ³³-³⁵]. When annealing the surface below room temperature, only the H₂O molecules are desorbed from surface with energy ranges between 0.7 eV to 0.8 eV depending on the water rate coverage [³⁶]. In the other hand, The H₂O has the possibility to dissociate on (Ti _5c ) sites, however, the second proton H⁺ transfer tothe neighbouring bridging oxygen (O _2c ) and forms two OH hydroxyl groups OH _t and OH _b respectively, which was discussed in details in Ref. [³] with dissociation energy toward 0.36 eV and 0.41 eV, when annealing the surface to room temperature under reaction (2) [³¹-³³] as shown in Fig. 3. This competition between molecular and dissociated adsorption is mostly governed by the basic strength of the bridging O atom receiving the H.

Figure 2 The adsorption of molecular water on free defect surface, dark (light) blue balls hydrogen (oxygen of water) atoms, K ₁, K ₂ are the rates constants of adsorption and desorption respectively. 

Figure 3 The dissociative adsorption of water on Ti _5c sites, where K ₃ and K ₄ are the rates constants of dissociation and recombination reaction, respectively.  

A missing of an oxygen atom, by removing the oxygen bridging (O _2c ) from the surface using electron bombardment or other methods, leads to the creation of a surface oxygen vacancy (O _V ) (none thermally). This was discussed in more details in Ref. [²].

With its concentration around 15% ML (Mono Layer) (1 ML is defined with respect to the coverage of Ti _5c sites of 5.2. 10¹⁴ cm^-2) from the surface of TiO₂ (110) [³⁴, ³⁵]. The water favoured dissociate in the oxygen defect at below temperature (130-200 K) and forms two OH hydroxyl groups in the O _V and the H⁺ proton of water transfer nearby bridging oxygen and creates the other hydroxyl for every vacancy as given in Fig. 4 [³⁶-³⁹], the recombination of the two OH occurs at temperature above 490 K [³, ¹⁸]. This reaction can be written as Eq. (3) indicates.

Figure 4 The dissociative adsorption of water on the surface oxygen vacancy, where the K ₅ and K ₆ are the dissociation and recombination rates constants, respectively 

4. Model and method

A mathematical description of these Equations (1)-(3) for the adsorption of water in different cases onto rutile TiO₂ (110) surface can be derived using an adaptation of the Langmuir adsorption model. This model is the most commonly applied of single component liquid-solid adsorption. Based on the kinetics principle, Langmuir isotherm model assuming that only monolayer adsorption exists; no impurities (CO₂, O₂...) are considered at the surface and to avoid interactions between water molecules the incident H₂O flux density can be supposed very weak and more addition the adsorbent surface is uniform with the same adsorption probability [²², ²⁴]. Therefore, the adsorption rate of each chemical reaction can be presented as following Eq. (4):

where Φ,θ,Pads,K are the flux, the surface sites coverage, the probability of the molecule will find adsorption sites and the desorption rate constant, respectively. The probability of adsorption is given by the following Eq. (5).

where K,na,ns,θa the adsorption rate constant (s^-1), the concentration of adsorption sites, the concentration of all atoms in the surface and the coverage of the adsorption sites respectively. The rate constant K _i can be expressed with an Arrhenius Eq. (6),

where K0,Ea,R and T are the attempt frequency (s^-1), the activation energy (Kj/mole, 1eV = 96.482 Kj), universal gas constant (8.314 J/mole.K) and the temperature K.

The different reactions (1) to (3) using Eq. (4) are organized below,

where θTi, θOH, θH2O,  θb and θv are the coverage of titanium, hydroxyl groups, water, bridging oxygen and oxygen vacancies on the surface respectively. The adsorption rate is proportional to the H₂O flux arrived on surface and the coverage of active sites (Ti _5c , O _V and O _2c ), desorption rate is proportional to the number of association and desorption adsorbed molecules, under such assumptions; in the adsorption process we have [³⁹].

The sum of all the different coverage at the rutile (110) surface are ∑‍θi=1 and the variation equal

We suppose the variation of the coverage’s remains the same in duration of the reaction, setting these rates to zero in the steady state gives the system (12) as indicated bellow. The stability of the set of equations is very important because any bifurcation parameter can dramatically change it and to avoid such behavior we took the rates constants Ki have similar thermal stability.

5. Results and discussion

We use a numerical method to solve this system Eq. (12) and the corresponding solution of coverages is given by the following equations:

It is clear that the solutions of each surface site strongly depends on the flux of water and the surface oxygen vacancies (O _V ). As indicated in Fig. 5 (a), OH hydroxyls increases and reaches a maximum for O _V = 0.04 then it decreases to the minimum for a higher value of O _V The same behavior for OH hydroxyls is illustrated in Fig. 5 (b) when this flux changes from 0 to 1. This graph indicates that the coverage of OH hydroxyls increases at low flux values and reaches a maximum around 0.7 (for a flux value around 0.4) afterward it decreases which means that OH hydroxyls production is slightly reduced.

Figure 5 a) The effect of oxygen vacancies on the production OH hydroxyls at H₂O flux equal 0.4. b) The effect of H₂O flux on the production OH hydroxyls at O _v around 0.05 in the steady state solution. 

To illustrate theses variations of OH hydroxyls on the surface versus H₂O flux and O _V , a three dimension (3D) is shown in Fig. 6. The coverage of OH hydroxyls grows with increasing the O _V and reaches to a maximum around 0.7 ML at O _V equal 0.05 ML. At high coverage of O _V up to 10 % the production of OH hydroxyls decays and gets a small value almost zero. In other hand, also the coverage of OH hydroxyls depends on the arriving H₂O molecules; when the H₂O flux arrived at surface increases, the production of OH hydroxyls increases and reaches a maximum at 0.7 ML for H₂O flux around 0.4 ML, despite, for H₂O flux takes value more than 0.4 ML and filling all the surface leading to no production of OH hydroxyls on the surface which is occupied only with H₂O molecules.

Figure 6 Kinetic curves for the OH hydroxyls at rutile TiO₂ (110) surface for various H₂O flux and concentration of O _V on the surface. 

From Eq. (16), the curve of the coverage of Ob for deferent concentrations of O _V and H₂O flux arrived on surface is shown in Fig. 7.

Figure 7 The coverage of bridging oxygen change versus: a) the coverage of O _v with the H₂O flux remains 0.4 ML, b) H₂O flux at the coverage of O _v around 0.05 ML on rutile TiO₂ (110) surface. 

Figure 7 (a) illustrates the variation of θOb as a function of θOv for the H₂O flux taken 0.4 ML. The graph clearly shows that θOb decreases when the concentration of O _V grows. Besides, in Fig. 7 (b) the coverage of Ob decays exponential-like with increasing the H₂O flux for the concentration of θOv equals 0.05 ML.

Figure 8 is plotted using Eq. (16); it is interesting to note some common relations; the coverage of O _b decreases exponential-like with increasing the coverage of O _V and tends to a small value for large value of O _V , also at high value for the H₂O flux, the coverage of O _b decreases.

Figure 8 The coverage of O _b as a function of H₂O flux and O _v defects on rutile TiO₂ (110) surface. 

This figure demonstrates that the O _b is a key factor in the dissociation and the adsorption of H₂O molecules.

The Eq. (15), variation of water coverage versus the rate of oxygen vacancies and the H₂O flux, indicating, the coverage of H₂O takes small values almost plateau when the H₂O flux and the concentration of O _V on surface is very small, afterward, the coverage of H₂O increases steadily with H₂O flux and the concentration of O _V , as illustrated in Fig. 9.

Figure 9 The coverage of H₂O on surface versus H₂O flux and the coverage of oxygen vacancies O _v . 

Figure 10 shows the ratio of (θOv/θH2O), from Eq. (15), versus the H₂O flux; indicating that the fraction of (θOv/θH2O) decreases with increasing the H₂O flux. This can be explained that increasing the coverage of H₂O at the surface is caused by the filling of oxygen vacancies sites.

Figure 10 The ratio of (θOv/θH2O) on rutile TiO₂ (110) surface versus H₂O flux. 

From these results, the interactions of H₂O with rutile TiO₂ (110) surface were heavily affected by the O _V and H₂O flux. This was expected, since the H₂O flux arrived on rutile TiO₂ (110) surface, H₂O molecules take up to Ti _5c sites (which is positively charged) rendering its able to bind H₂O molecules via electrostatic interactions. Many authors [⁵², ⁵⁴] have studied the effect of H₂O flux and the coverage of O _V on the behavior of H₂O molecules on rutile TiO₂ (110) surface. First, at low values of H₂O flux, is expected the mobility of H₂O molecules to be high which is helpful for the H₂O molecules to diffuse on the surface on direction [001] which dissociate at oxygen vacancies firstly to heal the vacancies and defect-free sites as well as a strong attractive of H proton to near bridging oxygen due to transfer the H proton and formed another OH hydroxyls, because dissociation at bridging oxygen vacancies is more favourable than H₂O adsorption on the Ti _5c sites [⁵⁵], as the H₂O flux increases the OH hydroxyls continue to increase till reach the maximum at 0.7ML for 0.4 of H₂O flux, this is in good accordance with the prediction by Lindan and Zhang [⁵⁶]. At a high H₂O flux, water may be adsorbed molecularly with a small fraction adsorbed dissociative on O _V because (i) at least one or two H₂O molecules uptake on Ti 5c sites and form a chains of H₂O molecules (ii) when the amount of H₂O molecules are large, hinders the diffusion of H₂O molecules and remains in molecular forms, therefore, H₂O molecular adsorption is more favourable at high coverage[³⁰] . As it is seen, the coverage of O V affects on the adsorption of water, however, the amount of OH hydroxyls formed on rutile TiO₂ (110) surface increases at low concentration of vacancies because the H₂O favourites the dissociation at oxygen vacancies [⁵⁴] reach to the maximum around 0.7 ML because the repulsive interaction between the functional groups, afterward, when the coverage of O _V increases beyond to 0.05 ML, the OH groups coverage decreases, because at high concentration of O _V affects on the structure properties, which lead to (2x1) reconstruction surface[⁴⁰], additionally, and we know the O _V ’s created from the bridging oxygen, this implies that the coverage of O _V increases due to decrease the bridging oxygen as illustrated in Fig. 8. In noting, when (NOb<NOv) deficient the dissociation in O _V and the defect-free sites, this is in a good agreement with previous works reported in [⁵⁷]. Exists due to the fact, the dissociation of H₂O molecules in either terminal hydroxyl or OH in vacancy need a neighbouring oxygen atoms for the H proton creates the second OH hydroxyls on O _b , besides, the coverage of bridging oxygen decreases with H₂O increases as Fig. 7 (a) improved that clearly. The formation of these radical groups is active to promote the charge separation process as well as the oxidation of organic substances and several applications (self-cleaning, pharmaceutical...) [⁴⁹] and references therein]. This inhibit the production of OH hydroxyls, meanwhile the molecular adsorption will be the predominant as Fig. 9 illustrates that; where the vacancies increasing the asymmetry for titanium atoms near vacancies that leads to enhance the water adsorption further the H₂O adsorbed molecular at high and low coverage [³⁸, ⁵³]. From the Eq. (14) it is clearly shown that the coverage of Ti _5c is independent to the flux, it depends only on the concentration of the surface O _V . Many authors overbalanced the increase to transfer the Ti _6c coordinate underneath the O _V to Ti _5c [⁵², ⁵⁸]. In other hand, H₂O adsorbed molecular on Ti _5c meanwhile diffuses toward the vacancy site where the dissociation takes place, thus, H₂O let behind a Ti _5c site, however, the coverage of Ti increases as illustrated by Fig. 11. Several literature papers conclude that mixed molecular/dissociate is the most stable configuration on rutile TiO₂ (110) surface [⁵⁹], besides, the behavior of the water (molecular or dissociative) can change dramatically even for the same materials as the structure and termination of the surface changes.

Figure 11 The coverage of Ti _5c on rutile TiO₂ (110) surface as a function of the concentration of oxygen vacancies (O _v ). 

6. Conclusion

The interactions H₂O / rutile TiO₂ (110) are intriguing subjects; therefore, several experimental and theoretical methods were used to understanding the reaction mechanisms. In terms of theoretical studies, the proposed model presented in this research, a system of coupled differential equations, described and solved in the steady state case taking into account the different reactions of H₂O / rutile TiO₂ (110). The findings strongly indicate that the interactions H₂O- rutile TiO₂ (110) have a crucial impact on the production of OH groups at the surface. As one might expect, the coverage of OH hydroxyls increases in the presence of O _V which can be applied to enhance the performance of the purification of waste water from organic and inorganic materials plus others applications, but when the concentration of O _V increases up to 0.05 leads to reduce the production of OH hydroxyls. This behavior is believed to be a good reason of a modification on surface structure, as well as the H₂O flux. By way of outlook for future challenge given a presence of chemically adsorbed water at rutile TiO₂ (110) interface and intriguing behavior, this shows the importance of understanding both the structure of rutile TiO₂ (110) surface and dynamics of H₂O molecules on them. Indeed, knowing the mechanism behavior of water molecules on metal oxides under deferent conditions is very important due to its applications in catalysis.