#
The Sticking of N_{2} on W(100) Surface: An Improvement in the Description of the Adsorption Dynamics Further Reconciling Theory and Experiment

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## Abstract

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## 1. Introduction

_{2}adsorption on a W(100) surface.

_{2}and H

_{2}, mainly with an iron surface [14].

_{2}on W(100), as obtained in experiments [9,10,11]. This improvement was obtained by adopting a PES determined by DFT calculations, including long-range interactions via the vdW-DF2 functional as implemented in the VASP code [16]. The authors of Ref. [7] ascribe the improvement in comparison with the experimental results to the fact that with the introduction of long-range interactions, the barrier in the entrance channel, revealed in the PES of Ref. [15], disappears.

_{2}-tungsten system to improve further the comparison between experiment and theory for the sticking probability and, at the same time, to provide an additional contribution to understanding reaction dynamics and the underlying energetics. To do this, we mainly focused on the evaluation of the long-range interaction strength, while also determining the ${\mathrm{C}}_{6}$ and ${\mathrm{C}}_{3}$ coefficients.

_{2}molecules impinging the W(100) surface in the low-collision-energy range. The surface temperature (T

_{S}) effect on the sticking probability, due to being related closely to the energy exchange with the phonons, explicitly considered in the adopted method, has also been investigated. The proper inclusion of surface phonons produces an agreement improved with the experimental results, whilst the correct treatment of the long-range interaction determining the precursor state suggests a new picture for the reaction dynamics.

## 2. Results and Discussion

#### 2.1. The ILJ Long-Range Potential of N and N_{2} Interacting with W(100)

_{3}coefficients, obtained in an internally consistent way from the ILJ ${\mathrm{C}}_{6}$ coefficients, with the results available in the literature. In particular, the parameters reported in Table 1 have been evaluated, according to Refs. [17,22], by exploiting the polarizability of the single, isolated N atom (1.1 Å

^{3}) and the effective polarizability value of N in N

_{2}(0.88 Å

^{3}). We also estimated the value of the effective polarizability of each W atom bound in the surface equal to 2.9 Å

^{3}and considered an atomic surface density (ρ) equal to 0.0632 atom/Å

^{3}. In Figure 1, the ${\mathrm{C}}_{3}$ coefficient for the nitrogen atom and molecule is reported as a function of their polarizability (α) in comparison with the values, determined with the same method, for rare gases interacting with W(100). In the same figure, the ${\mathrm{C}}_{3}$ coefficients for the interaction of rare gases with W(100), obtained with different methods [23], are reported for a useful comparison. Looking at the plot, we can infer both the linear correlation between the ${\mathrm{C}}_{3}$ coefficient and α, and a good comparison between the data reported in the literature and those calculated in this work for rare gases. Moreover, the right placement of points for N and N

_{2}in the plot makes us confident about the parameters used to obtain the long-range potential of interest for the present investigation.

_{2}impinging with the center of mass (CM) on the given site for two different orientations of the molecular axis with respect to the surface plane, parallel and perpendicular. In the same Figure, the ILJ potential for the N atom is also provided for the three sites.

#### 2.2. Potential Energy Surface Determination

_{N2}(R) and V

_{N}(R) represent (nitrogen molecule)– and (nitrogen atom)–surface interaction potentials, respectively. R is the distance of impinging species from the surface. The weight function fsw(r) switches the interaction potential between V

_{N2}and V

_{N}as r, the interatomic distance between the atoms in the molecule, increases and is given by

_{N2}and V

_{N}for r lower and higher, respectively, than the intramolecular distance considered critical for N

_{2}molecule dissociation (3 ÷ 3.5 Å).

_{N2}(R) has been obtained by combining the interaction potential on the three different sites (see Figure 2) on the surface through a switch function fswhbt, given in Equation (4):

_{at}in the assumed surface model lattice.

_{g}and Y

_{g}are the molecule CM coordinates on the X–Y plane of the assumed reference frame; a is the lattice constant of W(100), equal to 3.165 Å.

_{2}molecule impinging to the surface. Instead, for sites T and B, the ILJ potential was added to the fitting value of the potential determined in Refs. [6,26] for the incidence of N

_{2}in a perpendicular and parallel configuration, respectively. In fact, the authors of both papers determine the interaction potentials for distances up to 4.0 ÷ 5.0 Å, too short to control the asymptotic behavior at longer distances, responsible for the formation of the precursor state.

_{T}and V

_{B}, the expression, as a function of the CM distance along the normal to the surface of the incident molecule, is

_{2}molecule impinging with perpendicular and parallel orientation of its molecular axis and with CM on the three active sites on the W(100) surface is reported in Figure 3a,b.

_{N}is the same as Equation (3). For atomic nitrogen also, the interaction potential for the H site is obtained as the sum of short- and long-range contributions. The terms corresponding to the different sites are given similarly by the expression of Equation (6), while the corresponding parameters are reported in Table 3. The behavior of the obtained potential is reported in Figure 3c.

_{2}on W(100) results strongly corrugated. It is interesting to note that the molecular approach towards well-defined surface sites can introduce a small barrier for the transition from the physisorption well to the chemisorption well.

#### 2.3. The Sticking Probability for N_{2} Interacting on W(100)

_{2}collision energy (E

_{coll}) obtained by MD simulations are reported in Figure 4 for T

_{S}= 300 K in comparison with experimental data [10] and the results of previous simulations [7]. The values reported in the plot are obtained by including both dissociative and molecular adsorption events. Looking at the plot, it appears that the comparison with experimental results is improved significantly with respect to the comparison with the results of Ref. [7], mainly for the lower collision energies (E

_{coll}< 0.2 eV). In particular, in this energy range, the slope with which the sticking probability (P

_{Sticking}) decreases is almost the same as that of the experimental data. These findings, in comparison with those of Ref. [7], could indicate a different mechanism in the interaction dynamics due to the adoption of a dissimilar long-range interaction potential and/or that the treatment of the interaction with the phonons of the surface adopted here is able to describe better the energy exchanges between the incident species and the surface.

_{Sticking}can be explained in terms of a dynamic steering mechanism. In our picture of interaction dynamics, the rotational excitation counteracts the steering that favors the molecule path towards a direct dissociation and molecules remain trapped on the surface. In the past, the role of rotational effect in the adsorption of molecules on metals has been observed in the MB experiments [30,31] and studied in Refs. [32,33]. In addition to the dynamic steering, the results for P

_{Sticking}, for low collision energies, can also be ascribed to either the physisorption well appearing in the assumed PES and having a depth greater than or, at most, of the same order (see Section 2.2) as E

_{coll}, which prevents the molecule from being immediately scattered in gas-phase, and/or to the existence of a barrier that can be higher than the collisional energy. The dynamic mechanism just described can be observed in Figure 5, which shows one of the trajectories ending with molecular adsorption for E

_{coll}= 0.04 eV.

_{2}molecule with phonons is further extolled by considering, at a selected collision energy, the dependence of the trapping probability (P

_{trapping}) on T

_{S}. P

_{trapping}represents the probability for a molecule to become trapped on the surface or eventually, after a while spent bouncing on the latter, to be scattered in the gas-phase. We did this, and in Figure 6, we report the results of our study in comparison with those reported in Ref. [10], obtained by adopting a hard cube model with parameters chosen to provide results consistent with experimental measurements. Looking at Figure 6, we can conclude that, even in this case, there is a very good agreement, considering the error of our results and the approximations and parametrization of Ref. [10] to obtain the line in the plot based on experimental measurements.

_{trapping}, which appears to be more strongly correlated to the collision energy of the molecule and the energy exchanges occurring during the interaction. Such almost non-dependence on the surface temperature of P

_{trapping}can perhaps be attributed to the mass mismatch between the incident species and the substrate.

## 3. Methods

_{at}− 6 independent harmonic oscillators are perturbed by a linear force exerted between the species approaching the surface from the gas-phase and the solid substrate [19]. Details on the bulk potential and the density of phonon states can be found in Ref. [34].

_{3}coefficient is obtained through the well-known relationship (see Ref. [21] and references therein) that binds it to the ${\mathrm{C}}_{6}$ coefficient:

_{2}molecule interacting with W(100) surface is followed by solving self-consistently the relevant 3D Hamilton’s equations of motion with those of the lattice phonons, under given initial conditions:

_{i}and H is the Hamiltonian for a diatomic molecule impinging on a surface, given by

_{2}intramolecular interaction potential and ${\mathrm{V}}_{\mathrm{e}\mathrm{f}\mathrm{f}}(\mathrm{t},{\mathrm{T}}_{\mathrm{S}})$ the effective potential of mean field type, depending on time and surface temperature, formulated as

_{k}.$\text{}{\mathsf{\eta}}_{\mathrm{k}}$ are the “phonon excitation strengths” given in terms of the Fourier components ${\mathrm{I}}_{\mathrm{i},\mathrm{k}}$ of the external force:

_{2}molecules, assumed as a Morse oscillator [35], were analyzed in terms of the action-angle variables using the semiclassical quantization rules [36]. Therefore, the roto-vibrational states were determined as continuous variables, as we are unable to predict some features caused by quantum effects and selection rules.

_{coll}value, we propagated 30,000 trajectories, while T

_{S}was fixed to 300 K. The initial coordinates of impinging species were randomly generated at the beginning of each trajectory, in an aiming area coinciding with the unit cell. The molecule CM impinges with polar angle θ = 0°, defining the selected normal approach, and azimuthal angle (ϕ) of the molecular axis was randomly chosen at the beginning of each trajectory. We can describe and follow the different elementary surface processes occurring when the N

_{2}molecule impinges on the surface, as the assumed PES is reactive. The impact of a molecule on a surface can give rise to scattering (elastic or inelastic), adsorption of both atoms, desorption of just one atom with the other adsorbed on the surface or desorption of both atoms as a molecule or separated.

_{2}molecule and, at the same time, the distance between molecule CM and the surface is larger than 8.0 Å. On the contrary, if after the interaction with the tungsten surface the distance between N

_{2}CM and the surface is comparable to or smaller than the distance (≈5.0 Å) at which the potential approaches its asymptotic value, and r is lower than the dissociation distance, the molecule is considered trapped in the physisorption well. Further, a second “energy” criterion can be followed according to which the molecule is adsorbed when the energy available to escape from the potential well is less than the effective potential, accounting for the interaction with the surface phonons.

## 4. Conclusions

_{2}molecules impinging on the W(100) surface. The long-range interaction components, defining the asymptotic behavior of the PES, have been properly characterized and represented by an ILJ function. MD simulations, performed with a semiclassical collisional model, have been exploited to characterize basic details of the collision dynamics, including its dependence on collision energy, ranging from sub-thermal up to hyper-thermal conditions. In particular, we focused on the sticking probability, with its dependence on the collision energy; on the trapping probability, with its dependence on the surface temperature for a selected E

_{coll}; and on their comparison with the corresponding experimental determinations. This study proves that the asymptotic part of the interaction plays a crucial role in the molecular interaction dynamics since defining all relevant features of the precursor state controls the dynamics of basic phenomena occurring at the gas–surface interphase. Furthermore, the description adopted here for long-range forces is found to be more suitable than that obtained by using DFT methods with appropriate corrections.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**${\mathrm{C}}_{3}$ coefficients calculated for rare gases and nitrogen (atomic and molecular) as a function of polarizability (α) of the gaseous species interacting with W(100) in comparison with the data of Ref. [23]. The polarizability values of rare gas atoms and N

_{2}are taken from Ref. [24] and that of N from Ref. [25]. Note that while the relative ${\mathrm{C}}_{3}$ values vary almost linearly with the rare gas atoms’ polarizability, their absolute value also depends on the effective polarizability of W atoms bounded in the surface.

**Figure 2.**ILJ potential for N

_{2}impinging with its CM on T, B and H sites and with molecular axis being (

**a**) parallel and (

**b**) perpendicular to the surface plane. (

**c**) ILJ potential for nitrogen atom impinging on the three sites. On the right side of the figure is the top view of the unit cell of W(100) on which the three sites on the surface are located. The color of the curves is associated with that of the site label. In the plot, Z defines the component of R along the normal direction and Z = 0 corresponds to the surface first layer.

**Figure 3.**Interaction potentials for N

_{2}impinging with CM on T, B and H sites and with molecular axis (

**a**) parallel and (

**b**) perpendicular to the surface plane. (

**c**) Interaction potential for nitrogen atom impinging on the three sites. The correspondence between the colors of the curves and the sites is the same as in Figure 2.

**Figure 5.**A typical sticking trajectory for E

_{coll}= 0.04 eV. Time evolution of (

**a**) the coordinate of the two atoms (lines blue and red) in the N

_{2}molecule, along the Z normal direction, which ends with the adsorption; (

**b**) the rotational state j; (

**c**) the center of mass translational energy (E

_{CM}); (

**d**) the energy exchanged with the surface phonons; and (

**e**) the diffusion motion in the X–Y plane, of assumed reference frame, of the trajectory of two atoms (blue and red lines) in the N

_{2}molecule (orange squares indicate the starting point of each trajectory). The W atoms on the first (big black spheres) and second (small grey spheres) layers are also reported.

**Figure 6.**Trapping probability (P

_{trapping}) for N

_{2}interacting on W(100) as a function of surface temperature (T

_{S}) at E

_{coll}= 0.088 eV in comparison with results obtained in Ref. [10].

**Table 1.**ILJ adopted parameters and related dispersion coefficients. Note that ε, R

_{m}and ${\mathrm{C}}_{6}$ represent, respectively, the potential well, the minimum location and the dispersion attraction coefficient of each weak interacting effective atom-effective atom pair. β, related to the hardness of the partners, is a parameter defining the shape of the potential well. The ${\mathrm{C}}_{3}$ coefficient, defining the long-range atom/molecule–surface dispersion attraction, is proportional to ${\mathrm{C}}_{6}$ multiplied by the atomic surface density (for further details, see the Section 3).

Interaction | ε(meV) | ${\mathbf{R}}_{\mathbf{m}}$(Å) | β | ${\mathbf{C}}_{6}$(meV Å^{6}) | ${\mathbf{C}}_{3}$(meV Å^{3}) |
---|---|---|---|---|---|

N(N_{2})–W | 6.23 | 4.07 | 6.3 | 28,452 | 941.5 |

N–W | 7.40 | 4.07 | 7.0 | 33,635 | 1119 |

Site | ${\mathbf{D}}_{\mathbf{S}}$(eV) | ${\mathbf{b}}_{\mathbf{S}}$(Å^{−1}) | ${\mathbf{R}}_{\mathbf{S}}$ (Å) | ${\mathbf{x}}_{\mathbf{S}}$ (eV) | ${\mathbf{Z}}_{\mathbf{S}}$ (Å) | |||||
---|---|---|---|---|---|---|---|---|---|---|

A | B | A | B | A | B | A | B | A | B | |

T | 0.693 | −0.675 | 1.16 | 3.59 | 2.54 | −0.01 | 0.0079 | −0.0079 | 2.95 | 0.07 |

B | 0.390 | −0.155 | 1.25 | 1.1 | 2.89 | −0.63 | 0.0065 | −0.0057 | 2.72 | −0.03 |

Site | ${\mathbf{D}}_{\mathbf{S}}$ (eV) | ${\mathbf{b}}_{\mathbf{S}}$ (Å^{−1}) | ${\mathbf{R}}_{\mathbf{S}}$ (Å) | ${\mathbf{x}}_{\mathbf{S}}$ (eV) | ${\mathbf{Z}}_{\mathbf{S}}$ (Å) |
---|---|---|---|---|---|

T | 3.845 | 1.45 | 1.97 | 0.014 | 2.88 |

B | 2.35 | 1.75 | 2.08 | 0.0051 | 2.95 |

H | 1.69 | 1.50 | 2.42 | 0.018 | 2.30 |

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**MDPI and ACS Style**

Rutigliano, M.; Pirani, F.
The Sticking of N_{2} on W(100) Surface: An Improvement in the Description of the Adsorption Dynamics Further Reconciling Theory and Experiment. *Molecules* **2023**, *28*, 7546.
https://doi.org/10.3390/molecules28227546

**AMA Style**

Rutigliano M, Pirani F.
The Sticking of N_{2} on W(100) Surface: An Improvement in the Description of the Adsorption Dynamics Further Reconciling Theory and Experiment. *Molecules*. 2023; 28(22):7546.
https://doi.org/10.3390/molecules28227546

**Chicago/Turabian Style**

Rutigliano, Maria, and Fernando Pirani.
2023. "The Sticking of N_{2} on W(100) Surface: An Improvement in the Description of the Adsorption Dynamics Further Reconciling Theory and Experiment" *Molecules* 28, no. 22: 7546.
https://doi.org/10.3390/molecules28227546