Chemical modification is an effective method of tailoring the physical and chemical properties of graphene.^[1–5] With the decoration of foreign species of atoms or molecules, functional derivatives of graphene can be created. To date, at least three typical derivatives of graphene have been reported, namely: graphene oxide (GO), which is obtained by the decoration of hydroxyl and epoxy groups on graphene; graphane, an extended two-dimensional hydrocarbon with one-to-one molar ratio of carbon and hydrogen atoms, which was predicted first by first-principles calculations^[6] and synthesized experimentally later;^[2] and fluorographene, which was synthesized recently and is a two-dimensional counterpart of Teflon with one-to-one molar ratio of carbon and fluorine atoms.^[3] It is found experimentally that the hydrogenated graphene can rapidly lose the adsorbed H atoms at moderate temperatures,^[2] which casts doubt on the realistic application of graphane where the thermal stability is a prerequisite. On the contrary, due to the much stronger F–C bond than the H–C bond of graphane, fluorographene is observed to be inert and stable in air up to ∼400 °C.^[3]

The fluorination of graphene can open a finite gap in the energy band structure and thus tuning its electronic and optical properties from the original metallic state to semiconducting and even to insulating state.^[3,7] Furthermore, experimental^[8] and theoretical studies^[9] showed that local magnetic moment may appear in fluorinated graphene, resulting in the so-called d⁰ magnetism.^[10] It was shown recently that fluorination can tune the electronic and optical properties of the other two-dimensional (2D) systems such as bilayer graphene^[11] and 2D-SiC.^[12] On the other hand, pristine graphene is found to be stable at room temperature in the presence of the F₂ molecules.^[13] Such an inertness is surprising when considering the extremely strong oxidizing characteristics of F₂. However, for decades, the underlying mechanism has been unclear. In the present work, we attempt to resolve this puzzle at an atomic level by using first-principles calculations. We identify a kinetic barrier for the diffusion and adsorption of an F₂ molecule from the molecular state in gas phase to the atomic adsorption state on graphene surface. We further study the energy pathway for the diffusion of the adsorbed F₂ molecule on graphene, along which the key energy barriers are determined. The existence of such energy barriers of diffusion conduces to the understanding of the following experimental observations: 1) in the presence of molecular F₂, the inertness of graphene at room temperature and its reactivity at moderately higher temperatures; and 2) the high thermal stability of fluorographene.

All the calculations were carried out by using the Vienna ab initio simulation package (VASP),^[14,15] which is based on density functional theory (DFT). The electron wave function and the electron-ion interactions were respectively described by a plane wave basis set and the projector-augmented-wave (PAW) potentials.^[16,17] The exchange–correlation interactions of electrons were described by the PBE-type functional.^[18] The energy cutoff for plane waves is 600 eV. For the structural relaxation and total energy calculation of the F₂/graphene system, an 8 × 8 × 1 Monkhorst–Pack k-mesh^[19] was generated for sampling the Brillouin zone (BZ). The graphene sheet on which an F₂ molecule is adsorbed and diffuses is modeled by a (5 × 5) supercell of graphene which extends periodically in the x- and y-direction, separated by a vacuum layer of ∼15 Å in the z direction to minimize the artificial interactions due to periodic boundary condition employed in the simulations. The adsorption energy of an F₂ molecule is calculated as follows:

Shown in Fig. 1 are a number of typical adsorption configurations of one F₂ molecule (molecular and dissociative states) on graphene. The calculated adsorption energies are ∼1.81 eV, 1.65 eV, 1.03 eV, and 1.16 eV for configurations I, II, III, and IV, respectively. In contrast to the chemisorption of configurations I–IV, the F₂ molecule in configuration V is physically absorbed on the graphene sheet, with an adsorption energy of ∼0.40 eV. The adsorption energies and the related parameters describing the adsorption geometries are listed in Table 1. It is clear that configuration I, where the two F atoms of the F₂ molecule are separately adsorbed on the top sites of two C atoms which are the 3rd nearest neighbors of each other (Figs. 1(a) and 1(c)), is the most stable energetically. The second stable is configuration II, in which the two F atoms stay atop two C atoms which are the first nearest neighbors of each other. In gas phase, our calculations predict an F–F bond length of ∼1.42 Å and a bond energy of ∼2.71 eV, which are comparable to previous DFT-PBE calculations (1.41 Å, 2.35 eV).^[22] The difference of ∼0.36 eV in bond energy may arise from the different basis sets employed in DFT calculations: a plane wave basis set in this work and a localized basis set in Ref. [22].

Fig. 1.

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	Fig. 1. (color online) (a) Schematic diagram of some top-site adsorption configurations (I to IV) of an F2 molecule on graphene. Top and side view of (b) adsorption configuration V and (c) adsorption configuration I, where the transition is indicated by an arrow. For the clarity of illustration, only part of the (5 × 5) supercell is shown in Figs. 1(b) and 1(c).

Table 1.

Table 1.

Table 1. Adsorption energies and geometric parameters [the F–C distances: (two values), and the F–F bond lengths: ] describing the top-site adsorption configurations of an F2 molecule on graphene. . Configuration Eads/eV /Å /Å I 1.81 1.50; 1.50 3.06 II 1.65 1.46; 1.46 2.38 III 1.03 1.55; 1.58 4.45 IV 1.16 1.56; 1.56 2.94 V 0.40 3.00 1.64 F2 in gas phase — — 1.42

Table 1. Adsorption energies and geometric parameters [the F–C distances: (two values), and the F–F bond lengths: ] describing the top-site adsorption configurations of an F2 molecule on graphene. .

Compared with the molecules in gas phase, a physically adsorbed F₂ molecule stays at a height of ∼3 Å above the graphene sheet (Fig. 1(b), configuration V), with a slightly elongated bond length (∼1.64 Å, see Table 1). Considering the adsorption energy and the slight change in molecular structure, it is evident that an F₂ molecule will attach spontaneously from the gas phase to graphene. Therefore, configuration V serves as a good starting point for studying how an F₂ molecule diffuses and dissociates on the graphene surface, and finally reaches the most stable configuration I as schematically depicted in Figs. 1(b) and 1(c).

Figure 2 shows the optimal energy pathway for the diffusion and dissociation of an F₂ molecule from the initial molecular state (configuration V) to the final dissociated state on graphene (configuration I). Using the NEB method,^[18,19] we are able to locate some typical transition states along the diffusion pathway, as marked by capital letters A–D in Fig. 2. The dissociation of the F₂ molecule is evidenced by the elongation of the bond length and the adsorption is indicated by the decrease of its distance to the graphene surface, , as shown in Fig. 2.

Fig. 2.

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	Fig. 2. (color online) Calculated energy pathway (solid line) along which an F2 molecule diffuses from molecular state to dissociative state (DF2); and the distance (dashed line) from the geometric center of the F2 molecule to the graphene surface, denoted as . Configuration is abbreviated as cfg, and graphene is abbreviated as Gr.

An energy barrier on the diffusion pathway from configuration V to the transition state B, E_b ∼0.33 eV, is identified. It would be instructive to discuss the reaction rate of the transition from configuration V to configuration I. A common method of estimating the reaction rate is the Arrhenius equation: , where K is the rate constant, is the prefactor, is the activation energy, k_B is the Boltzmann constant, and T is the temperature. This empirical formula works well for the reactions in homogeneous systems such as gas phase. However, severe problems may be encountered in inhomogeneous systems such as dynamical process (including reactions) on surface. Previous researches^[23–25] have demonstrated that the experimentally fitted value of can be much smaller or larger than the normal value, which is . For instance, in the process of hydrogen desorption from amorphous hydrogenated silicon, the deduced prefactor from experimental data can differ by 14 orders of magnitude ().^[23] On the other hand, the value of is affected by many factors (available surface site, recombination, gas pressure, substrate temperature, translation and rotation motion of the adsorbates, …), which makes it difficult to theoretically evaluate with reference to experimental data. Therefore, our discussion will focus on the equilibrium probability of the transition states, , which is well established experimentally.

The diffusion from transition state B to the final state (configuration I) is expected to be spontaneous due to the downhill characteristics in the energy landscape. At room temperature (), the probability of surmounting such a kinetic barrier () is therefore estimated at . At , the probability is , increased by about one order of magnitude, at which detectable reaction between F₂ molecules and graphene comes into play, though a longtime of exposure is required to complete the reaction.^[3] When the temperature increases to ∼480 K (∼207 °C), the probability is , about two order of magnitude larger than that at room temperature. In fact, the significant increase in the reaction rate as expected by theoretical calculation has been experimentally demonstrated by the fluorination process of graphene, in which the reaction time decreases from weeks to hours when the sample was prepared at ∼200 °C.^[3] The existence of a kinetic barrier of diffusion explains why graphene is inert and stable in the atmosphere of F₂ at room temperature, and why it readily reacts with F₂ and gets fluorinated at temperatures well above 300 K.

According to the time of fluorination reaction provided by the experimental data,^[3] we can estimate the reaction rate r, rate constant K, and consequently the prefactor of the Arrhenius equation. To the first order, the reaction rate is , where θ is the probability of finding the 3rd nearest neighboring (3-NN) C sites for F₂ adsorption. Approximately, the value of θ can be calculated as the ratio of the solid angle occupied by the dissociated F₂ molecule (configuration I) to the whole space: . The factor 3 corresponds to the three equivalent 3-NN sites around one C atom. In our case, , where is the radius of F atom, which is ∼1.1 Å, and , is the distance between every two C atoms at the 3-NN site. It follows that . Assuming that , we have . At , the reaction time: . The reaction time at and 300 K are similarly estimated to be and ∼882.96 h (∼36.79 days, long enough to be considered as stable), respectively. The estimated time scales are in very good agreement with the orders of magnitude reported experimentally.^[3] Therefore, the prefactor in the fluorination reaction is estimated at . The abnormally small value of the prefactor may be explained by using the theory of large energy fluctuation of a small number of atoms in condensed phase, which was suggested to understand the abnormal kinetic process in Si-based systems.^[23] In our case, the large energy fluctuation of neighboring atoms on a transient time scale () will distort the local atomic structures and elevate the transient kinetic barrier for the dissociation of the adsorbed F₂. For instance, at T = 300 K, if the transient kinetic barrier is enlarged by ∼0.3 eV (, and ), then the corresponding will be a normal value (, i.e. atomic vibrational frequency) for the same rate constant. The estimated small value of in this work is actually the time-average of the kinetic process at equilibrium state, where the transient atomic vibration and energy fluctuation cancel each other. As a result, the transient larger values of and are averaged and thus becoming much smaller values which are estimated in an equilibrium state and on a macroscopic time scale.

As seen from Fig. 2, the bond length of the F₂ molecule gradually increases from the initial value of ∼1.64 Å to the final value of ∼3.06 Å. In the process of bond length elongation, the distance of the F₂ molecule to the graphene sheet decreases quickly and continuously, from the initial value of ∼3 Å to ∼2.2 Å at the transition state C, and decreases slightly to ∼2 Å when arriving at the final state. The atomic configurations for the typical transition states (A, B, C, and D in Fig. 2) are schematically shown in Fig. 3, where the elongation of the F₂ bond length and the decrease of F₂–graphene distance are clearly illustrated. We further study the electron transfer between the F₂ molecule and the graphene substrate, by making the Bader analysis^[26,27] of the charge density of the adsorption systems: the initial molecular state (cfg. V), the transition states (A, B, C, D), and the final dissociative state (cfg. I). Using such a method, the number of valence electrons can be intuitively assigned to each F atom. The results are listed in Table 2. For the molecular adsorption state, there is a net charge transfer of from the graphene substrate to each F atom, with reference to the number of valence electrons of free-state F being 7. From the transition states to the final dissociative state, the net charge transfer increases monotonically from to , indicating the enhanced F–graphene interactions, which lead the F–F bond to break.

Fig. 3.

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	Fig. 3. (color online) Top (upper) and side views (lower) of the transition states A–D in Fig. 2, which are one-to-one corresponding to panels (a)–(d). For the clarity only part of the (5 × 5) supercell is shown.

Table 2.

Table 2.

Table 2. Bader charges assigned to the two adsorbed F atoms (i.e., F1 and F2) on graphene, for the adsorption configurations V, A, B, C, D and I (Fig. 2). The corresponding distances () of the two F atoms to the graphene substrate are also listed. . V A B C D I F1 (e) 7.21 7.28 7.41 7.52 7.54 7.59 F2 (e) 7.21 7.29 7.41 7.54 7.58 7.58 /Å 3.00 2.84 2.71 1.66 1.54 1.49 /Å 3.18 2.86 2.23 1.48 1.49 1.49

Table 2. Bader charges assigned to the two adsorbed F atoms (i.e., F1 and F2) on graphene, for the adsorption configurations V, A, B, C, D and I (Fig. 2). The corresponding distances () of the two F atoms to the graphene substrate are also listed. .

We come to further study the diffusion of an F₂ molecule on the graphene surface, after its adsorption from gas phase. Our study will primarily focus on the diffusion from one most-stable configuration (configuration I) to another most-stable one at the nearest neighboring site, which is the most probable situation from the point of view of statistical mechanics. As schematically shown in Fig. 4(a), the diffusion of a pair of F atoms (dissociative state of F₂) from configuration I (denoted by the line segment ) to the neighboring most-stable configuration (denoted by the line segment ) will experience the intermediate metastable configuration II (denoted by the line segment ). The diffusion from the configuration to (the kinetic process is referred to as hereafter) involves surmounting an energy barrier of , and the diffusion from the configuration to (referred to as ) needs to cross an energy barrier of . Considering the geometric symmetry of the underlying lattice of C atoms, and are actually reverse processes to each other.

Fig. 4.

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	Fig. 4. (color online) Diffusion of F2 on graphene: (a) Schematics for the transition between two neighboring most-stable configurations, and (b) calculated energy pathway for the diffusion from configuration I to configuration II.

The energy landscape determined by the NEB method^[20,21] for diffusion process as a function of the bond length of F₂ molecule is shown in Fig. 4(b). The kinetic barrier associated with process is calculated to be ∼1.02 eV. The energy landscape for diffusion process is just a mirror reflection of the curve displayed in Fig. 4(b), with increasing F₂ bond lengths and a kinetic barrier of . More generally, the diffusion between any two most-stable configurations at arbitrary separation is simply a linear combination of the diffusion pathways studied here. The values of and indicate that diffusion on graphene surface is nearly prohibited and the adsorption configurations I and II are highly stable at room temperature. At higher temperatures, e.g., , the probability of surmounting is , and is for surmounting . Such a low probability conduces to understanding why fluorographene can be stable up to .^[3]

To make a comparison, we also study the diffusion of a single F atom on the graphene surface. Our calculations show that top site adsorption is the most stable configuration for an F atom: the top site adsorption energy is ∼0.25 eV larger than the bridge site adsorption energy, and is 0.35 eV larger than the hollow site adsorption energy. Therefore, the energetically favored diffusion pathway is simply from one top site to another along the direction of C–C bonds. Shown in Fig. 5 is the calculated energy pathway for the diffusion of an F atom from the top site of a C atom (configuration A) to its nearest neighboring top site (configuration E). Due to the symmetry of the graphene lattice, diffusion to the other top sites at arbitrary long distance away is simply a linear combination of the pathways studied here. The kinetic barrier for the diffusion of the single F atom is ∼0.25 eV, which is much smaller than for the diffusion of a pair of F atoms on graphene (). Such a difference implies that the interactions between the adsorbed F atoms play a key role in determining the kinetic barrier and the diffusion pathway. Therefore, except for the situation where very few F atoms are adsorbed (difficult to realize), the F–F interactions should be considered when studying their diffusion on graphene.

Fig. 5.

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	Fig. 5. (coloe online) Calculated energy pathway for the diffusion of a single F atom on graphene, from one top site (A) to another (E), via a number of transition states (B, C, D).

In this work, we study the adsorption and diffusion of F₂ molecules on graphene surface by using first-principles calculations. The calculated energy pathway for the diffusion from gas phase to the most-stable surface adsorption state reveals the existence of a kinetic barrier, which stabilizes the graphene in the atmosphere of F₂ molecules at room temperature. Meanwhile, the moderate value of the kinetic barrier (∼0.33 eV) opens the door to the activating and accelerating of the reaction between graphene and F₂ at moderately high temperatures. Our calculations of the energy pathway of the diffusion of F₂ molecules on graphene surface show the existence of high energy barriers (∼0.8 eV to 1 eV), which conduces to the understanding of the stability of fluorinated graphene under room and high temperature condition. These results shed new light on the interaction between graphene and F₂ molecules on an atomic level.