ABSTRACT

The present analytical review is devoted to the current problem of thermodynamic stability and related thermodynamic characteristics of the following graphene layers systems: 1) double-side hydrogenated graphene of composition CH (theoretical graphane) (Sofo et al. 2007) and experimental graphane (Elias et al. 2009); 2) theoretical single-side hydrogenated graphene of composition CH; 3) theoretical single-side hydrogenated graphene of composition C₂H (graphone); 4) experimental hydrogenated epitaxial graphene, bilayer graphene and a few layers of graphene on SiO₂ or other substrates; 5) experimental and theoretical single-external side hydrogenated single-walled carbon nanotubes, and experimental hydrofullerene C₆₀H₃₆; 6) experimental single-internal side hydrogenated (up to C₂H or CH composition) graphene nanoblisters with intercalated high pressure H₂ gas inside them, formed on a surface of highly oriented pyrolytic graphite or epitaxial graphene under the atomic hydrogen treatment; and 7) experimental hydrogenated graphite nanofibers-multigraphene with intercalated solid H₂ nano-regions of high density inside them, relevant to solving the problem of hydrogen on-board storage (Nechaev 2011-2012).

1. Introduction

As noted in a number of articles from 2007 through 2013, hydrogenation of grapheme—a single layer of carbon atoms arranged in a honeycomb lattice—as a prototype of covalent chemical functionality, and an effective tool to open the band gap of graphene is of fundamental importance [1,2].

It is relevant to the current problem of hydrogen onboard storage, and also to the related problems of thermodynamic stability and thermodynamic characteristics of the following systems: 1) double-side hydrogenated graphene (theoretical graphane of composition CH [3,4] and experimental graphane [5]); 2) theoretical single-side hydrogenated graphene of composition CH (SSHG) [6-8]; 3) theoretical single-side hydrogenated grapheme of composition C₂H (graphone) [9]; 4) experimental hydrogenated epitaxial graphene, bigraphene and a few layer graphene on SiO₂ or other substrates [5]; 5) experimental and theoretical single-external-side hydrogenated single-walled carbon nanotubes of composition about C₂H and experimental hydrofullerene C₆₀H₃₆ [10-14]; 6) experimental single-internal-side hydrogenated graphene nanoblisters possessing of a very high Young’s modulus (with intercalated high pressured H₂ gas) formed on the surface of highly oriented pyrolytic graphite (HOPG) or epitaxial graphene under the definite atomic hydrogen treatment [15-21]; and 7) experimental hydrogenated graphite nanofibers possessing of a high Young’s modulus with intercalated high density solid H₂ that is relevant to the problem of hydrogen on-board storage [18- 21].

In this analytical review, results of thermodynamic analysis and comparison of some theoretical and experimental data are presented, including those from the most cited works [3,5] and from the least non-cited works [18- 21].

In [8], the double-side hydrogenation of graphene is now well understood, at least from a theoretical point of view. For example, Sofo et al. predicted theoretically a new insulating material of CH composition called graphane (double-side hydrogenated graphene), in which each hydrogen atom is adsorbed on top of a carbon atom from both sides, so that the hydrogen atoms adsorbed in different carbon sublattices are on different sides of the monolayer plane [3]. The formation of graphane was attributed to the efficient strain relaxation for sp³ hybridization, accompanied by a strong (diamond-like) distortion of the graphene network [3,22]. In contrast to graphene (a zero-gap semiconductor), graphane is an insulator with an energy gap of E_g » 5.4 eV [4,23]. Only if hydrogen atoms adsorbed on one side of graphene (in graphane) are retained, we obtain graphone of C₂H composition, which is a magnetic semiconductor with E_g » 0.5 eV and a Curie temperature of T_c » 300 - 400 K [24].

As was noted in [6], neither graphone nor graphane is suitable for real practical applications, since the former has a low value of E_g, and undergoes a rapid disordering because of hydrogen migration to neighboring vacant sites even at a low temperature, and the latter cannot be prepared on a solid substrate [9].

Single-side hydrogenated graphene (SSHG) of CH composition is an alternative to graphane, in which hydrogen atoms are adsorbed only on one side [7,25]. In contrast to graphone, they are also adsorbed on all carbon atoms rather than on every second carbon atom. The value of E_g in SSHG is sufficiently high (1.6 eV lower than in graphane), and it can be prepared in a solid substrate in principle. However, this quasi-two-dimensional carbon-hydrogen theoretical system is shown to have a relatively low thermal stability, which makes it difficult to use SSGG in practice [6,7].

As seen in [7], it may be inappropriate to call the covalently bonded SSHG system sp³ hybridized, since the characteristic bond angle of 109.5˚ is not present anywhere, i.e., there is no diamond-like strong distortion of the graphene network, rather than in graphane. Generally in the case of a few hydrogen atoms interacting with graphene or even for graphane, the underlining carbon atoms are displaced from their locations. For instance, there may be the diamond-like local distortion of the graphene network, showing the signature of sp³ bonded system. However, in SSH Graphene all the carbon atoms remain in one plane, making it difficult to call it sp³ hybridized. Obviously, this is some specific sp³-like hybridization. Such model is taken into further consideration in this analytical study [10-21].

In a number of works, it shows that hydrogen chemisorption corrugates the graphene sheet in fullerene, carbon nanotubes [26], graphite [27] and graphene [28] and transforms it from a semimetal into a semiconductor [3,5]. This can even induce magnetic moments [29-31].

It is worth repeating the prediction for the double-side hydrogenated graphene (a free-standing membrane) that was partially confirmed by Elias et al. [5]. They demonstrated that graphene can react with atomic hydrogen, which transforms this highly conductive zero-overlap semimetal into an insulator of high thermal stability, and the double-side hydrogenation of graphene is reversible. The authors themselves expressed some doubts, relevant to the complete adequacy of the experimental graphane to the theoretical one [3]. Alternatively, they supposed that the experimental graphane (a free-standing membrane) produced by them may have a more complex hydrogen bonding than the one suggested by the theory, and that the latter may be as an “until now theoretical material”.

In the case of epitaxial graphene on substrates such as SiO₂ and others, hydrogenation occurs only on the top basal plane of graphene, and it is not accompanied with a strong (diamond-like) distortion of the graphene network, but only with some ripples. The first experimental indication of such a specific single-side hydrogenation came from Elias et al. [5]. The authors mentioned a possible contradiction with the theoretical results of Sofo et al. [3], which had down-played the possibility of a single side hydrogenation. They proposed an important facilitating role of the material ripples for hydrogenation of graphene on SiO₂, and believed that such a single-side hydrogenated epitaxial graphene can be a disordered material, similar to graphene oxide, rather than a new graphenebased crystal—the experimental graphane produced by them.

On the other hand, it is expedient to note that changes in Raman spectra of graphene caused by hydrogenation were rather similar (with respect to locations of D, G, D’, 2D and (D + D’) peaks) both for the epitaxial graphene on SiO₂ and for the free-standing graphene membrane [5].

As it is supposed by many scientists, such a single side hydrogenation of epitaxial graphene occurs, because the diffusion of hydrogen along the graphene-SiO₂ interface is negligible, and perfect graphene is impermeable to any atom and molecule [32]. But these two aspects are of the kinetic character, and therefore they can not influence the thermodynamic predictions [3,24,31].

Authors of [8] noted that their test calculations show that the barrier for the penetration of a hydrogen atom through the six-membered ring of graphene is larger than 2.0 eV. Thus, they believe that it is almost impossible for a hydrogen atom to pass through the six-membered ring of graphene at room temperature (from a private communication with H. G. Xiang and M.-H. Whangbo).

In the present analytical review, a real possibility is considered when a hydrogen atom can pass through the graphene network at room temperature. This is the case of existing relevant defects in graphene, i.e., in grain boundaries and/or vacancies [33-42]. This is related to further consideration of data in this analytical study as mentioned above.

Previous theoretical studies suggest that single-side hydrogenation of ideal graphene would be thermodynamically unstable [24,31]. Thus, it remains a puzzle why the single-side hydrogenation of epitaxial graphene is possible and even reversible, and why the hydrogenated species are stable at room temperatures [5,43]. This puzzling situation is also considered in the present analytical review. The main aim of this study is to show a real possibility, at least, from the thermodynamic point of view, of the existence of hydrogenated graphene-based nanostructures [18-21] possessing very high Young’s modulus, and also showing a real possibility of intercalation in nanostructures of solid molecular hydrogen under definite hydrogenation conditions relevant to the current problem of hydrogen on-board storage.

2. Analysis and Comparison of Data

2.1. Consideration of Data on Theoretical Graphanes

In work [3], the stability of graphane, a fully saturated extended two-dimentional hydrocarbon derived from a single graphene sheet with formula CH, has been predicted on the basis of the first principles and total-energy calculations. All of the carbon atoms are in sp³ hybridization forming a hexagonal network (a strongly diamond-like distorted graphene network) and the hydrogen atoms are bonded to carbon on both sides of the plane in an alternative manner. It has been found that graphane can have two favorable conformations: a chair-like (diamond-like, Figure 1) conformer and a boat-like (zigzag-

like) conformer [3].

The diamond-like conformer (Figure 1) is more stable than the zigzag-like one. This was concluded from the results of the calculations of binding energy () (i.e., the difference between the total energy of the isolated atoms and the total energy of the compounds), and the standard energy of formation () of the compounds () from crystalline graphite () and gaseous molecular hydrogen () at the standard pressure and temperature conditions [3].

For the diamond-like graphane, the former quantity is

and the latter one is

.

The latter quantity corresponds to the following reaction:

(1)

where is the standard energy (enthalpy) change for this reaction.

By using the theoretical quantity of, one can evaluate, using the framework of the thermodynamic method of cyclic processes [44], a value of the energy of formation () of graphane () from graphene () and gaseous atomic hydrogen () [3]. For this, it is necessary to take into consideration the following three additional reactions:

(2)

(3)

(4)

where, and are the standard energy (enthalpy) changes.

Reaction (2) can be presented as a sum of reactions (1), (3) and (4) using the framework of the thermodynamic method of cyclic processes [44]:

(5)

Substituting in Equation (5) the known experimental values of = −2.26 eV/atom and » −0.05 eV/atom, and also the theoretical value of = −0.15 эВ/atom, one can obtain a desired value of = −2.5 ± 0.1 eV/atom. The quantity of characterizes the break-down energy of C-H sp³ bond in graphane (Figure 1), relevant to the breaking away of one hydrogen atom from the material, which is

.

In evaluating the above mentioned value of ∆H₃, one can use the experimental data [45] on the graphite sublimation energy at 298 K

()and the theoretical data [25] on the binding cohesive energy at about 0 K for graphene

().

Therefore, neglecting the temperature dependence of these quantities in the interval of 0 - 298 K, and one obtains the value of. quantity characterizes the break-down energy of 1.5 C-C sp² bond in graphene, relevant to the breaking away of one carbon atom from the material. Consequently, one can evaluate the break-down energy of C-C sp² bonds in graphene, which is

.

This theoretical quantity coincides with the similar empirical quantities obtained in [18-21] from for C-C sp² bonds in graphene and graphite, which are

.

The similar empirical quantity for C-C sp³ bonds in diamond obtained from the diamond sublimation energy is

[18-21].

It is important to note that chemisorption of hydrogen on graphene was studied using atomistic simulations, with a second generation reactive empirical bond order of Brenner inter-atomic potential. As it has been shown, the cohesive energy of graphane (CH) in the ground state is (C). This results in the binding of hydrogen energy, which is

(H) [25].

The theoretical quantity characterizes the break-down energy of one C-H sp³ bond and 1.5 C-C sp³ bonds (Figure 1). Hence, by using the above mentioned values of and, one can evaluate the break-down energy of C-C sp³ bonds in the theoretical graphane, which is. Also, by using the above noted theoretical values of and, one can evaluate similarly the break-down energy of C-C sp³ bonds in the theoretical graphane, which is. Comparing the obtained values of, , , and show that the elastic and intrinsic strength properties, and particularly, the Young’s modulus of the theoretical graphanes is much less than those for perfect graphene, perfect graphite or perfect diamond [1,3,25].

2.2. Consideration of Data on Hydrogen Thermal Desorption from Theoretical and Experimental Graphanes

In [4], the process of hydrogen thermal desorption from graphane has been studied using the method of molecular dynamics. The temperature dependence for T = 1300 - 3000 K at the time () of hydrogen desorption onset (i.e., the time of removal ~1% () of the initial hydrogen concentration C₀ » 0.5 (in atomic fractions,) from the C₅₄H_{7(54 + 18)} clustered with 18 hydrogen passivating atoms at the edges to saturate the dangling bonds of sp³-hybridized carbon atoms have been calculated. The corresponding activation energy of E_a = 2.46 ± 0.17 eV and the corresponding near temperature independent frequency factor have also been calculated. The process of hydrogen desorption at T = 1300 - 3000 K has been described in terms of the following standard Arrhenius relationship:

(6)

where k_B is the Boltzmann constant. The authors predicted that their results would not contradict the experimental data [5], according to which the nearly complete desorption of hydrogen from a graphane membrane (Figure 2(b)) was achieved by annealing it in argon at T = 723 K for 24 hours (i.e.,).

By using Equation (6), the authors evaluated the quantity of for T = 300 K (~1∙10²⁴ s) and for T = 600 K (~2×10³ s). However, they note that the above two values of should be considered as rough estimates. Using Equation (6), one can evaluate the value of for T = 723 K, which is much less (by five orders) than the value [4].

In the framework of the formal kinetics approximation of the first order rate reaction [46], a characteristic quantity for the reaction of hydrogen desorption is 0.63–the time of the removal of ~ 63% () of the initial hydrogen concentration C0 (i.e.,) from the hydrogenated graphene. Such a first order rate reaction (desorption) can be described by the following equations [14,46]:

(7)

(8)

(9)

where is the reaction (desorption) rate constant, is the reaction (desorption) activation energy, and K₀ is the per-exponential (or frequency) factor of the reaction rate constant. In the case of a non-diffusion rate limiting kinetics, the quantity of K₀ may be the corresponding vibrational frequency (K₀ = n), and Equation (9) may be related to the Polanyi-Wigner value [14]. By substituting in Equation (8) the quantities of

and, one can evaluate the desired quantity. Hence, using Equation (9) results in the analytical quantity of A_an. = 2∙10¹⁵ s⁻¹.

Analogically, one can evaluate the desired quantity

which differs from by about three orders, by substituting in Equation (9) the quantity of

and supposing that

.

In such an approximation, one can evaluate the quantity for the experimenttal graphane membranes [5]. The obtained quantity of is less by one and a half orders of the vibrational frequency n_RD = 2.5∙10¹⁴ s⁻¹ corresponding to the D Raman peak (1342 cm⁻¹) for hydrogenated graphene membrane and epitaxial graphene on SiO₂ (Figure 2).

The activation of which in the hydrogenated samples authors attribute to breaking of the translation symmetry of C-C sp² bonds after formation of C-H sp³ bonds. Also, is less by one order of the vibrational frequency n_HREELS = 8.7∙10¹³ s⁻¹ corresponding to an additional HREELS peak arising from C-H sp³, and a stretching appears at 369 meV after a partial hydrogenation of the epitaxial graphene. The authors suppose that this peak can be assigned to the vertical C-H bonding, giving direct evidence for hydrogen attachment on the epitaxial graphene surface [47].

Taking into account n_RD and n_HREELS quantities, and substituting in Equation (9) quantities of

and

one can evaluate. In such approximation, the obtained value of coincides (within the errors) with the experimental value of the break-down energy of C-H sp³-like bonds in hydrofullerene C₆₀H₃₆

().

The above analysis of the related data shows that for the experimental graphene membranes (hydrogenated up to the near-saturation) can be used for the following thermodesorption characteristics, relevant to Equation (9), of the empirical character:

,

.

The analysis also shows that this is a case for a nondiffusion rate limiting kinetics, when Equation (9) corresponds to the Polanyi-Wigner [14]. Certainly, these tentative results could be directly confirmed and/or modified by receiving and treating within Equations (8) and (9) of the experimental data on t_0.63 at several annealing temperatures.

The above noted fact that the empirical quantity is much larger (by about 3 orders), than the theoretical one (), is consistent with that mentioned in [5]. The alternative possibility that the experimental graphane membrane (a free-standing membrane) may have a more complex hydrogen bonding, than the suggested by the theory, may point out for further theoretical developments.

2.3. Consideration of a Thermodynamic Probability of Existence of Hydrogenated Graphenes-Graphanes^* Possessing of a Very High Binding Energy

In connection with the above consideration, it seems expedient to consider a thermodynamic probability of existence of hydrogenated graphene-graphane^* ()

possessing of the values of [3- 5,12,13,18-21] and [18-21]. This corresponds to a very high binding (cohesive) energy (, in comparison with those considered above for theoretical graphanes. Because of such thermodynamic probability, it is necessary to take into consideration two more additional reactions:

(10)

(11)

where and are the standard energy (enthalpy) changes. Reaction (11) can be presented as a sum of reaction (2), applied for graphane^* as (2^*), as well as reaction (10), resulting in the following equation:

(12)

Substituting in Equation (12) above the considered values of

and

one can obtain a desired value of

.

The quantity of corresponds to the binding (cohesive) energy of graphane^*

().

The quantity characterizes the break-down energy of one (C-H) sp³-like bond and 1.5 (C-C) sp³-like bonds, relevant to breaking away of one hydrogen atom and one neighboring carbon atom from the material:

(13)

Hence, substituting in Equation (13) the above noted

value, one can evaluate the desired value of, which coincides (within the errors) with the analogical values for perfect graphene and perfect graphite. The same value of can be evaluated in a similar manner, i.e., for hydrogenated (up to composition C₂H) single-walled graphite nanotubes^* and hydrofullerene^* C₆₀H₃₆. Comparing of the obtained values of, , , , and

show that the elastic and intrinsic strength properties (and particularly Young’s modulus (E)) of graphane^*-like nanostructures can be closer to those for graphene. In connection with this, it is relevant to note that a unique experimental value from work [48] of a Young’s modulus of graphene is E_graphene = 1.0 terapascal.

As was noted in [5], when a hydrogenated graphene membrane has no free boundaries (a rigidly fixed membrane) in the expanded regions of it, the lattice is stretched isotropically by nearly 10% (i.e., the elastic deformation degree e_fix.membr. » 0.1) with respect to the pristine graphene. This amount of stretching (e » 0.1) is close to the limit of possible elastic deformations in graphene, and indeed it has been observed that some of their membranes rupture during hydrogenation. It is believed that the stretched regions are likely to remain non-hydrogenated. They also found that instead of exhibiting random stretching, hydrogenated graphene membranes normally split into domain-like regions of the size of the order of 1 mm, and that the annealing of such membranes led to complete recovery of the periodicy in both stretched and compressed domains [5].

By using the experimental value of the degree of elastic deformation () of the hydrogenated fixed graphene membranes, and the experimental value of a Young’s modulus of graphene (), one can evaluate (within Hooke’s law approximation) the stretching stress value

()

in the expanded regions (domains or grains) of the material [5,48]. This analytical result in this study is consistent with the analytical results of the related data considered from [15-21], relevant to the possibility of the existence of hydrogenated graphane^*-like nanostructures possessing of a Young’s modulus value close to that of grapheme ().

2.4. Consideration of Data on Hydrogen Desorption in the Hydrogenated Monoand Bi-Layer Epitaxial Graphene Samples

In [5], both the graphene membrane samples considered above, and the epitaxial graphene and bi-graphene samples on substrate SiO₂ were exposed to a cold hydrogen dc plasma for 2 hours to reach the saturation in the measured characteristics. They used a low-pressure (0.1 mbar) hydrogen-argon mixture of 10% H₂). Raman spectra for hydrogenated and subsequently annealed graphene membranes (Figure 2(b)) are rather similar to those for epitaxial graphene samples (Figure 2(a)), but with some notable differences. If hydrogenated simultaneously for 1 hour, and before reaching the saturation (a partial hydrogenation), the D peak area for a membrane was two factors greater than the area for graphene on a substrate (Figure 2, the left inset), which indicates the formation of twice as many C-H sp³ bonds in the membrane. This result also agrees with the general expectation that atomic hydrogen attaches to both sides of the membranes. Moreover, the D peak area became up to about three times greater than the G peak area after prolonged exposures (for 2 hours, a near-complete hydrogenation) of membranes to atomic hydrogen. The integrated intensity area of the D peak in Figure 2(b) corresponding to the adsorbed hydrogen saturation concentration in the graphene membranes is larger by a factor of about 3 for the area of the D peak in Figure 2(a), corresponding to the hydrogen concentration in the epitaxial graphene samples. This may be related to some partial hydrogenation localized in some defected nano-regions of the epitaxial graphene samples even after the prolonged (3 hour) exposures, i.e. after reaching their near-saturation [33-42,49]. It is expedient to note that in [5], the absolute values of the adsorbed hydrogen concentration (C₀) were neither considered for the hydrogenated graphene membranes, nor for the hydrogenated epitaxial graphene samples.

According to a private communication from D.C. Elias, a near-complete desorption of hydrogen

()

from a hydrogenated epitaxial graphene on a substrate SiO₂ (Figure 2(a)) has been achieved by annealing it in 90% Ar/10% H₂ mixture at T = 573 K for 2 hours (i.e.,). Hence, by using Equation (8), one can evaluate the value of, which is about six orders less than the evaluated value of.

Also, the changes in Raman spectra of graphene [5] caused by hydrogenation were rather similar in respect to locations of D, G, D’, 2D and (D + D’) peaks, both for the epitaxial graphene on SiO₂ and for the free-standing graphene membrane (Figure 2). Hence, one can suppose that

Then, by substituting in Equation (9) the values of

and, one can evaluate

Here, the case is supposed of a non-diffusion-ratelimiting kinetics, when Equation (9) corresponds to the Polanyi-Wigner one [14]. Certainly, these tentative thermodynamic characteristics of the hydrogenated epitaxial graphene on a substrate SiO₂ could be directly confirmed and/or modified by further experimental data on at various annealing temperatures.

It is now easy to state that: 1) these analytical results are not consistent with the mass spectrometry data (Figure 3) on thermal desorption of hydrogen from a specially prepared single-side graphane; and 2) they cannot be described in the framework of the theoretical models and characteristics of thermal stability of single-side hydrogenated graphene [6] or graphone [9]. According to the further considerations in this study, it may be a hydrogen desorption case of a diffusion rate limiting kinetics, when K₀ ¹ n, and Equation (9) does not correspond to the Polanyi-Wigner one [14].

By using the method of treatment for thermal desorption (TDS) spectra, relevant to the mass spectrometry data (Figure 3) on thermal desorption of hydrogen from a specially prepared single-side graphane (under heating from room temperature to 573 K for 6 minutes), one can obtain the following results: 1) the total integrated area of the thermal desorption spectra corresponds to ~2∙10⁻⁸ g of desorbed hydrogen; 2) the TDS spectra can be approximated by three thermodesorption (TDS) peaks (# 1, # 2 and # 3); 3) TDS peak # 1 (~30% of the total area, T_max#1 » 370 K) can be characterized by the activation energy of E_{TDS-peak # 1} = 0.6 ± 0.3 eV and by the per-exponential factor of the reaction rate constant; 4) TDS peak # 2 (~15% of the total area, T_{max # 2} » 445 K) can be characterized by the activation energy E_{TDS-peak # 2} = 0.6 ± 0.3 eV, and by the per-exponential factor of the reaction rate constant

; and 5) TDS peak # 3 (~55% of the total area, T_{max # 3} » 540 K) can be characterized by the activation energy E_{TDS-peak # 3} = 0.23 ± 0.05 eV and by the per-exponential factor of the reaction rate constant [14]. These analytical results show that all three of the above noted thermal desorption (TDS) processes (# 1_TDS, # 2_TDS and # 3_TDS) may be related to a hydrogen desorption case of a diffusion-ratelimiting kinetics, when in Equation 9 the value of and the value of, where D_0app is the per-exponent factor of the apparent diffusion coefficient, L is the characteristic diffusional size  (length), and Q_app. is the apparent diffusion activation energy. TDS process # 3_TDS may be related to TDS process (or peak) I in [14,18-21], for which the apparent diffusion activation energy is

and

.

Hence, one can evaluate the quantity of

which may be related to the linear size of the graphene specimens. Thus, TDS process # 3_TDS may be related to chemisorption models “H” and/or “G” (Figure 4) corresponding to TDS process (or peak) I in [14, 18-21].

TDS processes # 1_TDS and # 2_TDS may be related, in some extent to chemisorption models “H” and/or “G” (Figure 4). Model “H” corresponds to TDS process (or peak) II in [14,16,18-21], for which the apparent diffusion activation energy is Q_app.II » 1.2 eV that is comparatively close to E_{TDS-peak # 1, 2} and D_0app.II » 2∙10³ cm²/s. Obviously, chemisorption models “H” and/or “G” (Figure 4) can be applied only for the defected nano-regions in the epitaxial graphene flakes [5], for instance, in vacancies, grain boundaries (domains), and/or triple junc-

tions (nodes) of the grain-boundary network [33-42,49], where the dangling carbon bonds can occur.

It is important to note that in Items 2.1 - 2.3 chemisorption of atomic hydrogen on graphene membranes may be related to model “F^*” [14,16,18-21], which is relevant to chemisorption of a single hydrogen atom on one of the carbon atoms possessing of 3 unoccupied (by hydrogen) nearest carbons, but not two hydrogen atoms on two carbons, as seen in model “F”. Model “F^*” is characterized [14,16,18-21] by the quantity of

which coincides with the similar quantities of graphanes [3-5].

In work [5], the same hydrogenation procedures of the 2 hour long expositions have been applied, as well as bilayer epitaxial graphene on SiO₂/Si wafer. Bilayer samples showed little change in their charge carrier mobility and a small D Raman peak, compared to the single-layer epitaxial graphene on SiO₂/Si wafer exposed to the same hydrogenation procedures. The authors believe that higher rigidity of bilayers suppressed their rippling, thus reducing the probability of hydrogen adsorption.

In this study, further consideration must be given to some other known experimental data that on hydroge nation and thermal stability characteristics of mono-layer, bi-layer and three-layer epitaxial graphene systems play an important role in some defects found in graphene networks [33-42,49], relevant to the probability of hydrogen adsorption and the permeability of graphene networks for atomic hydrogen. The analytical results of Item 2 are presented in Table 1.

3. Analysis and Comparison of Data

3.1. Analysis of the Raman Spectroscopy Data on Thermal Desorption of Hydrogen from Hydrogenated Graphene Flakes

In [51], it is reported that the hydrogenation of single and bilayer graphene flakes by an argon-hydrogen plasma produced a reactive ion etching (RIE) system. They analyzed two cases: one where the graphene flakes were electrically insulated from the chamber electrodes by the SiO₂ substrate, and the other where the flakes were in electrical contact with the source electrode (a graphene device). Electronic transport measurements in combination with Raman spectroscopy were used to link the electric mean free path to the optically extracted defect concentration, which is related to the defect distance (L_def.). This showed that under the chosen plasma conditions, the process does not introduce considerable damage to the graphene sheet, and that a rather partial hydrogenation () occurs primarily due to the hydrogen ions from the plasma, and not due to fragmentation of water adsorbates on the graphene surface by highly accelerated plasma electrons. To quantify the level of hydrogenation, they used the integrated intensity ratio (I_D/I_G) of Raman bands. The hydrogen coverage (C_H) determined from the defect distance (L_def.) did not exceed ~0.05%.

In [51], they also performed the heating of the hydrogenated single graphene flakes (on the SiO₂ substrate) in a nitrogen environment, on a hot-plate, and with temperatures ranging from 348 K to 548 K, each time () of 1 min. As seen in Figure 5, heating results decrease the integrated intensity ratio (I_D/I_G) of Raman bands. Within a formal kinetics approach, the averaged kinetic data for samples of 10, 20 and 40 minute exposure can be treated by using Equation (7) transformed to a more suitable form (7’):

where Dt = 60 s, and C are determined from Figure 5. This resulted in finding 5 values of the reaction (desorption) rate constant (K) for 5 temperatures (T = 348, 398, 448, 498 and 548 K). Their temperature dependence is described by Equation (9). Hence, the desired quantities have been determined (Table 2) for the reaction (desorption) activation energy

and the per-exponential factor of the reaction rate constant. Hence, desorption time at 553 K is.

The calculated values of and are closer (within the errors) to those for TDS process # 3 (Table 1). These two desorption processes may be related to TDS process (or peak) I in [14,18-21], for which the apparent diffusion activation energy is

.

By taking into account the facts that the RIE exposure regime [51] is characterized by a form of (for), L_def. » 11 - 17 nm and the hydrogen concentration C_H £ 5∙10⁻⁴, one can suppose that the hydrogen adsorption centers in the single graphene flakes (on the SiO₂ substrate) are related in some point to nanodefects (i.e., vacancies and/or triple junctions (nodes) of the grain-boundary network) of diameter d_def. » const. In such a model, the quantity C_H can be described satisfactory as:

(14)

where n_H » const. is the number of hydrogen atoms adsorbed by a center;. It was also found that after the Ar/H₂ plasma exposure, the (I_D/I_G) ratio for bilayer graphene device is larger than that of the single graphene device. As noted in [51], this observation is in contradiction to the Raman ratios after exposure of graphene to atomic hydrogen and when other defects are introduced.

3.2. Analysis of the STM and STS Data on Reversible Hydrogenation of Epitaxial Graphene and Graphite Surfaces

In [52], the effect of hydrogenation on topography and electronic properties of graphene grown by CVD on top of a nickel surface and high oriented pyrolytical graphite (HOPG) surfaces were studied by scanning tunneling microscopy (STM) and spectroscopy (STS). The surfaces were chemically modified using 40 min Ar/H₂ plasma (with 3 W power) treatment (Figure 6). This determined that the hydrogen chemisorption on the surface of graphite/graphene opens on average an energy bandgap of 0.4 eV around the Fermi level. Although the plasma treatment modifies the surface topography in an irreversible way, the change in the electronic properties can be reversed by moderate thermal annealing (for 10 min at 553 K), and the samples can be hydrogenated again to yield a similar, but slightly reduced, semiconducting behavior after the second hydrogenation. The data shows that the time of desorption from both the epitaxial graphene/Ni samples and HOPG samples of about 99% of hydrogen under 553 K annealing is. Hence, by using Equation (8), one can evaluate the quantity, which is close (within the errors) to the similar quantity of for the epitaxial graphene flakes considered in the previous Section 3.1.

As noted in [53], before the plasma treatment, the CVD graphene exhibits a Moiré pattern superimposed to the honeycomb lattice of graphene (Figure 6(d)). This is due to the lattice parameter mismatch between the graphene and the nickel surfaces, and thus the characteristics of the most of the epitaxial graphene samples. On the other hand for the hydrogenated CVD graphene, the expected structural changes are twofold [53]. First, the chemisorption of hydrogen atoms will change the sp² hybridization of carbon atoms to tetragonal sp³ hybridization, modifying the surface geometry. Second, the impact of heavy Ar ions, present in the plasma, could also modify the surface by inducing geometrical displacement of carbon atoms (rippling graphene surface) or creating vacancies and other defects (for instance, grain or domain boundaries [33-42,49]). Figure 6(e) shows the topography image of the surface CVD graphene after the extended (40 min) plasma treatment. The nano-ordercorrugation increases after the treatment, and there are brighter nano-regions (of about 1 nm in height and several nm in diameter) in which the atomic resolution is lost or strongly distorted. It was also found that these bright nano-regions present a semiconducting behavior, while the rest of the surface remains conducting (Figures 6(g)-(h)) [52,53].

It is reasonable to assume that most of the chemisorbed hydrogen is localized into these bright nano-regions, which have a blister-like form. Moreover, it is also reasonable to assume that the monolayer (single) graphene flakes on the Ni substrate are permeable to atomic hydrogen only in these defected nano-regions. This problem has been formulated in Section 1 (Introduction). A similar model may be valid and relevant for the HOPG samples (Figures 6(a)-(c)).

It has been found out that when graphene is deposited on a SiO₂ surface (Figures 7 and 8), the charged impurities presented in the graphene/substrate interface produce strong inhomogeneities of the electronic properties of graphene. On the other hand, it has also been shown how homogeneous graphene grown by CVD can be altered by chemical modification of its surface by the chemisoption of hydrogen. It strongly depresses the local conductance at low biases, indicating the opening of a band gap in grapheme [53,54].

The charge inhomogeneities (defects) of epitaxial hydrogenated graphene/SiO₂ samples do not show long range ordering, and the mean spacing between them is L_def. » 20 nm (Figure 8). It is reasonable to assume that the charge inhomogeneities (defects) are located at the interface between the SiO₂ layer (300 nm thick) and the

graphene flake [53,54]. A similar quantity (L_def. » 11 - 17 nm, [51])) for the hydrogen adsorption centers in the single graphene flakes on the SiO₂ substrate has been considered in Section 3.1.

3.3. Analysis of the HREELS/LEED Data on Thermal Desorption of Hydrogen from Hydrogenated Graphene on SiC

In [55], hydrogenation of deuterium-intercalated quasifree-standing monolayer graphene on SiC (0001) was obtained and studied with low-energy electron diffraction (LEED) and high-resolution electron energy loss spectroscopy (HREELS). While the carbon honeycomb structure remained intact, it has shown a significant band gap opening in the hydrogenated material. Vibrational spectroscopy evidences for hydrogen chemisorption on the quasi-free-standing graphene has been provided and its thermal stability has been studied (Figure 9). Deuterium intercalation, transforming the buffer layer in quasi-freestanding monolayer graphene (denoted as SiC-D/QFMLG), has been performed with a D atom exposure of ~5∙10¹⁷ cm⁻² at a surface temperature of 950 K. Finally, hydrogenation up to saturation of quasi-free-standing monolayer graphene has been performed at room temperature with a H atom exposure >3∙10¹⁵ cm⁻². The latter sample has been denoted as SiC-D/QFMLG-H to stress the different isotopes used.

According to a private communication from R. Bisson, the temperature indicated at each point in Figure 9 corresponds to successive temperature ramp (not linear) of 5 minutes. Within a formal kinetics approach for the first order reactions [14,46], one can treat the above noted points at T_i = 543 K, 611 K and 686 K, by using Equation (8) transformed to a more suitable form (8’):

where t = 300 s, and the corresponding quantities C_0i and C are determined from Figure 9. It resulted in finding values of the reaction (hydrogen desorption from SiC-D/ QFMLG-H samples) rate constant K_i for 3 temperatures T_i = 543 K, 611 K and 686 K. The temperature dependence is described by Equation (9). Hence, the desired quantities

have been determined (Table 2) as the reaction (hydrogen desorption) activation energy

and the per-exponential factor of the reaction rate constant. The obtained value of is close (within the errors) to the similar ones (E_{TDS-peak # 1 [5]} and E_{TDS-peak # 2}) for TDS processes # 1 and # 2 (Item 2.4, Table 1). But the obtained value differs by several orders from the similar ones (and) for TDS processes # 1 and # 2 (Item 2.4, Table 1). Nevertheless, these three desorption processes may be (in some extent) related to chemisorption models “H” and/or “G” (Figure 4). Model “H” corresponds to TDS process II in [14,18-21], for which the apparent diffusion activation energy is Q_app.II » 1.2 eV.

In the same way, one can treat the points from Figure 9 at T_i = 1010 K, 1120 K and 1200 K, which one are related to the intercalated deuterium desorption from SiCD/QFMLG samples. This results in finding the desired quantities (Table 2): the reaction (deuterium desorption) activation energy

and the per-exponential factor of the reaction rate constant. Formally, this desorption process (of a diffusion-limiting character) may be described similarly to TDS process (peak) III in [14, 18-21], and the apparent diffusion activation energy may be close to the break-down energy of the Si-H bonds. As concluded in [55], the exact intercalation mechanism of hydrogen diffusion through the anchored graphene lattice, at a defect or at a boundary of the anchored graphene layer, remains an open question.

It is reasonable to assume that the quasi-free-standing monolayer graphene on the SiC-D substrate is permeable to atomic hydrogen (at room temperature) in some defect nano-regions (probably, in vacancies and/or triple junctions (nodes) of the grain-boundary network [33-42,49]). It would be expedient to note that the HREELS data [55] on bending and stretching vibration C-H frequencies in SiC-D/QFMLG-H samples (153 meV (3.7∙10¹³ s⁻¹)) and 331 meV (8.0∙10¹³ s⁻¹), respectively) are consistent with those considered in Section 2.2, related to the HREELS data for the epitaxial graphene.

The obtained characteristics (Table 2) of desorption processes [51,52,55] show that these processes may be of a diffusion-rate-controlling character [14].

3.4. Analysis of the Raman Spectroscopy Data on Thermal Desorption of Hydrogen from Hydrogenated Graphene Layers on SiO₂ Substrate

In [56], graphene layers on SiO₂/Si substrate have been chemically decorated by radio frequency hydrogen plasma (the power of 5 - 15 W, the pressure of 1 Tor) treatment for 1 min. As seen from the investigation of hydrogen coverage by Raman spectroscopy and micro-x-ray photoelectron spectroscopy characterization demonstrates that the hydrogenation of a single layer graphene on SiO₂/Si substrate is much less feasible than that of bilayer and multilayer graphene. Both the hydrogenation and dehydrogenation processes of the graphene layers are controlled by the corresponding energy barriers, which show significant dependence on the number of layers. These results [56] on bilayer graphene/SiO₂/Si are in contradiction to the results [5] on a negligible hydrogenation of bilayer epitaxial graphene on SiO₂/Si wafer, when obviously other defects are produced.

Within a formal kinetics approach [14,46], the kinetic data from Figure 10(a) for single layer graphene samples (1LG-5W and 1LG-15W ones) can be treated. Equation (7) is used to transform into a more suitable form (7’):

where = 1800 s, and DC and C are determined from Figure 10(a).

The results have been obtained for 1LG-15W sample 3 values of the I reaction rate constant for 3 temperatures (T = 373, 398 and 423 K), and 3 values of the II reaction rate constant for 3 temperatures (T = 523, 573 and 623 K). Hence, by using Equation (9), the following values for 1LG-15W samples have been determined (Table 3): the I reaction activation energy, the per-exponential factor of the I reaction rate constant, the II reaction activation energy

and the per-exponential factor of the II reaction rate constant. It also resulted in finding for 1LG-5W sample 4 values of the I_[56] reaction rate constant for 4 temperatures (T = 348, 373, 398 and 423 K), and 2 values of the II reaction rate constant for 2 temperatures (T = 523 and 573 K). Therefore by using Equation (9), one can evaluate the desired quantities for 1LG-5W specimens (Table 3): the I reaction activation energy

the per-exponential factor of the I reaction rate constant, the II reaction activation energy, and the per-exponential factor of the II reaction rate constant. A similar treatment of the kinetic data from Figure 10(c) for bilayer graphene 2LG- 15W samples resulted in obtaining 4 values of the II reaction rate constant for 4 temperatures (T = 623, 673, 723 and 773 K). Hence, by using Equation (9), the following desired values are found (Table 3): the II reaction activation energy, the per-exponential factor of the II_[56] reaction rate constant.

A similar treatment of the kinetic data from Figure 6(c) in [56] for bilayer graphene 2LG-5W samples results in obtaining 4 values for the I reaction rate constant for 4 temperatures (T = 348, 373, 398 and 423 K), and 3 values for the II reaction rate constant for 3 temperatures (T = 573, 623 and 673 K). Their temperature dependence is described by Equation (9). Hence, one can evaluate the following desired values (Table 3): the I reaction activation energy, the per-exponential factor of the I reaction rate constant

the II reaction activation energy

and the per-exponential factor of the II reaction rate constant.

The obtained characteristics (Table 3) of the desorption processes I and II show that these processes may be of a diffusion-rate-controlling character.

3.5. Analysis of TDS and STM Data on HOPG Treated by Deuterium

In [57], the results are present of a scanning tunneling microscopy (STM) study of graphite (HOPG) treated by atomic deuterium, which reveals the existence of two distinct hydrogen dimer states on graphite basal planes (Figure 11 and Figure 12(b)). The density functional theory calculations allow them to identify the atomic structure of these states and to determine their recombination and desorption pathways. As predicted, the direct recombination is only possible from one of the two dimer states. In conclusion, this results in an increased stability of one dimer species, and explains the puzzling double peak structure observed in temperature programmed desorption (TPD or TDS) spectra for hydrogen on graphite (Figure 12(a)) [57].

By using the described method of TPD (TDS) peak treatment (for the first order reactions), relevant to TPD (TDS) peak I (~65% of the total area, T_{max # I} » 473 K) in Figure 12(a), one can obtain values of the reaction I rate constant () for several temperatures (for instance, T = 458, 482 and 496 K) [14]. Their temperature dependence can be described by Equation (9). Hence, the desired values are defined as follows (Table 3): the reaction (desorption) I activation energy

and the per-exponential factor of the reaction I rate constant. In a similar way, relevant to TPD (TDS) peak II (~35% of the total area, T_max#II » 588 K)) in Figure 12(a), one can obtain values of the reaction II rate constant () for several temperatures (for instance, T = 561 and 607 K). Hence, the desired values are defined as follows (Table 3): the reac tion (desorption) II activation energy

,

and the per-exponential factor of the reaction II rate constant

The obtained characteristics (Table 3) of the desorption processes I and II show that these processes probably are of a diffusion-rate-controlling character [14]. In a diffusion-rate-controlling case, these processes can not be described by using the Polanyi-Wigner equation (as it has been done in [57]). The observed in “nano-dimer states” or “nano-protrusions” (Figure 11 and Figure 12(b)) may be related to the defected nano-regions, probably, as grain (domain) boundaries [49] and/or triple and other junctions (nodes) of the grain-boundary network in the HOPG samples. Some defected nano-regions at the grain boundary network (hydrogen adsorption centers #I, mainly, the “dimer B” structures) can be related to TPD (TDS) peak I, the others (hydrogen adsorption centers #II, mainly, the “dimer A” structures) to TPD (TDS) peak II.

In Figures 11(a) and 12(b), one can imagine some grain boundary network (with the grain size of about 2 - 5 nm) decorated (in some nano-regions at grain boundaries) by some bright nano-protrusions. Similar “nanoprotrusions” are observed and in graphene/SiC systems (Figures 13 and 14 from [58], and Figures 15 and 16 from [17]).

In Figures 13(a) and 14(b) [58], one can also imagine some grain boundary network (with the grain size of about 2 - 5 nm) decorated in some nano-regions at grain boundaries, by some bright nano-protrusions [33-42,49].

In [58], hydrogenation was studied by a beam of atomic deuterium 10¹² - 10¹³ cm⁻²∙s⁻¹ (corresponding to P_D » 10⁻⁴ Pa) at 1600 K, and the time of exposure of 5 - 90 s, for single graphene on SiC-substrate. The formation of graphene blisters were observed, and intercalated with hydrogen in them (Figures 13 and 14), similar to those observed on graphite [57] (Figures 11 and 12) and graphene/SiO₂ [17] (Figures 15 and 16). The blisters [58] disappeared after keeping the samples in vacuum at 1073

K (~15 min). By using Equation (8), one can evaluate the quantity of, which coincides (within the errors) with the similar quantity of evaluated for graphene/SiC samples [17] (Item 3.6, Table 3).

A nearly complete decoration of the grain boundary network [33-42,49] can be imagined in Figure 15(b). Also, seen in Figure 16, such decoration of the nanoregions (at the grain boundaries [33-42,49]) has a blister-like cross-section of height of about 1.7 nm and width of 10 nm order.

According to the thermodynamic analysis presented in Section 3.7, Equation (15), such blister-like decoration nano-regions (at the grain boundaries [33-42,49]) may contain the intercalated gaseous molecular hydrogen at a high pressure.

3.6. Analysis of PES and ARPES Data on Dehydrogenation of Graphene/SiC Samples

Atomic hydrogen exposures at a pressure of P_H » 1∙10⁻⁴ Pa and temperature T = 973 K on a monolayer graphene grown on the SiC (0001) surface are shown, to result in hydrogen intercalation [17]. This shows that the hydrogen intercalation induces a transformation of the monolayer graphene and the carbon buffer layer to bi-layer graphene without a buffer layer. The STM, LEED, and core-level photoelectron spectroscopy (PES) measurements reveal that hydrogen atoms can go underneath the graphene and the carbon buffer layer. This transforms the buffer layer into a second graphene layer. Hydrogen exposure (15 min) results initially in the formation of bilayer graphene (blister-like) islands with a height of ~0.17 nm and a linear size of ~20 - 40 nm, covering about 40% of the sample (Figures 15(b), 15(e), 16(a) and 16(b)). With larger (additional 15 min) atomic hydrogen exposures, the islands grow in size and merge until the surface is fully covered with bi-layer graphene (Figures 15(c), 15(f), 16(c) and 16(d)). A (√3 ×√3) R30˚ periodicity is observed on the bi-layer areas. Angle resolved photoelectron spectroscopy (ARPES) and energy filtred X-ray photoelectron emission microscopy (XPEEM) investigations of the electron band structure confirm that after hydrogenation the single p-band characteristic of monolayer graphene is replaced by two pbands that represent bi-layer graphene. Annealing an intercalated sample, representing bi-layer graphene, to a temperature of 1123 K or higher, re-establishes the monolayer graphene with a buffer layer on SiC (0001).

The dehydrogenation has been performed by subsequently annealing (for a few minutes) the hydrogenated samples at different temperatures, from 1023 to 1273 K. After each annealing step, the depletion of hydrogen has been probed by PES and ARPES (Figures 17 and 18). From this data, by using Equations (8) and 9), one can determine the following tentative quantities: (at 1023 K and 1123 K),

and (Table 3). These results can be interpreted so that the model of the interaction of hydrogen and silicon atoms at the graphene-SiC interface result in Si-C bonds at the intercalated islands. Obviously, the quantities of and correspond to those of the Polanyi-Wigner equation [14] relevant for the Si-C bonds [17].

3.7. Analysis of TDS and STM Data on HOPG Treated by Hydrogen

Atomic hydrogen accumulation in HOPG samples and etching their surface on hydrogen thermal desorption (TD) have been studied by using a scanning tunneling microscope (STM) and atomic force microscope (AFM). STM investigations revealed that the surface morphology of untreated reference HOPG samples was found to be atomically flat (Figure 19(a)), with a typical periodic structure of graphite (Figure 19(b)). Atomic hydrogen exposure (treatment) of the reference HOPG samples (30 - 125 min at atomic hydrogen pressure and a near-room temperature (~300 K)) to different atomic hydrogen doses (D), has drastically changed the initially flat HOPG surface into a rough surface, covered with nanoblisters with an average radius of ~25 nm and an average height of ~4 nm (Figures 19(c) and 19(d)) [15].

Thermal desorption (TD) of hydrogen has been found in heating of the HOPG samples under mass spectrometer control. As shown in Figure 20(a), with the increase of the total hydrogen doses (D) to which HOPG samples have been exposed, the desorbed hydrogen amounts (Q) increase and the percentage of D retained in samples (Q) approaches towards a saturation stage. After TD, no

nanoblisters were visible on the HOPG surface, the graphite surface was atomically flat, and covered with some etch-pits of nearly circular shapes, one or two layers thick (Figure 20(b)). This implies that after release of the captured hydrogen gas, the blisters become empty of hydrogen, and the HOPG surface restores to a flat surface morphology under the action of corresponding forces.

According to [15], nanoblisters found on the HOPG surface after atomic hydrogen exposure are simply monolayer graphite (graphene) blisters, containing hydrogen gas in molecular form (Figure 21). As suggested, atomic hydrogen intercalates between layers in the graphite net through holes in graphene hexagons, because of the small diameter of atomic hydrogen, compared to the hole’s size, and is then converted to a H₂ gas form which is captured inside the graphene blisters, due to the relatively large kinetic diameter of hydrogen molecules. However, such interpretation is in contradiction with that noted in Section 1 (Introduction) results [8,32], that it is almost impossible for a hydrogen atom to pass through the sixmembered ring of graphene at room temperature. It is reasonable to assume (as it’s been done in the previous Sections) that in HOPG [15] samples atomic hydrogen passes into the graphite near-surface closed nano-regions (the graphene nanoblisters) through defects (perhaps, mainly through triple junctions of the grain and/or subgrain boundary network) in the surface graphene layer. It is also expedient to note that in Figure 20(b), one can imagine some grain boundary network decorated by the etch-pits.

The average blister has a radius of ~25 nm and a height ~4 nm. Approximating the nanoblister to be a semi-ellipse form, results in the blister area and its volume. The amount of retained hydrogen in this sample becomes Q » 2.8∙10¹⁴ H₂/cm² and the number of hydrogen molecules captured inside the blister becomes. Thus, within the ideal gas approximation, and accuracy of one order of the magnitude, the internal pressure of molecular hydrogen in a single nanoblister at near-room temperature (T » 300 K) becomes. The hydrogen molecular gas density in the blisters (at T » 300 K and P_H2 » 1∙10⁸ Pa) can be estimated as

where M_H2 is the hydrogen molecule mass. It agrees with

data [60] considered in [18-21], on the hydrogen (protium) isotherm of 300 K. These results can be quantitatively described, with an accuracy of one order of magnitude, with the thermodynamic approach [44,46], by using the condition of the thermo-elastic equilibrium for the reaction of (), as follows [18]:

(15)

where is related to the blister “wall” back pressure (caused by)—the so called surface pressure [44] −P_H » 1∙10^-4 Pa is the atomic hydrogen pressure corresponding to the atomic flux [15], is the standard pressure [44,46], is the experimental value [45] of the dissociation energy (enthalpy) of one molecule of gaseous hydrogen (at room temperatures), is the dissociation entropy [45], is the apparent volume change, r_b » 30 nm is the radius of curvature of nanoblisters (at the nanoblister edge, Figure 21(b)), N_A is the Avogadro number, and T ≈ 300 K. The quantity of () is related to the work of the nanoblister surface increasing with an intercalation of 1 molecule of H₂.

The value of the tensile stresses (caused by) in the graphene nanoblister “walls” with a thickness of d_b and a radius of curvature r_b can be evaluated from another condition (equation) of the thermo-elastic equilibrium of the system in question, which is related to Equation (15), as follows [44,18]:

(16)

where is a degree of elastic deformation of the graphene nanoblister walls, and E_b is the Young’s modulus of the graphene nanoblister walls. By substituting the first part of Equation (16), the quantities of, r_b » 30 nm and d_b » 0.15 nm result in the value of.

The degree of elastic deformation of the graphene nanoblister walls, apparently reaches (Figure 21(b)). Hence, with Hooke’s law of approximation,

using the second part of Equation (16), one can estimate, with the accuracy of one-two orders of the magnitude, the value of the Young’s modulus of the graphene nanoblister walls:. It is close (within the errors) to the experimental value [48] of the Young’s modulus of graphene (E_graphene = 1.0 TPa).

The experimental data [15,59] on the thermal desorption of hydrogen from graphene nanoblisters in pyrolytic

graphite can be approximated by three thermodesorption (TDS) peaks, i.e., # I with T_{max # I} » 1123 K, # II with T_{max # II} » 1523 K, and # III with T_{max # III} » 1273 K. But their treatment, with using the above mentioned methods [14], is difficult due to some uncertainty relating to the zero level of the J_des quantity. Nevertheless, TDS peak # I can be characterized by the activation desorption energy, and by the per-exponential factor of the reaction rate constant of (Table 3). Analyses have shown that TDS peak I is related to TDS peak (process) III in [14,18-21], for which the apparent diffusion activation energy is Q_app.III = (2.6 ± 0.3) eV and D_0app.III » 3∙10^{−3 cm2}/s.

Hence, one can obtain (with accuracy of one-two orders of the magnitude) a reasonable value of the diffusion characteristic size of

which is related to the separating distance between the grapheme nanoblisters (Figure 21(b)) or (within the errors) to the separation distance between etch-pits (Figure 20(b)) in the HOPG specimens [15,59]. Thus, TDS peak (process) I is related to TDS peak (process) III in [14, 18-21], which is related to model “F^*” (Figure 4) considered in Item 2.4. Model “F^*” is characterized by the quantity that coincides with the similar quantities for graphanes (Table 1) [14,18-21].

Finally, it is reasonable to assume that the inner surfaces “walls” in the graphene nanoblisters in HOPG are hydrogenated, and that the graphene “walls” situation is related to some hydrogenated graphenes (Table 1). Obviously, such hydrogenation of the inner graphene surfaces in the nanoblisters occurs under action of the gaseous molecular hydrogen of a high pressure intercalated into the stressed (expanded) hydrogenated graphene nanoblister “walls” possessing of a high Young’s modulus [15,59].

As considered in the next Section, a similar (to some extent) situation may occur in hydrogenated graphite nanofibers (GNFs).

4. A Possibility of Intercalation of Solid H₂ into Hydrogenated Graphite Nanofibers Relevant to the Hydrogen On-Board Storage Problem

The possibility of intercalation of solid H₂ into hydrogenated graphite nanofibers (considered in [18-21]) is based on the following facts:

1) According to the data from Figures 22 and 23 (from [60]), a solid molecular hydrogen (or deuterium) of density of = 0.3 - 0.5 g/cm³ (H₂) can exist at 300 K, and an external pressure of P = 30 - 50 GPa.

2) As seen from data in Figures 19-21 and Equations (15) and (16), the external surface pressure of P = = 30 - 50 GPa at T » 300 K may be provided at the expense of the association energy of atomic hydrogen

(), into some closed hydrogenated (in gaseous atomic hydrogen with the corresponding pressure P_H) graphene nanostructures possessing of a high Young’s modulus (E_graphene £ 1 TPa).

2) As shown in [18], the treatment of the data (Figure 24 from [61]) on hydrogenation of graphite nanofibers (GNFs) results in the experimental value of the hydrogen density

(or (H₂-C-system)) of the intercalated high-purity reversible hydrogen (~17 mass. % H₂) corresponding to the state of solid molecular hydrogen at a surface pressure of P = » 50 GPa and T » 300 K, according to data from Figures 22 and 23 [61, 62].

3) Substituting in Equation (16) the quantities of » 5×10¹⁰ Pa, » 0.1 (Figure 24), the largest possible value of » 10¹² Pa, the largest possible value of the tensile stresses (» 10¹¹ Pa) in the edge graphene “walls” (of a thickness of d_b and a radius of curvature of r_b) of the slit-like closed nanopores of the lens shape, one can obtain the quantity of. It is reasonable to assume; hence, a reasonable value follows of d_b » 5 nm. A similar result can be obtained, supposing the quantity of » 10¹¹ Pa (as it is for the hydrogenated grapheme nanoblisters in HOPG [15], Section 3.7).

4) As noted in [18-21], a definite residual plastic deformation of the hydrogenated graphite (graphene) nanoregions is observed in Figure 24. Such plastic deformation of the nanoregins during hydrogenation of GNFs, may be accompanied with some mass transfer resulting in such thickness () of the walls.

5) A very important role of the spillover effect [63-69]

is the relevance to hydrogenation of GNFs [18-21,61, 62].

5) The related data presented in [70-73], and Figure 25 allows us reasonably to assume a break-through character of results [18-21], relevant for solving of the current problem [74,75] of the hydrogen on-board storage.

5. Conclusions

1) Consideration of some of the most cited works [3-9, 17,25,51-59,62] and the least non-cited works [18-21,61] on the thermodynamic stability of a number of hydrogenated grapheme layers systems (Tables 1-3) has shown expediency of further related (mainly experimental) studies for the determination of a complete and compatible set of thermodynamic characteristics of such systems.

2) It confirms that the alternative viewpoint of the experimental graphane (a free-standing membrane) may have a more complex hydrogen bonding than the one suggested by theory [3], and that the epitaxial graphene may be a different material, rather than the theoretical graphane (Table 1) [3,5].

3) A thermodynamic probability of existence of hydrogenated graphenes-graphanes^* [18-21] is in possessing of a very high binding (cohesive) energy, in comparison with the theoretical graphane [3] (Table 1).

4) It has been reasonably assumed (in the light of [70- 73] and Figure 25) a break-through character of analyticcal results [18-21] on the solid hydrogen intercalated in hydrogenated graphite nanofibers (Figure 24), relevant

for solving the current problem [74,75] of hydrogen onboard storage.

5) A constructive open discussion on the considered thermodynamic aspects of the hydrogenated graphene layer systems seems expedient, relevant to the promotion of further developments.

6. Acknowledgments

The authors are grateful to H.G. Xiang, M.-H. Whangbo, D.C. Elias, C. Casiraghi, A. Eckmann, N. Tombros, B.J. Van Wees, A. Castellanos-Gomez, R. Bisson, T. Yu, L. Hornekaer and Z. Waqar for helpful discussions, and valuable completing and/or reading of the related parts of the paper.

REFERENCES

NOTES

^*Corresponding author.