Vibrational properties and diffusion of hydrogen on graphene
Hydrogen and deuterium chemisorption on a single layer of graphene has been studied by path-integral molecular dynamics simulations. Finite-temperature properties of these point defects were analyzed in the range from 200 to 1500 K, by using a tight-binding potential fitted to density-functional calculations. On one side, vibrational properties of the adatoms are studied at their equilibrium positions, linked to C atoms. The vibrations display an appreciable anharmonicity, as derived from comparison between kinetic and potential energy, as well as between vibrational energy for hydrogen and deuterium. On the other side, adatom motion has been studied by quantum transition-state theory. At room temperature, quantum effects are found to enhance the hydrogen diffusivity on the graphene sheet by a factor of 20.
pacs:68.43.Pq, 68.43.Jk, 68.43.Fg, 81.05.Uw
In recent years there has been a surge of interest in carbon-based materials. Among them, those formed by C atoms with hybridization have been intensively studied, as is the case of carbon nanotubes, fullerenes, and graphene. The latter, in particular, is up to now the only real two-dimensional crystal, with exotic electronic properties.Geim and Novoselov (2007); Katsnelson (2007)
Carbon-based systems, in general, are considered as candidates for hydrogen storage.Dillon and Heben (2001) Also, chemisorption on two-dimensional systems, such as graphene, can be important for catalytic processes.Sluiter and Kawazoe (2003) The interest on hydrogen as an impurity in solids and on surfaces is not new, and dates back to many years. Even though this is one of the simplest impurities, a thorough understanding of its physical properties is complex due to its low mass, and requires the combination of advanced experimental and theoretical methods.Pearton et al. (1992); Estreicher (1995)
Experimental investigations on atomic, isolated hydrogen on graphene have been so far scarce, since H is difficult to detect. Recently, it has been clearly observed by transmission electron microscopy, and its dynamics were analyzed in real time.Meyer et al. (2008) In general, apart from its basic interest as an isolated impurity, an important property of hydrogen in solids and surfaces is its ability to form complexes and passivate defects, which has been extensively studied in the last twenty years.Pearton et al. (1992); Estreicher (1995); Zeisel et al. (1999)
From a theoretical viewpoint, atomic hydrogen on graphene has been studied by several authors using ab-initio methods.Sluiter and Kawazoe (2003); Yazyev and Helm (2007); Casolo et al. (2009); Boukhvalov et al. (2008); Duplock et al. (2004); de Andres and Vergés (2008) It is generally accepted that chemisorption of a single hydrogen atom leads to the appearance of a defect-induced magnetic moment on the graphene sheet, along with a large structural distortion.Yazyev and Helm (2007); Casolo et al. (2009); Boukhvalov et al. (2008) However, standard electronic-structure calculations, in spite of their quantum mechanical character, usually treat atomic nuclei as classical particles, and typical quantum effects as zero-point vibrations are not directly available from the calculations. Such quantum effects may be relevant for vibrational and electronic properties of light impurities like hydrogen, especially at low temperatures.
Finite-temperature properties of hydrogen-related defects in solids have been studied by ab-initio and tight-binding (TB) molecular dynamics simulations. In many earlier applications of these methods, atomic nuclei were treated as classical particles.Buda et al. (1991); Panzarini and Colombo (1994); Bédard and Lewis (2000) To consider the quantum character of the nuclei, the path-integral molecular dynamics approach results to be particularly suitable. In this procedure all nuclear degrees of freedom can be quantized in an efficient way, allowing one to include both quantum and thermal fluctuations in many-body systems at finite temperatures. Thus, molecular dynamics sampling applied to evaluate finite-temperature path integrals allows one to carry out quantitative studies of anharmonic effects in condensed matter.Gillan (1988); Ceperley (1995)
In this paper, the path-integral molecular dynamics (PIMD) method is used to investigate the role of the impurity mass on the properties of hydrogenic point defects. We study isolated hydrogen and deuterium (D) on a graphene sheet. Special attention has been laid upon the vibrational properties of these impurities, by considering anharmonic effects on their quantum dynamics. The results of these calculations show that such anharmonic effects lead to an appreciable deviation of the vibrational energy of the impurities, as compared to a harmonic approximation. Also, the atomic diffusion is found to be enhanced with respect to the classical limit, for both H and D. Path-integral methods analogous to that employed in this work have been applied earlier to study hydrogen in metalsGillan (1988); Makarov and Topaler (1995); Mattsson and Wahnström (1995) and semiconductors.Ramírez and Herrero (1994); Herrero and Ramírez (1995); Miyake et al. (1998); Herrero and Ramírez (2007) In connection with the present work, hydrogen has been studied inside and on carbon nanotubes by diffusion Monte Carlo.Gordillo et al. (2001); Gordillo (2007)
The paper is organized as follows. In Sec. II, we describe the computational methods employed in our calculations. Our results are presented in Sec. III, dealing with the energy of the defects, vibrational properties, and impurity diffusion. In Sec. IV we summarize the main results.
Ii Computational Methods
In this Section we present the computational methods employed in our simulations. On one side, in Sec. II.A we introduce the PIMD method used to obtain equilibrium properties related to the hydrogenic defects. On the other side, in Sec. II B we discuss a procedure to calculate rate constants for impurity jumps, in the context of transition-state theory.
ii.1 Path-integral molecular dynamics
Our calculations are based on the path-integral formulation of statistical mechanics, which is a powerful nonperturbative approach to study many-body quantum systems at finite temperatures. In this approach, the partition function is evaluated through a discretization of the density matrix along cyclic paths, composed of a number (Trotter number) of “imaginary-time” steps.Feynman (1972); Kleinert (1990) In the numerical simulations, this discretization gives rise to the appearance of “beads” for each quantum particle. Then, this method exploits the fact that the partition function of a quantum system can be written in a way formally equivalent to that of a classical one, obtained by substituting each quantum particle by a ring polymer consisting of classical particles, connected by harmonic springs.Gillan (1988); Ceperley (1995) Here we employ the molecular dynamics technique to sample the configuration space of the classical isomorph of our quantum system ( carbon atoms plus one impurity). Calculations were carried out in the canonical ensemble, using an originally developed software, which enables efficient PIMD simulations on parallel computers. The algorithms employed to integrate the equations of motion were based on those described in the literature.Martyna et al. (1996); Tuckerman (2002)
The calculations have been performed within the adiabatic (Born-Oppenheimer) approximation, which allows us to define a potential energy surface for the nuclear coordinates. An important question in the PIMD method is an adequate description of the interatomic interactions, which should be as realistic as possible. Since employing true density functional or Hartree-Fock based self-consistent potentials would restrict enormously the size of our simulation cell, we have derived the Born-Oppenheimer surface for the nuclear dynamics from an efficient tight-binding effective Hamiltonian, based on density functional (DF) calculations.Porezag et al. (1995) The capability of TB methods to reproduce different properties of molecules and solids was reviewed by Goringe et al.Goringe et al. (1997) In particular, the reliability of our TB Hamiltonian to describe hydrogen-carbon interactions in carbon-based materials has been checked in previous work,Herrero et al. (2006); Herrero and Ramírez (2007) and according to those results we expect that systematic errors in calculated diffusion barriers for H in these materials are less than 0.1 eV. An advantage of the combination of path integrals with electronic structure methods is that both electrons and atomic nuclei are treated quantum mechanically, so that phonon-phonon and electron-phonon interactions are directly taken into account in the simulation.
Simulations were carried out in the ensemble on a graphene supercell of size = 9.84 Å with periodic boundary conditions, containing 32 C atoms and one adatom. For comparison, we also carried out simulations of graphene without impurities, using the same supercell size. We have checked that larger supercells, i.e., give within error bars the same results as those derived below from the supercell. Sampling of the configuration space has been carried out at temperatures between 200 and 1500 K. For a given temperature, a typical run consisted of PIMD steps for system equilibration, followed by steps for the calculation of ensemble average properties. To have a nearly constant precision in the results at different temperatures, we took a Trotter number that scales as the inverse temperature, so that = 6000 K. For comparison with the results of our PIMD simulations, we have carried out some classical molecular dynamics simulations with the same interatomic interaction, which is achieved by setting = 1. The quantum simulations were performed using a staging transformation for the bead coordinates. Chains of four Nosé-Hoover thermostats were coupled to each degree of freedom to generate the canonical ensemble.Tuckerman and Hughes (1998) The equations of motion were integrated by using the reversible reference system propagator algorithm (RESPA), which allows us to define different time steps for the integration of the fast and slow degrees of freedom.Martyna et al. (1996) The time step associated to the DF-TB forces was taken in the range between 0.2 and 0.5 fs, which was found to be appropriate for the atomic masses and temperatures studied here. For the evolution of the fast dynamical variables, that include the thermostats and harmonic bead interactions, we used a time step . More details on the actual implementation of the simulation method can be found elsewhere.Ramírez et al. (2006); Herrero et al. (2006)
ii.2 Quantum transition-state theory
Classical transition-state theory (TST) is a well-established method for calculating rate constants of infrequent events. An important element in this computational method is the ratio between the probability of finding the system at a barrier (saddle-point) and at its stable configuration. A quantum extension of this theory has been developed in the context of path integrals, with the purpose of studying the kinetics of processes involving light atoms.Gillan (1988); Voth et al. (1989); Cao and Voth (1994) This quantum approach allows one to relate the jump rate to the probability density of the center-of-gravity (centroid) of the quantum paths of the jumping atom, defined as
being the coordinates of the “beads” in the associated ring polymer. Then, is related to the ratio between the equilibrium probability of finding the centroid at a saddle-point (say ) and at a stable site (say ).Gillan (1987); Voth et al. (1989) Namely:
where is a factor weakly dependent on temperature, taken to be the thermal velocity of the jumping impurity, and is the distance between and . Note that, apart of typical quantum effects, the factor in will favor a faster jump rate of the lighter atoms, as in classical TST. The probability ratio can be written as , being an effective free-energy barrier, given by the reversible work done on the system when the impurity centroid moves along a path from to :
where is the mean force acting on the impurity with its centroid fixed on at temperature :
Here is the potential energy, being in our case a -dimensional vector ( carbon atoms plus one impurity). The average value in Eq. (4) is taken over quantum paths with the centroid of the impurity fixed on at temperature . Thus, the jump rate (a dynamical quantity) is related to the free energy difference (a time-independent quantity), so that its temperature dependence can be obtained from equilibrium simulations without any direct dynamical information.Voth (1993) Quantum effects that may give rise to substantial deviations from the classical jump rate are taken into account within this kind of quantum TST. In this way, jump rates for kinetic processes can be obtained for realistic, highly nonlinear many-body problems. The reliability of this method to calculate free-energy barriers and jump rates has been discussed in Refs. Gillan, 1987; Makarov and Topaler, 1995. In particular, it was argued that this method can be inaccurate in the presence of asymmetric barriers at low temperatures. In our case, the error bar of the calculated barriers is expected to be less than the intrinsic error of the employed tight-binding potential.
In this kind of simulations, adatom motion was restricted along a reaction coordinate defined by the centroid of the quantum path. Note, for comparison, that in the equilibrium PIMD simulations described in the previous section, there is no restriction on the motion of the H (D) centroid. The force has been evaluated at 11 points along the reaction coordinate connecting the lowest-energy configuration (H linked to a C atom) with the saddle point of the energy surface, and then the integral in Eq. (3) was calculated numerically. For each point in the integration path, we generated 5000 configurations for system equilibration, and 2 configurations for calculating the mean force at a given temperature. More technical details can be found elsewhere.Gillan (1988); Herrero (1997); Noya et al. (1997)
iii.1 Vibrational energy
We first discuss the lowest-energy configuration for the hydrogenic impurities on a graphene sheet, as derived from classical calculations at , i.e., point atomic nuclei without spatial delocalization. The impurity binds to an C atom, which relaxes out of the sheet plane by 0.46 Å, with a bond distance between C and impurity of 1.17 Å. These results are in line with those reported in the literature, and in particular with the breaking of a bond and producing an additional bond, changing the hybridization of the involved C atom from to .Sluiter and Kawazoe (2003); Boukhvalov et al. (2008); Casolo et al. (2009) Assuming the host C atoms fixed in the relaxed geometry, one can calculate vibrational frequencies for the impurity in a harmonic approximation. Thus, we find for hydrogen a frequency = 2555 cm for stretching of the C–H bond (perpendicular to the graphene sheet), and = 1186 cm for vibrations parallel to the plane (twofold degenerate).
We now turn to the results of our simulations at finite temperatures, and will discuss the energy of the hydrogenic defects.. The internal energy of the system graphene plus impurity, , at temperature can be written as:
where is the potential energy for the classical material at (point-like atoms on their equilibrium positions), and is the vibrational energy of the whole system. Then, can be obtained by subtracting the energy from the internal energy derived from PIMD simulations. In Fig. 1 we show the temperature dependence of the vibrational energy for a graphene supercell including an H atom (circles). For comparison we also show for a pure graphene sheet (squares). At 300 K, the vibrational energy of graphene amounts to 6.18 eV per simulation cell, i.e., 0.19 eV/atom. As expected, increases as temperature is raised, and eventually converges to the classical limit at high .
An interesting characteristic of the different hydrogenic defects (H or D) is their associated vibrational energy. At a given temperature, this energy is defined as the difference = (32C + Imp) – (32C), where ‘Imp’ stands for H or D. Note that defined in this way is not just the vibrational energy of a given adatom on graphene, but it also includes changes in the vibrations of the nearby C atoms. Shown in Fig. 2 is the vibrational energy associated to the hydrogenic adatoms as a function of temperature. Symbols indicate results of PIMD simulations for H (squares) and D (circles), and dashed lines are guides to the eye. For comparison, we also present as solid lines the dependence of the vibrational energy of a single particle in a harmonic approximation with the frequencies and given above. At low the actual zero-point vibrational energy results to be smaller than that given in the harmonic approach [i.e., ], but this trend changes as temperature rises. In fact, at temperatures larger than 1000 K the vibrational energy derived from the simulations is larger than that obtained for the one-particle harmonic approach.
Another way of getting insight into the anharmonicity of the defect vibrations is by looking at the ratio between vibrational energies of point defects for both isotopes. At low temperature, this ratio should converge to in the harmonic approximation, going to unity at high temperatures. The results yielded by our simulations are shown in Fig. 3, along with the one-particle harmonic expectancy. The former lie somewhat lower than the latter in the whole temperature region under consideration. In particular, at 300 K we find = 1.364, vs a ratio of 1.394 derived in a harmonic approximation.
PIMD simulations allow us to separate the potential () and kinetic () contributions to the vibrational energy.Herman et al. (1982); Giansanti and Jacucci (1988); Gillan (1990) In fact, is related to the spatial delocalization of the quantum paths, which can be obtained directly from the simulations (see below). In Fig. 4, we present the kinetic and potential energy of the point defect associated to hydrogen, as a function of temperature. Symbols indicate results of our simulations for (squares) and (circles), whereas dashed lines are guides to the eye. The potential energy of the point defect is found to be clearly larger than the kinetic energy, indicating an appreciable anharmonicity of the whole defect (in a harmonic approach one has ). A solid line represents the expected dependence for both, kinetic and potential contributions, in a harmonic approximation with the frequencies and given above. At low temperature we find an appreciable change of the kinetic energy respect to the harmonic approach, contrary to the potential energy, which coincides within error bars with the harmonic expectancy. A qualitative understanding of this behavior can be obtained by analyzing the energy changes obtained through time-independent perturbation methods.Landau and Lifshitz (1965); Herrero and Ramírez (1995) Thus, assuming a perturbed one-dimensional harmonic oscillator (with perturbations of and type) at , the first-order change in the energy is totally due to a change in the kinetic energy, the potential energy remaining unaltered respect to its unperturbed value. A similar behavior has been obtained for hydrogen in silicon from path-integral Monte Carlo simulations.Herrero and Ramírez (1995) The main difference is that in that case the kinetic energy was found to increase respect to the harmonic value, contrary to the result obtained here for H on graphene. This seems to depend on the details of the interatomic interactions and the actual geometry of the point defect under consideration, but in both cases the potential energy at low temperature is very close to the value yielded by the harmonic approximation.
Path-integral simulations at finite temperatures describe quantum delocalization through paths of finite size. This means that the average extension of the paths is a measure of the importance of quantum effects in a given problem. One can define a kind of quantum delocalization as the mean-square “radius of gyration” of the ring polymers associated to the quantum particle under consideration.Gillan (1988, 1990) This means:
where is the position of the centroid of the paths defined in Eq. (1), and indicates a thermal average at temperature . Note that the total spacial delocalization of a particle includes an additional term, taking into account displacements of the center of gravity of the paths. This term is the only one surviving at high temperatures, since in the classical limit each path collapses onto a single point. For our problem of hydrogen on graphene, with well-defined H vibrations along three orthogonal axes, one can define a quantum delocalization of hydrogen along each of these directions ( and ). These delocalizations are displayed in Fig. 5 for vibrations parallel (circles) and perpendicular (squares) to the graphene plane, as derived from our PIMD simulations. decreases as the temperature is raised and the particle becomes more “classical”. For comparison, we also show in Fig. 5 the mean-square displacement expected for harmonic oscillators of frequencies and , which can be worked out analytically.Gillan (1988, 1990); Ramírez and Herrero (1993) The delocalization derived from the simulations follows closely the harmonic expectancy in the whole temperature range considered here. However, is higher than its corresponding harmonic result at 400 K.
iii.2 Hydrogen diffusion
Hydrogen is expected to diffuse on the graphene sheet, breaking a C–H bond and forming a new one with a nearby C atom. To check the appearance of this kind of hydrogen jumps, we have carried out some classical molecular dynamics simulations, which allow one to follow the adatom along its trajectory on the surface. As indicated above, this kind of classical limit is easily achieved with our PIMD code, by setting the Trotter number (one bead per atom). At temperatures lower than 1000 K, hydrogen jumps on the graphene surface result to be infrequent events, and are rarely observed along a simulation run. Then, a reliable estimation of the diffusion coefficient by this method is not possible. Moreover, at temperatures larger than 1000 K hydrogen begins to escape from the sheet, without remaining on it enough time for studying quantitatively the diffusion process. This situation is remedied by analyzing the hydrogen motion by TST, and in particular by the quantum version presented in Sect. II.B. This procedure allows us to obtain free energy barriers for impurity jumping, which include corrections due to the quantum character of the atomic nuclei, and in particular the renormalization of the barriers caused by zero-point motion. Also, phonon-assisted tunneling is included in the calculation, since motion of the C atoms takes into account a full quantization of the vibrational degrees of freedom. The main question we address here is the dependence of hopping rates on temperature and impurity mass. In connection with this, there is a vast literature about theoretical models for quantum diffusion of light particles in solids.Flynn and Stoneham (1970); Sugimoto and Fukai (1980); Schober and Stoneham (1988); Gillan (1988) Due to the complexity of this problem, such calculations have been typically based on model potentials for the impurity-lattice interactions.
Here, the jump rate of the impurity between two nearest equilibrium sites is derived from the free-energy barrier between those sites. To calculate this barrier, we first select a continuous path from one site to the other, minimizing the energy at the transition (saddle) point in a classical calculation (point-like atoms) at . In connection with this, it is worthwhile noting that seemingly simple atomic jumps can actually involve coupled barriers, as indicated in Ref. Ramamoorthy and Pantelides, 1996. Thus, to obtain the barrier for hydrogen jumps, we considered a coupled motion of H and the nearest C atom. For the optimal path, we obtained an energy barrier of 0.78 eV, with the saddle point corresponding to a symmetric (bridge) configuration of hydrogen between two C atoms and at a distance of 1.7 Å from the graphene plane. Once selected the best path, we performed finite-temperature path-integral simulations with the centroid of H fixed on several points along this path, as described in Sect. II.B. Then, the free-energy barrier is calculated from the mean force by using Eq. (3).
In Fig. 6 we present the free-energy barrier for adatom diffusion between adjacent C–H bonds. Data derived from line integration of the mean force are shown as a function of temperature for hydrogen (squares) and deuterium (circles). For comparison, we also show results derived from classical simulations (triangles). In this plot, one notices first that is higher for D than for H, and even higher for the classical limit, which in this respect may be considered as the large-mass limit .Ramírez et al. (2006); Herrero et al. (2006) Then, we observe an appreciable decrease in the free-energy barrier as the impurity mass is reduced. Second, one observes an increase in as temperature is raised in the three cases shown in Fig. 6. This increase is smaller, but not negligible, in the classical limit, which at converges to the potential energy barrier given above ( = 0.78 eV). At low temperature, the dependence of upon impurity mass is related to the change in internal energy of the defect complex along the diffusion path, since then the entropy contribution to the free energy becomes negligible. At 300 K we find = 0.79, 0.74, and 0.71 eV, for the classical limit, D, and H, respectively. This means that quantum effects renormalize the free-energy barrier at room temperature by about 6 % for D and 10 % for H. This reduction in decreases as temperature rises and the atoms become “more classical”. In fact, these free-energy barriers should converge one to the other in the high-temperature limit, which cannot be approached here due to the onset of adatom desorption.
Shown in Fig. 7 is the rate for impurity jumps on a graphene sheet as a function of the inverse temperature. Data were derived from the free-energy barriers displayed in Fig. 6, by using Eq. (2). Results are given for hydrogen (squares), deuterium (circles), and classical limit (triangles). At K, the jump rate for hydrogen results to be larger than that for deuterium, which in turn is higher than that found in the classical limit, as expected from the change in effective free-energy barrier discussed above. At 300 K, the jump rate for hydrogen is found to be 11.6 s, about 20 times larger than the value found in the classical calculation. The influence of quantum effects on hydrogen diffusivity increases as temperature is lowered, as could be expected, but the jump rate itself becomes very small. In fact, at 200 K the calculated rate for hydrogen, , is less than s. On the contrary, at high temperatures converges to the classical result, and the difference between both becomes unobservable at larger than 1000 K. At this temperature we find s.
The diffusion barriers obtained here for H and D on graphene are comparable to those found for hydrogen diffusion on graphite. Ferro et al.Ferro et al. (2004) used density-functional theory to study various diffusion barriers in graphite. For hydrogen diffusion on its surface, they obtained a (classical) barrier of 0.94 eV, somewhat larger than that found here at in the classical limit.
The diffusion of H and D on solid surfaces has been studied earlier using quantum TST, employing simulations similar to those presented here. Such studies were mainly focused on hydrogen diffusion on metal surfaces.Mattsson and Wahnström (1995); Rick et al. (1993) Our results are qualitatively similar to those obtained in these studies, with an appreciable enhancement of the jump rate of the impurity in comparison to a classical model. At temperatures lower than those studied here ( 50 K), a temperature-independent diffusion was found on metal surfaces.Mattsson and Wahnström (1995, 1997) This is also a possibility for hydrogen diffusion on graphene, and remains as a challenge for future research. PIMD simulations at those temperatures would need the use of Trotter numbers larger than 100, that together with the interaction potentials employed here requires computational resources out of the scope of the present work.
The main advantage of PIMD simulations of hydrogen on graphene is the possibility of calculating defect energies at finite temperatures, including a full quantization of host-atom motions, which are not easy to take into account from fixed-lattice calculations and classical simulations. Isotope effects can be readily explored, since the impurity mass appears as an input parameter in the calculations. This includes the consideration of zero-point motion, which together with anharmonicity may cause appreciable non-trivial effects. Our results indicate that hydrogen adsorbed on graphene cannot be accurately described as a particle moving in a harmonic potential. Even if anharmonicities of the interatomic potential are taken into account, a single-particle approximation is questionable as a realistic description of impurity complexes at finite temperatures. It is then necessary to treat the defect as a many-body problem with anharmonic interactions.
Ab-initio theoretical techniques to calculate defect energies in solids have achieved an excellent precision in recent years. However, zero-point motion is a factor limiting the accuracy of state-of-the-art techniques to predict energy bands and total energies of solids.Ramírez et al. (2006) The same happens for defect levels caused by impurities in solids, since their energy may change appreciably as the impurity mass is varied. Here we have illustrated how anharmonicities in the atomic motion cause an appreciable difference between kinetic and potential energy of the defect (about 20 % for H at 300 K), and have quantified the effect of the impurity mass on anharmonic shifts in the energy.
Due to the large relaxation of the nearest C atoms, hydrogen migration requires important motion of these atoms. Then, an adatom jump has to be viewed as a cooperative process involving a coupled motion of the impurity and the nearest host atoms. This picture is similar to that described in the literature as “opening of a door”,Boucher and DeLeo (1994) which favors impurity diffusion. Thus, quantum motion of both adatom and C atoms helps to renormalize the diffusion barriers respect to the classical expectancy. At 300 K we find a hydrogen diffusivity 20 times larger than that derived from classical barriers.
Acknowledgements.This work was supported by Ministerio de Ciencia e Innovación (Spain) through Grant No. BFM2003-03372-C03-03.
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