ISSN 0015-4628, Fluid Dynamics, 2009, Vol. 44, No. 3, pp. 475–479. © Pleiades Publishing, Ltd., 2009. Original Russian Text © V.L. Kovalev, A.N. Yakunchikov, 2009, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2009, Vol. 44, No. 3, pp. 167–171.
Simulation of Hydrogen Adsorption in Carbon Nanotubes V. L. Kovalev and A. N. Yakunchikov Received November 12, 2008
Abstract—The processes of physical hydrogen adsorption by carbon nanotubes are simulated by considering the molecular dynamics. The interactions are described by the Lennard–Jones potential and quantum effects are neglected. The dependences of the relative mass content of adsorbed hydrogen on the pressure and temperature are obtained. The formation of a second adsorption layer at low temperatures is detected. This leads to a higher stored hydrogen content. DOI: 10.1134/S0015462809030168 Keywords: molecular dynamics, hydrogen storage, adsorption, carbon nanotubes.
Hydrogen is a highly efficient and ecologically clean energy source. The main obstacle to the static and mobile use of hydrogen is the lock of efficient storage methods. The tank transport and storage of hydrogen in the gaseous or liquid states involves safety problems. For constructing high-pressure hydrogen storage vessels it is necessary to use high-strength steels and special materials for leakage prevention. When hydrogen is stored in tanks, its mass is equal to approximately 3–5% of the mass of the tank itself. When hydrogen is stored in the liquid state, there are losses relating to both the cooling of the system during filling and hydrogen evaporation during storage. One way of solving this problem is to store hydrogen in the adsorbed state in carbon nanotubes, which are chemically stable and relatively inexpensive and have large surface areas and insignificant masses. These properties make them an ideal material for hydrogen storage. In recent years many experimental and theoretical studies devoted to hydrogen adsorption in carbon nanostructures have been published [1, 2]; however, the results do not agree. Earlier experiments [3] indicated a high 5–10% relative mass hydrogen content in carbon nanotubes at room temperature and pressures of the order of atmospheric. In subsequent experiments [4, 5] the hydrogen content at room temperature was found to be significantly lower. In theoretical studies, molecular dynamics [6] and Monte–Carlo statistical simulation [7] methods have been used for estimating the adsorptivity of nanotube bundles. In the present study hydrogen adsorption on an individual carbon nanotube is directly simulated numerically at various pressures and temperatures using molecular dynamics methods. 1. We will consider a carbon nanotube with the chirality (10, 10) surrounded by molecular hydrogen at a pressure p and temperature T . We will assume that the processes of physical adsorption take place on the outer surface of the tube since without special treatment the tube ends are closed. In this case two types of interaction are taken into account, namely, the interaction between H2 molecules and C atoms and the interaction between the H2 molecules themselves. These interactions can be described by means of the Lennard–Jones 12–6 potential: 6 12 σ σ − , U (r) = 4ε r r where r is the distance between particles. For each of these interactions the parameters ε and σ borrowed from [6] are given in table. The internal degrees of freedom in the H2 molecule and quantum effects are neglected. 475
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Table Interaction C–H2 H2 –H2 T–H2
˚ σ, A 3.17 2.92 2.6
ε , meV 2.76 3.18 36.8
ε /k, K 32.0 36.9 428
Source [6] [6] Present study
In order to determine the adsorption energy Ea we will calculate the potential energy surface describing the interaction between a carbon nanotube and a hydrogen molecule. The minima of the potential energy ˚ the nanotube surface opposite to the centers Ea = 36.8 meV = 428 kJ are located at a distance of 3.1 Afrom of the hexagons of carbon atoms. The dependence of the potential between the tube and a hydrogen molecule was approximated by the Lennard–Jones 8–4 potential which depends only on the distance between the tube surface and the hydrogen molecule UTH (r) = 4εTH 2
σ 2
TH2
8
r
−
σ
TH2
4
r
.
The parameters εTH and σTH are given in table. In the direct simulation we calculated the trajectories 2 2 of a large number of particles simulating the gas molecules m
d 2 ri = FTH (ri ) + 2 dt 2
∑ FH H (ri j=i
2
2
− r j ),
where m is the molecular mass, ri are the coordinates of the ith molecule, FTH (r) is the force exerted 2 on the molecule by the nanotube, and FH H (ri − r j ) is the force exerted on the ith molecule by the jth 2 2 ˚ around which the molecule. At the initial instant of time a fragment of nanotube with a length of 50 A, ˚ computation hydrogen molecules are regularly distributed, was positioned at the center of the 80× 80× 50 A domain. The molecular velocities were distributed in accordance with the Maxwell equilibrium function. The equations were solved with a constant time step Δt = 5 × 10−15 s. Macroparameters such as the density, pressure, and temperature were found from the distributions of the corresponding molecular quantities. 2. A phenomenological model based on the ideal Langmuir adsorbed layer theory is proposed for investigating the adsorptivity of the carbon nanotubes. It is assumed that the adsorbed particles are linked with certain local centers on the adsorbent surface, that each center can attach only a single particle, and that the energy of the adsorbed particles at all the centers on the surface is the same. Within the framework of this theory, hydrogen molecule adsorption is an equilibrium process and for the function of occupied active surface centers we have Ja θ= , Ja (1 − θ ) = Ja θ , Ja + Jd where Ja is the adsorbed molecule flux for completely free active centers and Jd is the desorbed molecule flux for completely occupied active centers. The active centers were assumed to be located at the above-mentioned points of minimum potential energy of the interaction between the hydrogen molecule and the tube. However, our calculations showed that under the action of the already adsorbed molecules every second center in the longitudinal direction of the tube turns out to be unusable. As a result, for every four hydrogen atoms there is a single center with the adsorption energy Ea = 36.8 meV= 428 kJ. For complete occupation of the surface this corresponds to 4.2 relative mass percent. It was assumed that for adsorption it is sufficient for a molecule to impinge on a free active center. Assuming that in the gas phase the molecular velocities are distributed in accordance with a Maxwellian FLUID DYNAMICS
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equilibrium function, we have ∞ ∞ ∞ Ja = n −∞ −∞ 0
f (u, v, w) =
p nkT =√ , u f (u, v, w) du dv dw = √ 2π mkT 2π mkT
β3 exp(−β 2 (u2 + v2 + w2 )), π 3/2
β=√
1 , 2RT
where n is the number density of the hydrogen in the gaseous phase and m is the hydrogen molecular mass. The adsorbed molecules are assumed to oscillate about the equilibrium position in which their velocities u are distributed according to a Maxwellian distribution function in conformity with the tube surface temperature. For desorption the molecule must surmount a potential barrier of height Ea , i.e., have a radial velocity greater than u∗ = 2Ea /m; therefore, the desorbed molecule flux has the form: ∞ ∞ ∞ Jd = na
ue (u) f (u, v, w) du dv dw, −∞ −∞ u∗
where na is the number density of the adsorbed molecules when the active centers are completely occupied and ue is the molecular velocity remaining after the potential barrier Ea has been surmounted. The dependence ue (u) can be determined from the energy conservation law mu2 mu2e = − Ea . 2 2 The amplitude of the thermal oscillations of an adsorbed molecule is determined by its velocity u in the equilibrium position, i.e., A(u) = r2 (u) − r1 (u), where r1 and r2 are the coordinates of the extreme positions in the molecular oscillations determined from UTH (r1,2 ) = −Ea + 2
mu2 . 2
The following estimate for the number density of adsorbed hydrogen was used: na =
N α = , A(u) ¯ × S A(u) ¯
α=
N , S
1 1 u¯ = √ , πβ
where N is the total number of active centers, S is the total nanotube surface area, α is the surface density of the active centers, and u is the absolute value of the thermal velocity in a single direction. 3. Using the direct simulation method, we found that two layers of adsorbed molecules are formed at low temperatures (T = 80 K) and a pressure higher than 30 atm (Fig. 1). The density of the second layer is significantly less then the density of the first layer but due to the greater radius the second layer significantly increases the mass content of adsorbed hydrogen. At room temperature and pressures up to 100 atm no second layer formation was detected. At room temperature the values of the relative mass content of adsorbed hydrogen obtained phenomenologically and by the direct simulation method are in agreement over a broad pressure interval (Fig. 2). When T = 80 K these values differ significantly for p > 20 atm, since the phenomenological model only describes monolayer adsorption. When T = 298 K the mass content of adsorbed hydrogen found in [7] by means of the Monte–Carlo method for individual tubes (curve 5) is significantly higher than the values obtained in the present study. The area of the outer surface of the bundle is approximately six times less than the area of the outer surface of all the tubes that form it [4]. If we assume that adsorption takes place only on the outer surface of the bundle, then the relative mass hydrogen content in the bundles will be less by a factor of six than for FLUID DYNAMICS
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Fig. 1. Hydrogen density as a function of the distance from the nanotube surface obtained by the direct numerical simulation method at T = 80 K and p = 50 atm.
Fig. 2. Relative mass hydrogen content as a function of the pressure at T = 80 K (a) and T = 298 K (b). Curve 1 corresponds to the Langmuir isotherm, curve 2 to direct simulation for individual tubes, curve 3 to an estimate for a tube bundle, 4 to experiment [4], and curve 5 to calculations [7] for individual tubes.
individual nanotubes. In Fig. 2 curve 3 was obtained by multiplying the simulation results for individual tubes (curve 2) by a factor of 1/6. This estimate is in good agreement with the experimental data for bundles [4] at pressures lower than 50 atm. At higher pressures the nature of the experimental curve changes. The authors of [4] attribute this to a change in the bundle geometry and the internal penetration of hydrogen. Summary. An efficient method of direct numerical simulation of the processes of hydrogen adsorption by carbon nanotubes is developed. The dependence of the relative mass content of adsorbed hydrogen on the pressure at temperatures of 80 K and 298 K which is in good agreement with the available experimental data is obtained. It is found that at T = 80 K and p > 30 atm two layers of adsorbed molecules are formed. This significantly increases the mass hydrogen content. The formation of a second adsorbed layer could not be detected at room temperature and pressures up to 100 atm. In this case a phenomenological model based on the ideal adsorbed Langmuir layer is proposed for simulating the process of multilayered adsorption. This model is in agreement with the results of direct simulation over a broad pressure interval.
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REFERENCES 1. B.P. Tarasov and N.F. Gol’dshleger, “Hydrogen Sorption by Carbon Nanostructures,” Al’ternativnaya Energetika i Ekologiya, No. 3, 29–39 (2002). 2. A.V. Eletskii, “Sorption Properties of Carbon Nanostructures,” Uspekhi Fiz. Nauk 174, No. 11, 1191–1231 (2004). 3. A.C. Dillon, K.M. Jones, T.A. Bekkedahl, C.H. Kiang, D.S. Bethune, and M.J. Heben, “Storage of Hydrogen in Single-Walled Carbon Nanotubes,” Nature 386, No. 6623, 377–379 (1997). 4. Y. Ye, C.C. Ahn, C. Witham, B. Fultz, J. Liu, A.G. Rinzler, D. Colbert, K.A. Smith, and R.E. Smalley, “Hydrogen Adsorption and Cohesive Energy of Single-Walled Carbon Nanotubes,” Appl. Phys. Lett. 74, No. 16, 2307–2309 (1999). 5. J. Lawrence and Gu Xu, “High Pressure Saturation of Hydrogen Stored by Single-Wall Carbon Nanotubes,” Appl. Phys. Lett. 84, No. 6, 918–920 (2004). 6. S. Maruyama and T. Kimura, “Molecular Dynamics Simulation of Hydrogen Storage in Single-Walled Carbon Nanotubes,” in 2000 ASME Int. Mechanical Engineering Congr. and Exhibit., 405–409 (2000). 7. N. Hu, X. Sun, and A. Hsu, “Monte Carlo Simulation of Hydrogen Adsorption in Alkali-Doped Single-Walled Carbon Nanotubes,” J. Chem. Phys. 123, No. 4, 044708.1-044708.10 (2005).
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