Hydrogen diffusion and effect of grain size on hydrogenation kinetics in magnesium hydrides X. Yaoa) Australian Research Council (ARC) Center for Functional Nanomaterials, University of Queensland, QLD 4072, Australia; and School of Engineering, James Cook University, Townsville, QLD 4811, Australia
Z.H. Zhu Australian Research Council (ARC) Center for Functional Nanomaterials, University of Queensland, QLD 4072, Australia
H.M. Cheng National Laboratory of Materials Science, Institute of Metals Research, Shenyang 110015, China
G.Q. Lu Australian Research Council (ARC) Center for Functional Nanomaterials, University of Queensland, QLD 4072, Australia (Received 22 May 2007; accepted 23 July 2007)
Hydrogenation and dehydrogenation of metal hydrides are of great interest because of their potential in on-board applications for hydrogen vehicles. This paper aims to study hydrogen diffusion in metal hydrides, which is generally considered to be a controlling factor of hydrogenation/dehydrogenation. The present work first calculated temperature-dependent hydrogen diffusion coefficients by a theoretical model incorporated with experimental data in a Mg-based system and accordingly the activation energy. The grain size effect on diffusion in nanoscale was also investigated. I. INTRODUCTION
Economical and safe hydrogen storage is critical to the viability of a future hydrogen economy. Magnesium and magnesium-based alloys have been considered among the most promising materials for hydrogen storage because of their low cost and potentially high capacity, and they have been extensively studied in recent years.1–18 Several novel approaches have successfully been proposed to improve the hydrogen absorption and desorption kinetics, which is one of the key limitations of application of Mg-based materials, such as nanostructuring and defect inducing by the ball-milling technique,1–7 doping with catalytic elements,4–12 alloying,13,14 and hybriding with other hydrogen-storage materials.15–17 Yao et al.18 reported that enhancement of diffusion by carbon nanotubes (CNTs) additives increased the absorption kinetics significantly at low temperatures, which indicated that the hydrogenation of MgH2 is diffusion limiting at low temperatures. Recently, the dissociative behavior of molecular hydrogen at the surface of magnesium, which is believed to be an important step for hydrogenation, was studied by the first-principle calculations.19–22 Results also show that diffusion of the dissociated hydrogen atoms into the bulk magnesium is the controlling step in a)
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[email protected] DOI: 10.1557/JMR.2008.0063 336
J. Mater. Res., Vol. 23, No. 2, Feb 2008
the formation of hydrides. However, the diffusion behavior of the hydrogen atoms and the effect of grain size on hydrogenation have not been reported so far. This work aims to develop a mathematical model that describes the diffusion of hydrogen atoms in magnesium and firstly predicts the temperature-dependent hydrogen diffusion coefficient in magnesium hydrides.
II. MODEL DESCRIPTION
With assumptions of: (i) the geometry of grains inside Mg particles is spherical; (ii) the hydrogen diffusion along grain boundaries is much faster than diffusion inside grains so that the initial hydrogen concentration around each grain is same and controlled by hydrogen pressure around the Mg particle; and (iii) the diffusion of hydrogen in magnesium is much faster than inside magnesium hydride: hydrogen diffusion is very fast in magnesium, e.g., Dm ⳱ 6.0 × 10−9 m2/s,23 and this value is extremely large compared with that in MgH2. This statement can be implicitly supported from experimental results. The H atom diffusion into Mg became virtually 0 when the thickness of Mg hydrides covered around the Mg particles exceeded a critical thickness of several tens nm.24 Accordingly, a simple physical model is presented for hydrogen diffusion in Mg/MgH2 system, schematically shown as Fig. 1, in which we can only consider the hydrogen diffusion inside MgH2 and neglect it in Mg © 2008 Materials Research Society
X. Yao et al.: Hydrogen diffusion and effect of grain size on hydrogenation kinetics in magnesium hydrides
where Cn =
2 n , n = n R0 − R*
.
Assuming that the phase transition from magnesium to hydrides occurs at its nominal concentration, e.g., 7.6 wt% of hydrogen and need no supersaturation, the C*H ⳱ 7.6%. Additionally, the flux of solute H diffuses through the Mg/MgH2 interface equals the solute H in the volume of MgH2 formed per time per unit (assuming the same density of Mg and MgH2). With the first-order approximation, it is given by −DH
冏 冏 ⭸CH ⭸r
= VC* H
(5)
,
r=R*
where V is the velocity of the advancing MgH2/Mg interface. V can determined by the relationship between the fraction of MgH2, ␣, and the time from experimental measurement of hydrogen absorption kinetics curve if V is assumed to be constant. It can be given by26 V = R0共1 − 共1 − ␣兲1 Ⲑ 3兲 Ⲑ t
FIG. 1. Hydrogen diffusion model for a spherical geometry.
matrix. The governing equation for diffusion in hydrides within grains is then given by 1 ⭸CH ⭸2CH 2 ⭸CH = for 0 艋 r 艋 R0, t ⬎ 0 + DH ⭸t r ⭸r ⭸r2
, (1)
with the initial and boundary conditions of CH共r,t0兲 = C0, CH共R*,t兲 = C* H, for R* 艋 r ⬍ R0, t ⬎ 0 , (2) and
冉 冊
0 exp − DH = DH
⌬G RT
,
(3)
where t is time, CH and DH are the concentration and diffusion coefficient of hydrogen atoms in MgH2, D0H is the diffusion constant for hydrogen, ⌬G is the diffusion barrier or activation energy, T is the temperature, and R is the Boltzmann constant (* denotes the parameters at the MgH2/Mg interface). The solution by Crank25 for diffusion in a hollow spherical geometry can be applied to solving Eqs. (1) and (2). The solution of the governing equations is given by R*C* 共R0Ci − R*C* H H兲共r − R*兲 + r r共R0 − R*兲 ⬁ Cn + 关R0共Ci − C0兲 cosn − R*共C* H − C0兲兴 r n=1 × sin n共r − R*兲 exp (4) 共−2nDHt兲 for R* ⬍ r 艋 R0 ,
CH共r,t兲 =
冱
.
(6)
Using Eq. (5) as coupling condition, it is thus obtained from Eq. (4) with assumption of C0 ⳱ 0: −
冉冱
R0Ci cos 共n兲 − R*C* H exp 共−2nDHt兲 R0 − R* R0Ci − R*C* H − C* + = VC* . (7) H H R0 − R*
DH 2 R*
冊
Accordingly, the hydrogen profile can be determined by Eq. (7) with known diffusion coefficient, or by calculating the diffusion coefficient with known hydrogen profile in MgH2 by experimental measurement. III. CALCULATED RESULTS AND EXPERIMENTAL VALIDATION
To simplify the calculation of Eq. (7), the relationship between the fraction of hydrides formed and the time for hydrogenation can be approximately considered linear during the initial period based on experimental measurements,4 as seen in Fig. 2 (reproduced from Fig. 3 in Ref. 4). Figure 2 shows the experimentally measured kinetic curves by using an automated Sievert apparatus under a hydrogen pressure of 2.0 MPa at different temperatures of 100, 150, 200, and 300 °C for a Mg–FeTi–CNTs system (detailed experimental procedure is referring to Ref. 4). Synergistic catalytic effects have been achieved in Mg–FeTi–CNTs systems. The transition metals of FeTi dramatically reduced the hydrogen dissociative energy barrier to realize the hydrogenation possible thermodynamically at low temperatures,4,27 while CNTs significantly increased the hydrogen atomic diffusion along Mg subsurface28 and in bulk Mg matrix.4,29 The data of
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X. Yao et al.: Hydrogen diffusion and effect of grain size on hydrogenation kinetics in magnesium hydrides
FIG. 2. Experimentally measured hydrogenation kinetic curves in a Mg-based system (reproduced from Ref. 4).
Mg–FeTi–CNTs system rather than a pure Mg were selected to calculate and validate the present model due to valid data of Mg–FeTi–CNTs at low temperatures. From Fig. 2, the measured hydrogenation rates are 0.0168, 0.00173, 0.00044, and 0.000076 wt% H per second at 300, 200, 150, and 100 °C, respectively (as the slope of tangent of each measured kinetic curve). The velocity of the advancing MgH2–Mg interface is then calculated, and the results show that V remains almost constant at each temperature during the initial growth period, which fulfills the requirement of Eq. (5). Supposing that the residual Mg particles with average size of 20 nm covered by Mg hydrides, it is obtained that the average size of milled Mg particles before hydriding is about 30 nm with a hydrogen capacity of 5.8 wt%.4,18 With such a relationship at different temperatures in the synthesized nanocomposites with an average grain size of R0 ⳱ 30 nm, the diffusion coefficients of hydrogen inside the magnesium hydrides at different temperatures can be easily derived by Eq. (7) as shown in Fig. 3. Figure 3 shows that the diffusion coefficient of hydro-
FIG. 3. Diffusion coefficient of hydrogen in Mg hydrides. 338
gen within MgH2 is in a range of 10−18 to 1024 m2/s at 300 to 100 °C. The coefficient for H atoms diffusing in MgH2 is very small due to the microstructure of MgH2. In MgH2, all the vacancies within the Mg structures have already been occupied by H atoms that formed Mg–H bonding. For further H diffusion, new vacancies have to be created through the breakdown of existing Mg–H bonding and then the movement of the disbranched H atoms. From the data of the diffusion coefficients at different temperatures, the activation energy for hydrogen diffusion in Mg hydrides can be easily calculated to be 107.9 kJ/mol by Eq. (3). This value agrees well with the data in a recent review by Sholl.30 If we use his method to calculate the diffusion coefficient with an energy barrier of 107.9 kJ/mol that was derived by our present model, D0H ⳱ 1.5 × 10−9 m2/s. Using Eq. (3) and the energy barrier of 107.9 kJ/mol, it is easy to calculate that the diffusion coefficients at 300 and 100 °C are 2.19 × 10−19 m2/s and 1.17 × 10−24 m2/s, respectively. It should be noted that the value of hydrogen diffusion coefficients and the activated energy is for Mg–FeTi–CNTs system, in which the diffusion coefficient should be higher than in pure Mg, while the activated energy is smaller. This can be supported by recent investigations of Vegge et al.31,32 that the activated energy of the controlled step is as high as ∼300 kJ/mol, although the difficulty of hydrogen diffusion inside MgO was involved in their experiments. Now we look into the effect of grain size on hydride formation with the calculated diffusion coefficients at various temperatures. Assuming that the velocity of the MgH2/Mg advancing interface is controlled by diffusion, we calculated the fraction of MgH2 with time at grain sizes of 3 and 300 nm, respectively, using the diffusion coefficients in Fig. 3. The results are presented in Fig. 4. It is obvious that a decrease in crystallite size remarkably increases the rate of hydride formation, i.e. the hydrogenation rate. At a high temperature of 300 °C, magnesium grains in a diameter of 300 nm are very difficult to hydrogenate, compared with experimental data of the hydrogenation rate of a Mg grain that is 30 nm in diameter [Fig. 4(a)].14 Interestingly, the Mg will be fully hydrided in a few seconds if the grain size is at 3 nm. This suggests that a decrease of the Mg grain size can gain higher hydrogen storage capacity and faster hydrogenation rate for practical hydrogen storage. At a lower temperature, the decrease of grain size enables hydrides formation at a faster kinetics. As an example, the hydrogenation rate with a smaller grain size of 3 nm at 150 °C is almost four times that with a larger grain size of 30 nm at 200 °C, even though the absorption temperature for the former is 50 °C lower. Further, at as low as 100 °C, a Mg grain of 3 nm exhibits a relatively fast hydrogenation (0.04 wt% H per minute) which is about 10 times the hydrogenation rate of Mg at a grain size of 30 nm.
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X. Yao et al.: Hydrogen diffusion and effect of grain size on hydrogenation kinetics in magnesium hydrides
FIG. 4. Effect of grain size on magnesium hydrides formation at different temperatures: (a) 300 °C, (b) 200 °C, (c) 150 °C, and (d) 100 °C.
IV. CONCLUSIONS
In summary, an analytical model for hydrogen diffusion in Mg and Mg hydrides has been developed to successfully estimate hydrogen diffusion coefficients in Mg hydrides at various temperatures as well as the diffusion activation energy. This model can also be used to study the relationship between hydrides formation, e.g., hydrogenation rate and the grain size of Mg. It has been shown that the practicality of using Mg as hydrogen storage could be enhanced by reducing the Mg grain size in nanoscale.
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ACKNOWLEDGMENT
Financial support from the Australian Research Council is gratefully acknowledged.
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