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)
Address all correspondence to this author.
e-mail: [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
J. Mater. Res., Vol. 23, No. 2, Feb 2008
337
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.
J. Mater. Res., Vol. 23, No. 2, Feb 2008
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.
6.
7.
8.
9.
ACKNOWLEDGMENT
Financial support from the Australian Research Council is gratefully acknowledged.
10.
11.
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