First-Principles Theory of Hydrogen Diffusion in Aluminum
Hakan Gunaydin1, Sergey V. Barabash2, K. N. Houk1,3, and V. Ozoliņš*2,3
1
Department of Chemistry and Biochemistry,
University of California, Los Angeles, California, 90095
2
Department of Materials Science and Engineering,
University of California, Los Angeles, California, 90095
3
California NanoSystems Institute,
University of California, Los Angeles, California 90095
E-mail: vidvuds@ucla.edu
Quantum effects on H diffusion
Tunneling corrections for the hydrogen diffusion in Al in the temperature range of our study
(650 to 850 K) were computed using the Wigner[1] and Eckart[2] tunneling methods. We used
the values of the activation energy and imaginary frequency at the transition state for a TdOh
jump computed by Wolverton, Ozolins and Asta in Ref. [3]. The computed corrections to the
classical tunneling rate are given in Fig. S1, which shows that the inclusion of tunneling
corrections increases the rate of hydrogen diffusion in Al by about 1 to 3 percent at the
temperature range of our simulations. This is consistent with the results of other first-principles
studies, which typically find that quantum corrections become very small at temperatures above
400 K.[4-7]
Figure S1. Tunneling contributions to the diffusion rate of H in Al computed with the Wigner
and Eckart methods.
S1
More elaborate quantum tunneling methods have been developed by Fermann and Auerbach [6],
who include the effect of quantization of vibrational energy levels. Fermann and Auerbach also
introduce the concept of crossover temperature used to separate the quantum from the classical
regime. The crossover temperature refers to a temperature below which the tunneling corrections
to the rate of H diffusion cannot be neglected in comparison with the classical hopping
dynamics. These authors show that even in the case of covalently bound hydrogen (binding
energy of ~1 eV and an imaginary frequency of ~1500i cm-1), the crossover temperature is only
368 K, above which tunneling corrections do not affect the rate of H migration significantly. In
addition, Fermann and Auerbach show that above the crossover temperature their derived
tunneling correction gives results that are nearly identical to the Wigner tunneling correction.
Determination of Hydrogen Residence Times
The diffusion coefficient of H in Al was calculated using the residence time method. Within this
approach, one needs to determine the average time that H spends in the Oh and Td sites; these
residence times were obtained directly from the computed ab initio molecular dynamics (AIMD)
trajectories. At each time step, the hydrogen was assigned either to a Td or an Oh site based on
distance criteria. When the hydrogen-Td site distance was smaller than the hydrogen-Oh site
distance, hydrogen was considered to reside at the Td site, and when the hydrogen-Oh distance
was smaller than the hydrogen-Td distance, hydrogen was considered to reside at the Oh site. The
residence time histograms obtained at 800 K are given in Fig. S2. As seen from Fig. S2,
hydrogen residence times in the Td site are much longer than the hydrogen residence times in the
Oh site. The longer average residence time in the Td site corresponds to a higher binding energy
of hydrogen in the Td site in comparison with the Oh site.
S2
Figure S2. Oh (left panel) and Td (right panel) residence times obtained from AIMD simulations
at T=800 K. Yellow lines represent the Poisson fits to the data.
Uncorrelated H jumps at 800 K
To ascertain the possibility that successive TdOh and OhTd jumps are uncorrelated, we
calculated the probability for the hydrogen to return to the same site after two consecutive jumps
at T=800 K. According to our AIMD simulations, these probabilities are 5/50 and 12/48 for
jumps originating from the Td and Oh sites, respectively. The fact that these values are very close
to the theoretical probabilities of 1/8 and 1/4 suggests that two consecutive jumps are
uncorrelated and the diffusion of H in Al can be calculated assuming that the direction of each
TdOh (OhTd) jump is independent of the preceding OhTd (TdOh) jump.
Timeline for the diffusion of H in Al
Figure S3 shows the residence times that were obtained for the diffusion of H in Al at T=800 K.
The blue strides correspond to Td site occupations and yellow strides correspond to Oh site
occupations. As seen from Fig. S3, residence times are uncorrelated.
Figure S3. Timeline for the Td site and Oh site occupations for H atom at T=800 K. Yellow time
strides correspond to Oh site occupations and blue time strides correspond to Td site occupations.
The total length of this simulation is 20 ps.
S3
Calculating Diffusivity from Residence Times
To calculate the diffusion coefficient from the measured AIMD residence times, we use the fact
that the Oh sites form a simple face-centered cubic lattice and that diffusion away from any given
Oh site necessarily goes through one of the 12 neighboring Oh sites (see Fig. S4).
Figure S4. A schematic diagram of Td  Oh jumps. Oh site is shown green, and the Td site is
red. The twelve nearest-neighbor Oh sites are on the edges of the cubic unit cell.
Therefore, one can use the standard expression for diffusion on cubic lattices, D  2 6 t NN ,
where   a
2 is the separation between nearest-neighbor sites and t NN is the average time
for a hydrogen to jump from an Oh site to any one of the neighboring Oh sites via one or more
intermediate residences on Td sites. The average Oh to Oh jump time t NN
is calculated as
follows. Let’s assume that the average residence times are t1 and t2 for Oh and Td sites,
respectively, and that the probability to have a residence time, t, in the Oh site follows the
Poisson distribution with an average t1:
 O (t) 
h
1  t t1
e .
t1
(1.1)
A hydrogen atom in an Oh site can jump to any one of the eight surrounding Td sites with a
probability distribution given by Eq. (1.2) below. The probability to make a jump at time t from
an Oh site to a chosen neighboring Td site is:
pOh Td (t) 
 O (t)
h
8
,
(1.2)
where the factor 8 accounts for the number of nearest-neighbor (NN) Td sites. A hydrogen atom
in a Td site has four jump possibilities to any one of the surrounding Oh sites. The corresponding
S4
jump and residence probabilities for the Td site can be obtained from equations analogous to
Eqns (1.1) and (1.2) above. In particular, the jump and residence probabilities for the Td site are
given by
pTd Oh (t) 
 T (t ) 
d
 T (t)
d
4
,
1  t t2
e ,
t2
(1.3)
(1.4)
where the factor 4 in Eq. (1.3) accounts for the fact that each Td site has 4 neighboring Oh sites.
The H atom in an Oh site can return to the same Oh site after two jumps or diffuse to one of the
neighboring Oh sites at a distance   a
2 away from the original Oh site (where a is the lattice
parameter). The probability that two jumps will take hydrogen back to the original Oh site is
8 (4 * 8)  1 4 . Hence, the probability distribution for the diffusion of hydrogen to a NN_Oh site
via a two jump route (Oh Td NN_Oh) can be obtained from:
(1.5)
where the factor 24 accounts for 24 different routes of reaching a NN_Oh site via one of the NN
Td sites. Eq. (1.5) gives the total probability for a hydrogen atom to move from an Oh site to a
NN_Oh site in time interval t via one intermediate residence on a Td site.
The total probability or reaching the NN_Oh site is a sum over all possible pathways of diffusing
into a NN_Oh site. First, the hydrogen can jump to a NN_Oh site via the Oh Td NN_Oh route
considered above. Second, the hydrogen can reach a NN_Oh site via a four-jump route, Oh
Td Oh Td NN_Oh , i.e., it can jump into one of the Td sites, then back into the original Oh
site, then to one of the neighboring Td sites, followed by a jump to a NN_Oh site. The probability
distribution for this four-jump process is given by:
t
t
t
0
1
2
p4 (t)  192  d1  d 2  d 3 pOh Td (1 )pTd Oh ( 2  1 )pOh Td ( 3   2 )pTd Oh (t   3 ) , (1.6)
where  is the time of the first OhTd jump, 2 is the time of the return TdOh jump, and 3 is
the time of the second Oh Td jump. The factor 192 accounts for the fact that there are 8 ways to
perform the first two jumps which bring hydrogen back to the same Oh site, and 24 ways to get
out of it after two more [see Eq. (1.5)].
S5
The third contribution comes from a 6 jump route (Oh Td Oh Td Oh Td NN_Oh):
t
t
t
t
t
0
1
2
3
4
p6 (t)  8  8  24  d 1  d 2  d 3  d 4  d 5 pOh Td ( 1 )pTd Oh ( 2   1 ) 
(1.7)
pOh Td ( 3   2 )pTd Oh ( 4   3 )pOh Td ( 5   4 )pTd Oh (t   5 )
The total probability to jump out of the Oh site into a NN_Oh site in time t is the sum over all
these processes:

p NN (t )   p2 n (t ) .
(1.8)
n 1
Integrals like (1.6) and (1.7) were computed analytically by using Mathematica. The total jump
probability satisfies

p
NN
(t)dt  1 .
(1.9)
0
The mean time required to jump to a NN_Oh site is given by

t NN   tpNN (t) dt 
0
4
t1  t2 .
3
(1.10)
Since the Oh sites form a simple FCC lattice, the diffusivity can be can be calculated from the
conventional expression, D  2 6 t NN
for random walk on cubic lattices, resulting in the
following expression:
(1.11)
which was used to convert from the residence times to diffusion coefficients in Table I of the
manuscript.
Trapping of Hydrogen in Vacancies and Non-Vacancy Trap Sites
The effect of H trapping in vacancies and non-vacancy trap sites can be quantified by using a
grand-canonical formalism. We assume that the concentration and spatial distribution of
vacancies remain constant (and uniformly disordered) due to the slow kinetics of vacancy
diffusion under typical experimental conditions. These assumptions hold during typical lowtemperature high-fugacity charging with hydrogen (such methods are commonly used to increase
H concentration before hydrogen desorption experiments measuring the hydrogen diffusivity)
S6
and for hydrogen evolution experiments due to the much faster diffusion rates of hydrogen in
comparison with vacancy diffusion. For instance, in Ref. [10] aluminum samples were charged
over a time period of 48 hours at 90 oC. The rate of Al self-diffusion at 90 oC is approximately
10-19 m2/s, whereas the rate of diffusion of H in Al at 90 oC is 10-10 m2/s. This corresponds to a
penetration depth/diffusion-mean-distance of ~200 nm for aluminum vacancies and ~1 cm for
hydrogen atoms during the charging process. Similarly, hydrogen desorption typically occurs
over short time spans (on the order of minutes) during which the vacancy diffusion, with a
diffusion-mean-distance of a few tens or hundreds of Angstroms, can be safely neglected. Figure
S5 shows the relative rates of H and vacancy diffusion in Al: the diffusion of H is many orders of
magnitude faster than the diffusion of Al vacancies under both charging and desorption
conditions. Therefore, under the conditions relevant to the discussion vacancies do not reach
thermal equilibrium.
Figure S5. The relative diffusivities of H and Al vacancies.
The presence of trap sites such as vacancies, dislocations and grain boundaries at low H
concentrations can significantly alter hydrogen diffusivity in metals. In the case of H in Al, the
calculated hydrogen-vacancy binding energies are higher than the calculated hydrogen diffusion
barriers, and we expect that vacancies and other trap sites will have a significant contribution to
the effective diffusion coefficient.
In order to describe the hydrogen distribution between the lattice and trap sites quantitatively, we
adopt a grand-canonical formalism for a solid in equilibrium with a reservoir of hydrogen at a
given chemical potential, . The total concentration of hydrogen in the solid, CH, can be
S7
controlled by varying the hydrogen chemical potential (CH increases with increasing ). We
assume that hydrogen in the trap sites is always in local equilibrium with hydrogen in the lattice,
and therefore the whole system can be characterized by using the same value of chemical
potential. The validity of this assumption was checked by estimating the hydrogen residence
time in a vacancy using the calculated H-vacancy binding energies and diffusion barriers from
Ref. [3], and was found to be on the order of microseconds, i.e. much faster than the time scale
of diffusion experiments. Strict mathematical criteria for the validity of the assumption of local
equilibrium have been derived by Oriani [8], and they appear to hold for most systems and
experimental conditions of practical interest.
We write the total hydrogen concentration as a sum of H content in the lattice (CL) and H content
in the vacancies and non-vacancy trap sites (Ctrap)
CH  CL  Ctrap .
(2.1)
Following Oriani’s treatment of hydrogen diffusion in a lattice with trap sites, we assume that
only the lattice hydrogen contributes to the measured hydrogen flux (i.e. vacancy diffusion with
trapped hydrogen is much slower and can be neglected on the time scale of typical experiments):
J  DL CL ,
(2.2)
where DL is the lattice diffusion coefficient without traps. Most experiments do not measure DL
directly since they have no means of measuring the gradient in the concentration of lattice
hydrogen. Instead, they measure the relation between the gradient of the total hydrogen
concentration [given by Eq. (2.1)] and the flux:
J   DC H ,
(2.3)
where D is the apparent diffusivity. Due to the local equilibrium condition between the lattice
hydrogen and hydrogen in the trap sites, some hydrogen is released from (absorbed by) the trap
sites upon decrease (increase) in the lattice concentration CH due to compositional fluxes. This
process changes the concentration of the lattice hydrogen and decreases the concentration
gradient C L in Eq. (2.2), leading to a decrease in the apparent diffusivity.
Using the assumption of local equilibrium, the gradient of the total hydrogen concentration can
be related to the gradient of the lattice concentration by the following relation:
S8
C trap 

C H  C L  1 
,
C L 

(2.4)
where the derivative in the parentheses describes the release and absorption of hydrogen in the
trap sites in response to a change in the lattice concentration due to hydrogen fluxes. Introducing
the chemical potential , using the mathematical identity
Ctrap
CL
1
 C  Ctrap
, and
 L
  

combining it with Eqs. (2.2) and (2.3), we obtain that the experimentally measured apparent
diffusivity D is related to the true lattice diffusivity DL by the following relation:
D
DL
1
 C  Ctrap
1  L 
   
.
(2.5)
Since the hydrogen population is an increasing function of the chemical potential, the
denominator in Eq. (2.5) is always larger than 1, and the apparent diffusivity is smaller than the
lattice diffusivity. Only in the case when all traps are empty (very small binding energies) or
completely saturated (high binding energies or very high lattice concentrations and
correspondingly high ) does the observed diffusivity equal the lattice diffusivity. From Eq. (2.5)
it is seen that D is an implicit function of the temperature and local hydrogen concentration in the
lattice (which in turn is determined by the local value of the chemical potential). To evaluate this
dependence, we turn to constructing a model for H trapping using the first-principles calculated
hydrogen-vacancy binding energies from Refs. [3] and [9].
We consider a system with N lattice sites and a uniform vacancy concentration CV, which does
not change appreciably during the charging and discharging processes due to the relatively slow
rate of vacancy diffusion. We account for non-vacancy trap sites (dislocations, grain boundaries,
etc.) by introducing an effective concentration CD of such sites. In what follows, we assume that
the concentration of vacancies, trap sites and lattice hydrogen is sufficiently small (on the order
of 10-4; similar to the values reported by Young and Scully [10]) to be treated in the ideal
solution approximation where all lattice sites are independent and hydrogen atoms do not interact
with each other. The total concentration of hydrogen is given by a sum of lattice hydrogen
(occupying Td and Oh sites), H in vacancies ( C VH ) and H in other non-vacancy traps sites ( C HD ):

S9

C H  CTd  COh  CHV  C HD .
(2.6)
The total free energy of the system is given by
G  kBT lnQ ,
(2.7)
where the statistical partition function is
  Q  Q  Q 
Q  QTd
2N
N
NCV
Oh
NCD
V
D
.
(2.8)
Here, QTd , QOh , QV and QD are partition functions for the tetrahedral Td and octahedral Oh lattice
sites, vacancies, and other trap sites, respectively. These can be expressed in terms of the site
energies, temperature, and hydrogen chemical potential as follows:
 E  
QTd  1 STd e  Td ,
(2.9)
 E  
QOh  1 SOh e  Oh ,
12
QV  1  SnH-Ve
(2.10)

  EnH V n
,
(2.11)
n1
QD  1  SD e  ED    .
(2.12)
The energies of hydrogen in the Td and Oh sites, ETd, and EOh, are measured relative to fcc Al,
while the hydrogen-vacancy energy EnH-V is measured with respect to fcc Al with a vacancy (i.e.,
the vacancy formation energy is not included in EnH-V ). Sum over n in Eq. (2.11) accounts for the
possibility that up to 12 hydrogen atoms can be housed in one vacancy. Non-vacancy trap sites
are described by using an effective hydrogen absorption energy ED. In all cases, the reference
state for hydrogen is the energy of an H2 molecule at T=0 K. The chemical potential  therefore
includes all effects due to pressure, temperature and charging conditions. Symmetry factors STd,
SOh, SnH-V and SD account for the degeneracy of each type of site, e.g. S=1 for the Oh and Td sites,
while S1H-V=8 since the hydrogen ion can be in any of the eight neighboring Td sites next to a
vacancy. Fractional occupations of each type of site can be found by taking the derivatives of the
free energies with respect to the chemical potential:
STd e  Td
1  GTd 
fTd 

2N    T
QTd
 E
1  GV 
1
fV 

NCV    T QV
S10
12

,
 nS
n1

(2.13)

H-V
H-V   En  
n
e
,
(2.14)
with similar expressions for the other hydrogen sites. Hydrogen concentrations are found by
multiplying
these
occupations
with
the
number
of
sites
of
each
type:
CTd  2 fTd , COh  fOh , CHV  CV fV , and CHD  CD fD . These formulae are used to convert from the
difficult-to-measure chemical potential in local equilibrium to the total hydrogen concentration.
Finally, we evaluate the derivatives entering the denominator of the expression for the apparent
diffusivity, Eq. (2.5). By taking the derivative of Eq. (2.13), we obtain
 CL 
2
2
    2  fTd   fOh ,
T
where  fTd 2  fTd 2  fTd
2
(2.15)
is the variance of the site occupations. Similarly, for the trap sites
we obtain
 Ctrap 
2
2
    CV  fV  CD  fD ,
T
(2.16)
and the expression for the apparent diffusivity becomes
D
DL
.
CV  fV  C D  f D 2
2
1
(2.17)
2  fTd 2   fOh 2
We evaluated the derived expressions using the first-principles data taken from Ref. [9] and [3].
The value for hydrogen binding to non-vacancy trap sites is not known from calculations, and
therefore we use an experimental estimate of 0.273 [10]. The symmetry factors for most
configurations (except the most highly symmetric ones, such as corresponding to n=1, 2, 6 and
12 H atoms in a vacancy) are guessed with an upward bias. We checked that the results are not
particularly sensitive to the precise values of these guesses, since the energies of these
nonsymmetric configurations are far from the convex hull in the energy-versus-occupation plot
(see Fig. 2 in Ref. [9]), and therefore occur with negligibly small probabilities.
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[1]
[2]
[3]
[4]
E. Wigner, Z. Phys. Chem. B19, 203 (1932).
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C. Wolverton, V. Ozolins, and M. Asta, Phys. Rev. B 69, 144109 (2004).
B. Bhatia, and D. S. Sholl, Phys. Rev. B 72, 224302 (2005).
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[5]
[6]
[7]
[8]
[9]
[10]
P. G. Sundell, and G. Wahnstrom, Phys. Rev. B 70, 081403 (2004).
J. T. Fermann, and S. Auerbach, J. Chem. Phys. 112, 6787 (2000).
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