富山大学水素同位体科学研究センター研究報告 20：21-29, 2000.

論文
Pd-Pt 合金による水素吸収曲線の解析
金 瑩, 原 正憲, 渡辺国昭
富山大学水素同位体科学研究センター〒930-8555 富山市五福 3190

Analysis of Hydrogen Absorption Kinetics for Pd-Pt Alloys
Y. Jin, M. Hara, K. Watanabe
Hydrogen Isotope Res. Centr.,
Toyama Univ., Gofuku 3190, Toyama 930-8555, Japan(Received February 9, 2001; Accepted March 15, 2002)
ABSTRACT

Hydrogen absorption kinetics of palladium alloys was studied by using a vacuum microbalancesystem. The data analysis procedures were shown in detail in the present paper for a sphericalsample as an example, where the second order hydrogen adsorption and desorption were assumed tohave occurred on the sample surface. The factors that may influence hydrogen absorption curves
and kinetic parameters are discussed.

1. Introduction
Hydrogen absorption by palladium has long been the topics of a number of investigations, andsophisticated studies can be found in literature published as early as 1930s. The thermodynamic
data are well described by statistical thermodynamics and quantum mechanics [1,2]. In contrast
with the well-defined thermodynamic features, however, kinetic results from different researcherswith different experimental procedures showed little agreement [3~5]. This can be attributed tosensitive nature of kinetic processes on working conditions, because such conditions have been
proven to include a number of factors difficult to control. This is also a consequence of complex
reaction mechanisms comprising adsorption and desorption on the surface, the transport of absorbedhydrogen atoms between the surface and the bulk layer just beneath the surface, and hydrogen
diffusion into/from the bulk.
To avoid the complexity, very thin films (below 3000Å) with clean surfaces were used as
specimen for studying the reaction kinetics and mechanisms [3,4,6]. Recent efforts on Pd-H
interaction studies have mostly devoted to the surface physics on well-defined surfaces of only
several layers thickness. Such studies are surely prospective in obtaining information about thesubtle surface processes and diffusion in the bulk, revealing the reaction mechanism, and specifying
the rate controlling step in order to predict the dependence of the reaction rate on the key
parameters such as temperature, hydrogen pressure, and surface and bulk conditions of the metal aswell as its hydrogenization history.

21

Y. Jin, M. Hara, K. Watanabe

They are, however, still far from being able to provide significant help for prediction of themacroscopic kinetic features for technological applications, because Pd and its alloys are mostly
used in the form of powder with partly poisoned surfaces. In the case of gas chromatography forhydrogen isotope separation, for example, the absorption and desorption take place throughcontaminated powder surfaces in a flow of carrier gas. To describe the separation efficiency, one has
to know the absorption and desorption kinetics of hydrogen isotopes under such operation
conditions.
The purpose of our effort is to undertake an investigation on the hydrogen absorbing behavior of
Pd-Pt alloys, especially those in the form of powder, and to find a set of kinetic parametersdescribing the absorption and desorption kinetics of hydrogen isotopes for Pd and its alloys informing α-phase solution. The absorption of hydrogen isotopes was measured by using a vacuum
microbalance in the range of temperature from 0 to 80℃ and of hydrogen pressure from 0.1 to 10
Torr. The specimens were filings of alloy ingots below 200 meshes. Details of experimental
procedures and principal results were described in a separate paper [7]. The present paper focuses
an attention to data analysis and factors requiring special cautions for the analysis.
2. Simulation principle and procedures
2.1 Bulk region 1<i<imax
The specimen powder was assumed spherical. The sample sphere was separated into imax layers
along the radius direction with dx, the thickness of each layer. Inside each layer, the hydrogenconcentration was supposed to be uniform. The surface corresponds to the imax-th layer (Fig.1).
The diffusion in the bulk was described by
the Fick’s second law as∂c∂ 2 c 2 ∂c
= D( 2 +
)
∂t
x ∂x
∂x

a

(1)

or

dx

∂u ∂ 2u
(2)
=
2
∂
T
∂
X
i=imax+1
where c is the hydrogen concentration in
Surfacecc0
atomic fraction [H/M], x the distance from theSurface layer, i=imax
center [cm], a the radius of sphere [cm], D the
cimax
c1
diffucion constant [cm2/sec], t the time [sec],
Fig.1. The schematic diagram of simulation model.
X=x/a, u=cx=caX and T=Dt/a2. Similar to thederivation of plane samples [8], the above equation can be written in difference scheme as− rui −1, j +1 + (2 + 2r )ui , j +1 − rui +1, j +1 = rui −1, j + (2 − 2r )ui , j + rui +1, j(3)

where, r=dT/(dX)2 and the total error is O[dT+(dX)2]. In order to solve Eq.3, two boundary
conditions should be defined, which are explicated separately in the following sections, along with22

Analysis of hydrogen absorption kinetics for Pd-Pt alloys

the initial condition that the hydrogen concentration is zero throughout the specimen.
2.2 Surface, i=imax
Similar to Ali-Khan’s approach [9], which has already been widely accepted in analyzinghydrogen permeation through membrane, hydrogen transport between the surface and the bulk justbeneath the surface was ignored in the present analysis. Namely the surface concentration is
described by the hydrogen concentration in the outermost bulk layer, which is assumed to be in a
local equilibrium with the surface concentration. Accordingly, the absorption and desorption rates
are described by a pseudo rate constants.
The flux of hydrogen flowing through the sample surface at any moment can be presented by the
Fick’s first law. In the case of the second order adsorption and desorption, it is given by−D∂c= k1 p (c* − c) 2 − k1r c 2∂x[cm·s-1](4)

where p is the partial pressure of hydrogen. The pseudo rate constants k1 and k1r are in [cm·Torr-1s-1] and [cm · s-1], respectively. c* and c (all in [atomic fraction of H to Pd]) are theconcentrations of total available active sites for hydrogen adsorption and those sites alreadyoccupied by H atoms in the surface layer, respectively, and the former was assumed to be 1.0.
This equation can be described in a difference form as
ci max +1, j = ci max, j +dx
2[k1 p(1 − ci max, j ) 2 − k1r ci max, j ]D(5)The concentration c was converted into u (c=u/x) for sphere model, where ci was defined as thehydrogen concentration in i-th layer, which is given by the average concentration at (i-1/2)dx asshown in Fig.1, thus cimax=uimax/(a-dx/2), cimax+1=uimax+1/(a+dx/2). Non-dimensionalization of thespace coordinate by x=aX gives
1 + dX / 2
dX (1 + dX / 2)2
(6)
k1 p a (1 − dX / 2) − u i max, j − k1r u i2max, j
u i max +1, j =u i max, j +2
1 − dX / 2
D (1 − dX / 2)

{ [

]

}

Similarly, on account of the relationship between uimax+1,j+1 and uimax,j+1 for time (j+1), thenon-dimensional equation for the surface is described as(2 + r −{[rdX
r dX (1 + dX / 2)⋅
)u i max, j +1 − r (u i max −1, j +1 + u i max −1, j ) −1 − dX / 2
D(1 − dX / 2) 2

k1 p a (1 − dX / 2) − u i max, j +1
= (2 − r +] + [a(1 − dX / 2) − u2]}2i max, jrdX
r dX (1 + dX / 2)
⋅ k1r (u i2max, j +1 + u i2max, j ))u i max, j −2
1 − dX / 2
D(1 − dX / 2)

2.3 Center, i=1
At center of the sample, the boundary condition is written in terms of symmetric property as

23

(7)

Y. Jin, M. Hara, K. Watanabe

c1, j = c0, j(8)

at any time, j. Therefore,u1, jadX / 2=

u 0, j− adX / 2 ,∴u0, j = −u1, j(9)This relation is valid irrespective of sample shapes (spherical, cylindrical or plane). Substitution of
the above equation and a similar equation for the time, j+1, into the difference equation of thesecond law (Eq.3) at i=1 leads to the non-dimensional boundary condition at the center as(2 + 3r )u1, j +1 = r (u2, j + u2, j +1 ) + (2 − 3r )u1, j(10)

3. Analysis of experimental curves
During an experimental run, the weight gain of the sample was recorded with a given sampling

frequency by microbalance controller. Then one can obtain a weight gain vs time curve by exposing
the sample to hydrogen atmosphere. The weight gains of the sample were converted to hydrogen
atom concentrations in atomic fraction to metal atoms.
If the used sample is extremely thin or small so that the sample is only made from a monolayer ofatoms with a thickness equal to λ, the hydrogen adsorption rate observed in experiment can be
described as Eq.11,

λ∂c= k1 p (1 − c) 2 − k1r c 2∂t[cm·s-1] (11)However, for a sample with finite size, the obtainable experimental data from microbalance is theaveraged weight gain of the whole metal sample through hydrogen absorption, i.e., the averagehydrogen concentration (the atomic fraction to metal atom) and not the surface information. This isespecially important to analyze the very initial stage of absorption, because hydrogen should beabsorbed only in the surface region of the sample. Then some correction is required for applying thesurface boundary condition. The correction adopted here was based on a geometrical consideration.
Accordingly the surface boundary condition was expressed by using the apparent hydrogenconcentration as
∂capp
∂t

=
t →0

A ∂c
⋅λ
V
∂t t → 0[s-1] (12)where A and V are the surface area and the volume of the sample, respectively. The apparent
hydrogen ab/desorption rate constants, kapp ([Torr-1s-1]) and kapp,r ([s-1]), could be directly
determined from experimental results by Eq.13 and 14 as described elsewhere [7].

24Analysis of hydrogen absorption kinetics for Pd-Pt alloys

∂capp
∂t

= k app p

(13)

t →0

k app,r = k app p / ce2(ce << c* )(14)where ce is the equilibrium hydrogen concentration. From Eq.12~14, it is clear that the relation between
apparent rate constants and real rate constants are

k1 =V
k app ,A

k1r =V
k app , rA

(15)

These relations do not refer to the sample shape (Eq.11~15) and can be applied to any shape ofsamples. The values of V/A are a/3, a/2, a for spherical, cylindrical and plane samples, respectively,
where a is the radius of sphere, half thickness of the wire or the foil samples, respectively. The
non-dimensionalized difference schemes for the boundary conditions of plane and cylindricalsamples are omitted in this paper. One can refer to Crank's book for diffusion equations in bulk [8].
Data analyses were carried out as follows. First, the apparent hydrogen ab/desorption rate
constants were estimated from the initial hydrogen adsorption rate and the equilibrium state (Eq.13,
14) of an experimental curve. Hydrogen absorption measurements were carried out at several
different hydrogen pressures under each temperature to determine reliable apparent hydrogenab/desorption rate constants. Subsequently, the hydrogen ab/desorption rate constants used in

Atomic fractionｃ ([H]/[M])simulation (k1, k1r) were calculated through Eq.15, adopting V/A=a/3 for spherical particles. During
simulation, the hydrogen ab/desorption rate constants were regulated in small extent to reproduce
the experimental result better. Accordingly, the above boundary conditions, Eq.7 and Eq.10, can besolved together with the equations from Eq.3
(2≤i≤imax-1) as coupled equations, in which the
0.006
10Torrdiffusion coefficient of hydrogen inside the0.005
bulk was treated as a fitting parameter to
0.004
5Torr
reproduce the experimental results.
0.003
2Torr
The original computation outputs are
0.002
described in u-T-X space. Then, u, T and X
1Torr
0.001
were converted back to hydrogen concentration
0in atomic fraction, real time and real distance,
0
200
400
600
800
1000
respectively, to obtain the results in c-t-x space,
Time /sshowing the time dependence of hydrogen
Fig.2. Time-course of protium uptake under different
concentration distribution in the sample. The
hydrogen partial pressures at 312K by Pd-4%Pt (points)
and the corresponding computer simulation (lines). kapp,average H-concentration of the sample at j
kapp,r and D are valued to be 5.75×10-6 [Torr-1s-1], 1.36
[s-1] and 4.6×10-9 [cm2s-1], respectively.

proportion of mesh i in the space coordinate.
25

moment isi max∑ vi ci , j , where vi is the volumei =1

Y. Jin, M. Hara, K. Watanabe

Figure 2 is an example of the simulations (lines) with kapp, kapp,r and D equaling to 5.75×10-6[Torr-1s-1], 1.36 [s-1] and 4.6×10-9 [cm2s-1], respectively, together with the corresponding
experimental curves (points) of protium uptake under different pressures at 312K by Pd-4%Pt. It
shows that the present hydrogen absorption model and simulation could describe well the hydrogen
absorption process and determine the hydrogen ab/desorption rate constants and diffusion
coefficient as well as their activation energies by examining the temperature dependences.
4. Factors requiring cautions
Under a certain experimental condition, hydrogen absorption by a sample might be controlled bysurface ab/desorption processes, while hydrogen transfer in the bulk is fast enough to keephydrogen concentration at the center the same as that on the surface at any moment. Under someother conditions, hydrogen absorption is controlled by both surface and bulk processes. In the thirdcase, mass transfer in the bulk is the rate-limiting step of hydrogen absorption, i.e., the surface
concentration of hydrogen reaches an equilibrium value instantaneously. In the first case, no exact
information on the diffusion can be obtained. The absorption process is relatively easy to beunderstood, and the apparent hydrogen ab/desorption rate constants and the corresponding
activation energies can be estimated. In the second case, the diffusion coefficient of hydrogen atomin the sample could be evaluated by the simulation presented in this paper. In the third case, one willfind that the hydrogen absorption curve is not sensitive to the surface ab/desorption rate constants;
instead the diffusion coefficient of hydrogen in the metal can be well defined.
Besides the above situations, there exist some factors that should be avoided in experiments andsimulation or be kept in mind when comparing the kinetic data from different researchers andexperimental conditions. These factors are 1) inadequate selection of geometrical factor, 2) too
small imax and jmax value for sectioning, 3) surface contamination, 4) too slow hydrogen diffusion ingas phase, 5) insufficient system sensitivity of detecting absorption rate and amount. All of thesefactors can lead to deviation or error in the estimated diffusion coefficients, hydrogen ab/desorption
rate constants or even misunderstanding the mechanism. In the following sections, the effects ofgeometrical factors and surface contamination are discussed respectively.
4.1 Geometrical factors
1.) Sample shape
The apparent rate constants are given as

k app =A
k1 ,V

k app , r =

A
k1r
V

(16)

where the real rate constants k1 and k1r are independent of sample geometrical feature. While, it isnoted from the above equation that the apparent rate constants are proportional to the26Analysis of hydrogen absorption kinetics for Pd-Pt alloys

surface/volume ratio of the sample. As mentioned
above, the apparent hydrogen ad/desorption rate1
0.8
2
constants are 3/a, 2/a and 1/a times of real rate3
0.6
constants for spherical, cylindrical and flat
sample, respectively. Then there will be 50%
0.4
1__spherical sample
deviation at most for evaluating the rate constants
2__cylindrical sample
0.2
3__plane sample
if experimental data are analyzed by assuming0
sphere for the sample powder consisting of
0
100
200
300
400
500
cylinder-like particles. Figure 3 shows theTime /s
Fig.3. Hydrogen absorption simulations of spherical,
simulations of time dependences of hydrogen
cylindrical and plane samples with a values all equal
absorption reacted fraction by spherical,
to 0.005cm, in which k1 and k1r are valued to becylindrical and plane samples with a all equal to
7.18×10-8 [cm · Torr-1s-1] and 3.68×10-3 [cm · s-1],
respectively.
0.005cm, in which k1 and k1r are valued to be
-8
-1 -1
-3
-1
7.18×10 [cm·Torr s ] and 3.68×10 [cm·s ], respectively, where the hydrogen absorptions arelimited by surface reactions.
Reacted fraction C / Ce

12.) Particle size distribution
For samples in different size, one may observe quite different hydrogen absorption curves. The
difference is not only caused by different apparent hydrogen ab/desorption rate constants predicted
by Eq.16, but also by the change in the rate-limiting step with sample size.
The effect of different average sample size is easy to be noticed. However, concerning the particle
sample, there exist some particles apart from average size. Thus the possible differences amonghydrogen absorptions will be discussed for the samples with different size distributions.
1

4000

1 2

1
2
3
4

600
500
400

Reacted fraction c/ce

Count Distribution

3000

0.00010.00080.0018
0.005

300
200

0.8

3 4

0.6

0.4

0.2

100
0
0.000

0
0.002

0.004

0.006

0.008

0.010

0

Radius of the particles / cm

100

200

300

400

500

Time /s
Fig.5. Hydrogen absorption simulations of thesamples with the size distributions shown in Fig.4,
in which k1 and k1r are valued to be 7.18×10-8 [cm·Torr-1s-1] and 3.68×10-3 [cm·s-1], respectively.

Fig.4. Four normal distributions N(μ, σ 2) ofparticle size with same average value and different
scatter coefficients. μ=0.005cm and σ=0.0001,
0.0008, 0.0018 and 0.005 as shown in the graph,
respectively.

27

Y. Jin, M. Hara, K. Watanabe

The curves in Fig.4 indicated four different particle size distributions with exactly the same
average particle size, and all the four cases show normal distributions in size with different standard
deviations. Simulations of the time dependences of hydrogen absorption reacted fraction by thesesamples of a given amount were instructed in Fig.5, in which k1 and k1r are also valued to be
7.18×10-8 [cm·Torr-1s-1] and 3.68×10-3 [cm·s-1], respectively, where the hydrogen absorptions are
all limited by surface reactions. It is seen that the sample with the most uniform particle size shows
the highest apparent hydrogen absorption rate constant than those samples with rather scattered
distributions. In this present example, the highest apparent absorption rate is because of the largest
specific geometrical surface area of sample 1. However, for diffusion controlled hydrogen
absorption, the hydrogen diffusion inside the particles with different size causes the different
apparent hydrogen absorption rates.
It should be noted, however, that as for the temperature dependence of the apparent rate constants
the geometrical factors thus considered only affect the pre-exponential factors and the activationenergies do not depend on the geometrical factors being inconsistent with the real shape and size
distribution of samples.

Atomic fraction c ([H]/[M])4.2 Surface poisoning
Besides the above problems, the same sample can change its apparent rate constants because of
the reduction in surface cleanness during a series of experiments. In some cases, the absorption
becomes to be totally limited by the surface reactions. Among the influencing factors discussed in
the present paper, the cleanness factor f (i.e., the surface coverage of active sites) is the one the mostdifficult factor to be controlled in kinetic
studies.
0.002
Figure 6 shows an example of this problem

observed for hydrogen absorption by Pd-8at%Pt
alloy exposed in 1 Torr H2 at 293K. In the seriesof experiments, the absorption rate decreased
step by step by repeating the measurementswith the same sample. Since no particularsurface cleaning treatment was executed
between the experimental runs, the reduction in
the absorption rate can be attributed to surface
poisoning by impurities in the residual gas ofthe vacuum system and/or in the used H2-Ar
mixture gas. It was found that the hydrogen
absorption was controlled by surface reactions
in all the cases, and the ratio among theapparent rate constants were 1.0, 0.50, 0.40,

0.00150.001

0.0005

50

100

150

200

250

300

Time /s
Fig.6. Experimental and simulation results of hydrogen
absorption by Pd-8%Pt alloy at 293K, 1Torr H2. Points
are experimental results and lines are simulations.
Diffusion is fast enough to keep the uniform H
concentration inside the alloy particle. k1 and k1r are
valued to be 3.25×10-8 [cm·Torr-1s-1] and 7.78×10-3[cm·s-1], respectively. The ratio of f between the fivecases are determined to be 1.0:0.50:0.40:0.28:0.19
from upside down.

28Analysis of hydrogen absorption kinetics for Pd-Pt alloys

0.28, 0.19 for the absorption curves from the top to the bottom. Data at different temperatures gave
no clear straight lines for them, indicating both the pre-exponential factor and the activation energy
cannot be obtained from the samples with varying cleanliness factor in a series of experiments; onecan predict no absorption curves under any experimental conditions in this case.
5. Conclusions
Analyses of hydrogen absorption curves were described in detail as well as discussion of thepossible influences such as geometrical factors and surface conditions of the sample that may cause
deviation in evaluating hydrogen ab/desorption rate constants and activation energies. By using asample with well-defined shape (plane, cylindrical or spherical), one can estimate the hydrogenab/desorption rate constants, diffusion coefficient of hydrogen in the sample and theircorresponding activation energies. As for the isotope effect on the absorption, one can obtain thereliable parameters in terms of rate constant and diffusion coefficient ratios at given temperature
under given experimental conditions.

Acknowledgements
This work was partly supported by a Grand-in-Aid for Scientific Research from the Ministry of
Education, Science and Culture of Japan.

References[1] H. Noh, T. B. Flanagan, T. Sonoda and Y. Sakamoto, J. Alloys and Compounds, 228 (1995) 164.
[2] T. Yasumatsu, J.L. Wan, M Matsuyama, K. Watanabe, J. Alloys and Comp., 293/295 (1999) 900.
[3] R.V. Bucur, V. Mecea, T.B. Flanagan, Surface Science, 54 (1976) 477.
[4] R.V. Bucur, V. Mecea, Surface Technology, 11 (1980) 305.
[5] W. Auer, H.J.Grabke, Ber. Bunsenges. Physik. Chem., 75 (1971) 1130; 78 (1974) 58.
[6] Y. Li, Y. Ts. Cheng, Int. J. Hydrogen Energy, 21/4 (1996) 281.
[7] J.L. Wan, M. Hara, Y. Jin, K. Watanabe, Ann. Rept. HRC, 19 (1999) 9.
[8] J. Crank, The Mathematics of Diffusion (Sec. Ed.), Oxford University Press, 1975.
[9] I. Ali-Khan, K. J. Dietz, F. G. Waelbroeck, P. Wienhold, J. Nucl. Mater., 76-77(1978) 337.

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