Grain-Boundary Diffusion of Helium in Palladium with
Submicron-Grained Structure
Zhiganov A.N., Kupryazhkin A.Ya.
Department of Molecular Physics, Urals State Technical University, Yekaterinburg, Russia
Abstract. Helium diffusion in polycrystalline palladium sample with submicron-grained structure (the grain size
is ∼150 nm) was investigated with the method of gas thermal desorption from previously saturated in gas phase sample.
The dependences of the helium solubility in the sample upon the saturation pressure have the typical shape of saturation
curve with clear expressed “plateau”. The obtained dependences of the effective helium diffusion coefficient have high
temperature (400÷508) K (1) and low temperature (293÷400) K (2) branches that can be described with
(
)
D1,2 = D0 exp − E1D,2 kT exponents. In the range of linear dependence of the helium solubility upon the saturation
pressure (2.5 bar) the low temperature branch is featured with the migration energy E 2D =(0.0036±0.0015) eV, when the
high temperature branch corresponds to the migration energy E1D =(0.33±0.03) eV. The migration energies at the
“plateau” of the dependence of the helium solubility upon the saturation pressure (20 bar) are E 2D =(0.0052±0.005) eV in
the low temperature region and E1D =(0.18±0.01) eV in the high temperature region. The helium diffusion and solubility
mechanisms are discussed.
DESCRIPTION OF THE SAMPLE AND THE MEASUREMENT PROCEDURE
The study of the gas diffusion in ultra small pores when the size of a grain boundary is comparable with the sizes
of the atoms, is interesting in connection with investigation of the gas-surface interactions. The present work studies
helium diffusion in polycrystalline palladium with the submicron size of the grains.
For the present investigations we have chosen a palladium sample of purity 99.99 % which was kindly submitted
for the researches by prof. R.R. Mulyukov. The submicron-grained structure of the sample was prepared by severe
plastic deformations to the true logarithm power e=7 by the torsion technique under the quasi-hydrostatic pressure
with a device similar to the Bridgman's anvil [1].
Investigations of the microstructure of the samples analogous to the one used in this work were performed using
the transmission electron microscope JEM 2000EX. From the transmission electron microscopy data it was
obtained, that after the severe plastic deformation the samples get highly dispersed structure saturated with
dislocations and with the average grain size of 150 nm and the width of the grain boundaries up to 0.6 nm.
The sample had shape of a plate with thickness h = (6.1±0.3) 10-3 cm, total area of the surface S = (2.5±0.1) cm2
and mass m = (90.9±0.1) mg.
The technique of experiment [2] involves saturation of a sample in the saturation chamber at desired temperature
T and saturation pressure P, tempering the dissolved helium in the sample by sharp cooling down to room
temperature, transferring the sample from the saturation chamber to the desorption chamber and measurement of the
helium desorption at the same temperature T. The measuring system is constructed on the basis of modernized mass
spectrometer MI-1201 B. To increase the helium sensitivity of the system we used a quasi-static (with respect to
helium) exhaust schedule of the mass spectrometer achieved by the use of a getter pump, which provides absorption
of all the residual gases except for helium up to 10-6 Pa [3]. The calibration of the measuring system was performed
by the method of double expansion of known quantity of helium from the calibrated chamber [3]. The time interval
of the sample saturation was determined experimentally.
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
The effective helium diffusion coefficient Deff was determined from processing of the desorption curves
according to the solution of the desorption problem for gas desorption from a saturated sample having form of an
infinite plate, into vacuum (1).
J( t ) =
8SC eff Deff ∞
h
 π 2 (2k + 1)2 D
eff
∑ exp −
k =0
h2


t 


(1)
EXPERIMENTAL RESULTS
The measurements were carried out at various saturation temperatures T and saturation pressures P. This had
allowed taking dependencies of the effective diffusion coefficient Deff and the effective solubility Ceff upon the
saturation temperature. To eliminate the influence of the high temperature annealing on the measurement results the
reiterated measurings were performed at the low temperatures. The error of determination of the coefficient Deff was
≤ 7%. The helium solubility in the sample was calculated from the data on complete desorption of the sample. The
error of determination of the solubility was ≤ 10%.
The experimental dependencies of Ceff in grain boundaries of submicron-grained palladium and Deff upon the
saturation pressure are given on Fig. 1, 2 for helium desorption from the previously saturated sample.
Two temperature branches may be observed at the curves of dependence of the diffusion coefficient (Fig. 3) and
the solubility of helium upon the saturation temperature. The high temperature branch corresponds to the
(400÷508) K region and the low temperature branch corresponds to the (293÷400) K region. The diffusion
coefficient Deff and the solubility of helium Ceff dependences upon saturation temperature follow the exponents (2).
(
exp(− E
)
kT )
Deff 1,2 = Deff 0 exp − E1D,2 kT
C eff 1,2 = C eff 0
P
1,2
(2)
The values of the parameters of the helium diffusion in the palladium sample with submicron-grained structure
are shown in Table 1.
Ceff ,
1016 cm-3
2
-1
-2
-3
-4
1
0
10
20
P, bar
FIGURE 1. Dependence of the effective helium solubility in the sample upon the pressure of saturation at the saturation
temperatures: 1 - 387 K; 2 - 403 K; 3 - 433 K; 4 - approximation.
D
-9
eff
,
2
10 cm s
-1
3
2
-1
-2
-3
-4
1
0
10
20
P, bar
FIGURE 2. Dependence of the effective coefficient of helium diffusion in the sample upon the pressure of saturation at the
saturation temperatures: 1 - 387 K; 2 - 403 K; 3 - 433 K; 4 – approximation.
ln(Deff )
500
400
300 T, K
-19
Deff , cm2s-1
-8
10
-1
-2
-3
-20
-9
10
-21
2.0
2.4
2.8
3.2 1000/T, 1/K
FIGURE 3. Dependence of the effective coefficient of helium diffusion on the sample upon the temperature of saturation at the
saturation pressures: 1 – P = 20 bar; 2 - P = 2.5 bar; 3 – approximation.
TABLE 1. The Values of Parameters of the Helium Diffusion in the Palladium Sample with Submicron-Grained
Structure
∆T, K
P, bar
293÷400
2.5
400÷508
2.5
293÷400
20
400÷508
20
Deff 0, cm2s-1
(0.98 )⋅ 10
(1.1 )⋅10
(9.3 )⋅10
(3.9 )⋅10
+1.1
−0.09
+0.9
− 0.5
+1.6
−1.4
+1.1
−0.9
ED, eV
−9
0.0036 ± 0.0015
−5
0.33 ± 0.03
−9
0.052 ± 0.005
−9
0.18 ± 0.01
The solution energies of helium were determined from processing of dependences of the helium solubility upon the
saturation
E 2S
temperature
and
are
E1S = (− 0.025 ± 0.008) eV
at
the
low
temperature
branch
and
= (0.086 ± 0.008) eV at the high temperature branch.
DISCUSSION
According to the data of the research of palladium samples prepared by the same technique as for this work, by
means of the positron lifetime measurements [4] and the magnetic susceptibility [5], these samples have sufficiently
high concentration of vacancy and vacancy clusters (up to 6-12 vacancies [5]) in comparison with the undeformed
polycrystalline samples. In accordance with the data of [5], there are two typical anneal temperatures. The annealing
at the temperature T=473 K leads to almost double increase of the grain size, because of annealing of the lineages of
the grains, and, according to [4], after the annealing at this temperature the vacancy cluster concentration decreases
down to the detection limit of the method. The second typical annealing temperature is ∼823 K. The assembling
recrystallization is observed at this temperature, and, therefore, according to [5], the vacancy concentration
decreases down to the values which are typical for the undeformed palladium polycrystalline samples.
According to our measurements, values of Deff and Ceff at temperatures up to 508 K do not depend on sample
annealing and, correspondingly, on the grain size. Therefore, it seems that helium solubility in the lineages of the
grains is negligibly small. The helium atom solution energies for palladium determined from the lattice static
approach [6] are 0.52 eV for the solution in vacancy and 3.68 eV for the solution in interstitial position. This allows
us to neglect the dissolution of helium atoms in interstitial positions of the palladium lattice (in the grain volume)
and realization of the interstitial mechanism of helium diffusion in our experiments.
The saturated positions in the palladium sample with submicron-grained structure may be positions of defectless
grain boundaries, vacancies, divacancies and vacancy clusters in the large tilt grain boundaries, which have been
created during the deformation and do not anneal at the low temperatures. The effective solubility is sum of the
helium solubilities in all the solution positions:
C eff (T , P ) = C gb + C v + C div + C nv .
(3)
Here C v , C div , C vn , C gb are the helium solubilities in vacancies, divacancies, vacancy clusters in grain boundary
and in the defectless grain boundaries of polycrystalline, respectively.
As it follows from the measurements presented in this work, helium solubility isotherms have typical shape of
the saturation curve with clear expressed “plateau” (Fig. 1). Such behavior of the effective solubility dependence
upon the saturation pressure indicates complete filling of certain traps (positions of dissolution) at the surface or the
near-surface region of grain boundaries. We can write the dependence of Ceff upon the saturation pressure and
temperature, which correspond to the curves observed in the experiment:
Ceff (P ,T ) =
4 C * Γ (T )P
∑ 1 +k Γk (T )P ,
k =1
(4)
k
Here summation is performed over all types of the solution positions (vacancies, divacancies, vacancy clusters in
grain boundary and in the defectless grain boundaries of polycrystalline), C *k is the concentration of the solution
positions of type k, Γk is constants for the solution positions of type k that does not depend on pressure.
The effective volume of the polycrystalline grain boundaries is relatively large. This is especially true for the
polycrystalline with submicron-grained structure. Because of the large values solution energy for helium atoms in
the defectless grain boundary Γgb (4) are much smaller then values of Γk corresponding to the other solution
positions. So in the whole range of the used saturation pressures, the full saturation of the defectless grain
boundaries was not achieved and the condition C *gb >> C gb (P ) was true.
From processing of experimental curves we can determine only the effective values for the case when only one
type of the solution positions dominates. Assuming this, we get the approximation of equation (4) when only one
term of the sum remains. Then, C *k =1 = C *eff = (2.0±0.2)⋅1016 cm-3 is concentration of the traps mainly saturated with
helium. The C *eff is the same for the all investigated saturation temperatures.
Because the concentrations C* of the positions saturated with helium, do not depend on temperature (see Fig. 1),
it seems that
C * ≈ C nv , which is also confirmed by the value of the helium solution energy
E1P = (− 0.025 ± 0.008) eV at the low saturation temperatures (293÷400 K). The latter may be compared with the
helium adsorption energy at solid-state surfaces [7]. At the higher saturation temperatures the total solubility of
helium is affected with the solution in simpler defects, including vacancies and/or defectless grain boundaries. The
solution energy in these defects is higher, and, therefore, the effective helium solution energy in the sample
increases to E2P = (0.086 ± 0.008) eV .
The diffusion of gas atoms can be explained by several mechanisms: the vacancy diffusion, the diffusion of the
vacancy clusters, the divacancy diffusion mechanism and, at last, the grain boundary diffusion. It seems, that the
most likely diffusion mechanisms are the monovacancy diffusion, the divacancy diffusion and the diffusion via the
defectless grain boundaries featured by the trapping of the atoms in the vacancy clusters at the large tilt boundaries.
These clusters do not anneal at the investigated temperatures [4]. Assuming realization of these mechanisms in the
experiment, we can, similarly to [8], obtain the equation for the effective diffusion coefficient from the random walk
theory:
D gb C gb + Ddiv C div + Dv C v
DC
Deff = ∑ i i ≈
.
C eff
i C eff
(5)
Here C i C eff is multiplier, which equals fraction of the time a gas atom spends in solution position of type i; Di is
the coefficient of helium diffusion via the positions of type i. In our case considerable contribution to the effective
diffusion coefficient comes from the terms D gb , which corresponds to the diffusion via the defectless grain
boundaries and Ddiv , Dv , which describe the diffusion via the divacancies and monovacancies of the near-surface
region of the grains. The value of the coefficient for diffusion via vacancy clusters in the grain boundary is
negligibly small because of the clusters has low mobility; therefore we neglect this term in the numerator of (5).
Similarly to [8], we can find, that
(
)
(
)
Ci
Ci* − Ci (P ) Li exp − EiS kT
=
,
3
C eff
S
*
(
)
−
−
C
C
P
L
exp
E
kT
∑ k k
k
k
(
k =1
)
(
)
(6)
where summation in the denominator is over all the position types (monovacancies, divacancies, vacancy clusters
and defectless grain boundaries); C k0 is concentration of the positions of type k; E kS is the energy of the helium
solution in the position of type k; Lk is multiplier which does not depend on pressure.
At low occupancy of the solution positions (at the low saturation pressure), so that C *k −C k (P ) ≈ C *k , where
C k (P ) corresponds to expression (4) for the k-type solution position, the effective diffusion coefficient Deff (Fig. 2)
does not strongly vary with saturation pressure. As the saturation temperature increases, the equation (6) predicts
transition from diffusion via divacancies at the low temperatures to the monovacancy diffusion and/or defectless
grain boundaries diffusion at the higher temperatures, which corresponds to (5) (Fig. 3, curve 2).
As the saturation pressure increases, value of denominator in (5) decreases in accordance with (6), along with the
increase of the vacancy clusters saturation. At the same time, Deff increases (Fig. 3, curve 1) and achieves the
“plateau” (Fig. 2).
The analysis of the change of the migration energy requires additional investigations.
CONCLUSION
It is shown, that the method of the helium thermal desorption from the polycrystalline samples previously
saturated in gas phase, which is used in this work, has high sensitivity, that allows determination of the
concentration of the saturated positions from helium solubility in the sample.
From the carried out analysis of solution mechanisms, it seems, that helium solution in polycrystalline palladium
with submicro-grained structure at 293÷508 K occurs mainly in vacancy clusters at the large tilt grain boundaries. It
is confirmed by the solution energy value, which may be compared with helium adsorption energy at solid-state
surfaces. At low occupancy of the solution positions (the low saturation pressures) and the low temperatures the
helium diffusion is conditioned mainly by the divacancy mechanism with trapping of the atoms in the vacancy
clusters at the grain boundaries. At the high temperatures the main mechanism is the monovacancy diffusion and/or
defectless grain boundaries diffusion with the same traps.
The filling of the traps with helium atoms as the saturation pressure increases leads to smoothing of the potential
relief for the diffusing helium atoms and to increase in the effective diffusion coefficient.
Improvement of understanding of the diffusion mechanisms on atomistic level requires additional investigations.
ACKNOWLEDGMENTS
The authors are grateful to prof. R.R. Mulyukov for the polycrystalline sample kindly submitted for the research.
REFERENCES
1.
2.
3.
4.
5.
6.
Mulyukov, R.R., and Starostenkov, M.D., Acta Met. Sinica, 13, 301-309 (2000).
Dudorov, A.G., and Kupryazhkin, A.Ya., Zhurnal Tekhnicheskoi Fiziki, 68, 85-89 (1998).
A.Ya. Kupryazhkin and A.Yu. Kurkin. Fizika Tverdogo Tela, 35, 3003-3007 (1993).
Würschum, R., Kübler, A. et. al., Ann. de Chim. - Sci. des Mat, 21, 471-482. (1996).
Rempel, A.A., Gusev, A.I. et. al. Doklady Akademii Nauk, 345, 330-333 (1995).
Wilson, W.D., and Johnson, R.A., “Rare gas in metals”, In Interatomic potentials and simulation of lattice defects. edited by
P.C. Gehlen et. al., Plenum Press, New York – London, 1972, pp. 375-385.
7. Mezhfazovaya granitsa gaz – tverdoe telo. edited by A.E. Flood, Mir, Moscow, 1970. p. 430.
8. Lidiard, A.B., Rad. Effects., 53, 133-140 (1980).