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Comments on “a mathematical model describing the effects of microvoids upon the diffusion of hydrogen in iron and steel”
Scripta
METALLURGICA
Vol. 8, pp. 7 6 3 - 7 6 8 , P r i n t e d in the U n i t e d
1974 States
Pergamon
Press,
Inc.
COMMENTS ON "A MATHEMATICAL MODEL DESCRIBING THE ~I~'~'~:CTSOF MICROVOIDS UIDON THE DIFFU~!ON OF HYDROGEN IN IRON AND STEEL"*
W. J. Kass Sandia Laboratories, Albuquerque, New Mexico 87115
(Received
April
8,
1974)
Recently, Alien-Booth and Hewitt (i) have proposed a model describing the effect of microvoids on hydrogen diffusion in iron and steel. trapped in voids in the metal. the lattice.
The basis of the model is that hydrogen gas is
This gas is assumed to be in local equilibrium with hydrogen in
After some discussion concerning the nonideality of hydrogen gas and the varia-
tion of the void volume as a function of internal hydrogen pressure~ they obtain the relationship
fm~
p
-V0
,,here P is the hydrogen pressure in the voids, Hx is concentration of hydrogen in the voids, and V 0 is the void volume at zero pressure.
The total hydrogen concentration is H' = H + H x,
where H is the hydrogen concentration in the lattice.
The diffusion equation for this situa-
tion has been derived by McNabb and Foster (2) and is Eq. (7) in Ref. (I).
1
This equation may be reduced to
D
(8)
~t
if the simplifying assumption is made that HX/H = ~ (a constant). diction of the equilibrium equation governing this process.
Supported by the U.S. Atomic Energy Commissions
763
This assumption is a contra-
The model described by Allen-Booth
764
C O M M E N T S ON " E F F E C T S OF M I C R O V O I D S U P O N D I F F U S I O N OF H IN Fe A N D STEEL"
and Hewitt (i) corresponds to linear absorption into traps.
Vol.
8, No.
7
This type of absorption would be
that characteristic of au atomic hydrogen trap such as described by McNabb and Foster (2) in the limit of infinite dilution, where the fraction of occupied trapping sites is small. The process by which hydrogen in the lattice tends toward equilibrit~nwith hydrogen gas in a void should be expressed as:
2H~H~
(gas)
.
(I)
Z2 At equilibrium
where it should be mentioned again that H refers to the hydrogen concentration "-'(atoms/~n3). It is immediately recognized that this is the reverse of Sievert's law, which was the means by which hydrogen was originally dissolved in the metal. stant°
S is the Sievert's solubility con-
The hydrogen concentration in the voids in atoms/cm j is given by
z~° =
RT
ki~ - ½
zv°
zv°
~
""~"-
RT
s2
where Z = 2 hydrogen atoms/hydrogen molecule.
(m) '
The lattice hydrogen concentration may also be
expressed as a function of the hydrogen pressure in a void°
H = s,/~
.
(~)
The diffusion equation is
(
•~ - ~
H + -~-
9
= V2H
The diffusion equation may be cast in simpler form by making the substitutions
ZVoP0
ZVoC 0 u = H/C 0 and RT
S2
-
RT C O
where a is the ratio of the hydrogen concentration in voids to the concentration in the lattice,
P0 is the reference external pressure, and CO = S ~ P O is a reference concentration whlchwould be the concentration just inside the upstream surface for the case of diffusion in a slab or
Vol.
8, No.
7
C O M M E N T S ON " E F F E C T S OF M I C R O V O I D S UPON D I F F U S I O N OF H IN Fe A N D S T E E L "
the initial uniform saturation concentration in a desorption experiment°
765
The differential
equation for diffusion is now
V2u= ~1 ~ (u+~u 2)
(v)
.
The important differences between Eqs° (8) and (V) have been pointed out by Crank (3)° tion (8) behaves as if D were replaced by D/I+~.
Equa-
This implies that the shape of the permea-
tion or desorption experimental curves versus time is not changed but is simply scaled in time.
Time scaling alone would allow experimental desorption data plotted versus log t to be
fit to the theoretical curve by a shift of the abscissa, corresponding to a change in the t scaling factor.
However~ Foster, McNabb, and Payne (4) have analyzed desorption data using
their (2) trapping model and have found more than just choosing an appropriate time scaling factor necessary to fit the data. Equation (V) is fundamentally different from Eqo (8) and the solutions of Eq. (5) must be obtained by numerical methods °
Figure 1 shows the flux of hydrogen atoms through a slab,
for several values of a, for the permeation boundary and initial conditions below: H:
0
H = C0 H=0
O
t=
0
X : a~ H=O ~
t>O
It can be seen that, as ~ increases, the shape of the transient flux deviates more and more from the shape expected for pure interstitial diffusion (a = 0).
The range of applicability
of this model may be obtained by using the method of McNabb and Foster (2) to determine the permeation time lag t i (5)° t i = ~ a2 where a is the sample thickness.
1 +
,
(VI)
i D If we use a 20% deviation of t i from its normal value of [ -~
as a criterion of deviation from normal diffusion behavior, ~ must be greater than 0.4. Figure 2 shows log H plotted versus IOO0/TK for hydrogen in the lattice and in voids°
a For
temperatures in which 0°4 H < H x, a will be nonnegligible and the normal diffusion behavior will be perturbed.
It can be seen from Armbruster's
(6) solubility data for hydrogen in iron
that this occurs at approximately 100°C for a void volume of l0 -3 cm3/cm 3 metal and a saturation pressure of 1 arm.
This void volume has been easily obtained in cold worked steels (7).
Also shown in Fig. 2 are data for hydrogen solubility in gold (8), copper (9), and an austenitic stainless steel (lO).
It is seen that trapping in voids will be important in gold and
copper until relatively high temperatures are reached. in both these systems (ll,12).
Anomalous effects have been observed
For the austenitic steels, however, void trapping will not be
important but atomic hydrogen traps (2) may be the source of abnormal diffusion behavior.
766
COMMENTS ON "EFFECTS OF M I C R O V O I D S UPON D I F F U S I O N OF H IN Fe AND STEEL"
Vol.
I. 000
Jo 0. I00
Jlo
- O [cq (HICo)~ \(3 (x/a) /L
(a : I00)
o.J
O. 0
0
1 O. 10
O. O1
~ 1. O0
10. O0
Dt/a 2 FIG. 1 Hydrogen flux (J) through a permeation sample as a function of time. Jo is the normal flux in the absence of any trapping.
1020
800
400
T°C 200
100
25
O
I
I
]
I
I
I
H/3%S S. S
.
~
-- - - -
i019 1018
~ lO17
x •
/
Vo : 10-3 cm31cm3 , Po = IATM
1016 I
/
1015
x
~
/
('
~,Vo : 10 -4 cm31cm 3 ~,,~( f < --
\
H
N Po : I A I M "
1014 1013
i
i
1
I
I
2 IO00/TK
i
I
3
i
I
4
FIG. 2 Lattice hydrogen concentration (H) and hydrogen in the gas phase (Hx) as a function of temperature. H is at a pressure of 1 arm.
8, No.
7
Vo ]~
8,
X'o.
7
COMMENTS ON "EFFECTS OF i~IICROVOIDS UPON DIFFUSION OF H rN Fe AND STEEl,"
767
In conclusion, gas phase hydrogen trapped in voids may be an Janportant contributor to anomalous behavior in a wide variety of metal/hydrogen systems, especially in systems where the hydrogen solubility is lowo
However, the equilibrium law which determines hydrogen pres-
sure in internal voids must be the same as that one which determines hydrogen solubility in the lattice.
Since the model presented by Allen-Booth and Hewitt (2) is actually an atomic
trapping model but leads to the same conclusions about the temperature dependence of the diffusion coefficient as does the void (molecular) trapping model, it will be necessary to apply both models to transient diffusion behavior to deter~aine the applicability of either. References io
D. Mo Allen-Booth and Jo Hewitt~ Acts. Met. 22, 171 (1974)o
2.
A. McNabb and P. K. Foster, TranSo Met. Soc. AIME 227, 618 (1963).
3.
J. Crank, The Mathematics of Diffusion, ist. Ed., Chapto 8, Oxford University Press (1956).
4.
P. K. Foster, A. McNabb, and C. M. Payne, Trans. Met. Soc. AIME 233, 1022 (1965).
5o
See for exsanple, Wo Jost, Diffusion in Solids, Liquids, and Gases, p. 44, Academic Press, New York (1960).
6.
Marion H. Armbruster, J. Am° Chem. Soco 65, 1043 (1943)o
7.
G.M.
8.
R. B. McLellan, Jo Phys. Chem. Solids 34, i137 (1973).
9.
W. Eichenauer and Ao Pebler, Z. Metallkunde 48, 373 (1957).
Evans and E. Co Rollason, J. Iron and Steel Inst. 207, 1591 (1969).
i0.
Wo J. Kass and Wo Jo Andrzejewski, The Permeation of Hydrogen and Deuterium in 309S Stainless Steel, Sandia Laboratories Development Report, SC-DR-720136, Sandia Laboratories, Albuquerque, New Mexico (April, 1972).
Iio
Go Ro Caskey, Jr. and W. L. l>illinger, Effects of Trapping on Hydrogen Permeation in Copper, SRL DP-MB-73-53, du Pont de Nemours (E.I.) & Co., Aiken, South Carolina (1973)o
12.
Unpublished data on hydrogen and deuterium diffusion in a Au 20~0 Ag alloy by Wo Jo Kasso
METALLURGICA
Vol. 8, pp. 7 6 3 - 7 6 8 , P r i n t e d in the U n i t e d
1974 States
Pergamon
Press,
Inc.
COMMENTS ON "A MATHEMATICAL MODEL DESCRIBING THE ~I~'~'~:CTSOF MICROVOIDS UIDON THE DIFFU~!ON OF HYDROGEN IN IRON AND STEEL"*
W. J. Kass Sandia Laboratories, Albuquerque, New Mexico 87115
(Received
April
8,
1974)
Recently, Alien-Booth and Hewitt (i) have proposed a model describing the effect of microvoids on hydrogen diffusion in iron and steel. trapped in voids in the metal. the lattice.
The basis of the model is that hydrogen gas is
This gas is assumed to be in local equilibrium with hydrogen in
After some discussion concerning the nonideality of hydrogen gas and the varia-
tion of the void volume as a function of internal hydrogen pressure~ they obtain the relationship
fm~
p
-V0
,,here P is the hydrogen pressure in the voids, Hx is concentration of hydrogen in the voids, and V 0 is the void volume at zero pressure.
The total hydrogen concentration is H' = H + H x,
where H is the hydrogen concentration in the lattice.
The diffusion equation for this situa-
tion has been derived by McNabb and Foster (2) and is Eq. (7) in Ref. (I).
1
This equation may be reduced to
D
(8)
~t
if the simplifying assumption is made that HX/H = ~ (a constant). diction of the equilibrium equation governing this process.
Supported by the U.S. Atomic Energy Commissions
763
This assumption is a contra-
The model described by Allen-Booth
764
C O M M E N T S ON " E F F E C T S OF M I C R O V O I D S U P O N D I F F U S I O N OF H IN Fe A N D STEEL"
and Hewitt (i) corresponds to linear absorption into traps.
Vol.
8, No.
7
This type of absorption would be
that characteristic of au atomic hydrogen trap such as described by McNabb and Foster (2) in the limit of infinite dilution, where the fraction of occupied trapping sites is small. The process by which hydrogen in the lattice tends toward equilibrit~nwith hydrogen gas in a void should be expressed as:
2H~H~
(gas)
.
(I)
Z2 At equilibrium
where it should be mentioned again that H refers to the hydrogen concentration "-'(atoms/~n3). It is immediately recognized that this is the reverse of Sievert's law, which was the means by which hydrogen was originally dissolved in the metal. stant°
S is the Sievert's solubility con-
The hydrogen concentration in the voids in atoms/cm j is given by
z~° =
RT
ki~ - ½
zv°
zv°
~
""~"-
RT
s2
where Z = 2 hydrogen atoms/hydrogen molecule.
(m) '
The lattice hydrogen concentration may also be
expressed as a function of the hydrogen pressure in a void°
H = s,/~
.
(~)
The diffusion equation is
(
•~ - ~
H + -~-
9
= V2H
The diffusion equation may be cast in simpler form by making the substitutions
ZVoP0
ZVoC 0 u = H/C 0 and RT
S2
-
RT C O
where a is the ratio of the hydrogen concentration in voids to the concentration in the lattice,
P0 is the reference external pressure, and CO = S ~ P O is a reference concentration whlchwould be the concentration just inside the upstream surface for the case of diffusion in a slab or
Vol.
8, No.
7
C O M M E N T S ON " E F F E C T S OF M I C R O V O I D S UPON D I F F U S I O N OF H IN Fe A N D S T E E L "
the initial uniform saturation concentration in a desorption experiment°
765
The differential
equation for diffusion is now
V2u= ~1 ~ (u+~u 2)
(v)
.
The important differences between Eqs° (8) and (V) have been pointed out by Crank (3)° tion (8) behaves as if D were replaced by D/I+~.
Equa-
This implies that the shape of the permea-
tion or desorption experimental curves versus time is not changed but is simply scaled in time.
Time scaling alone would allow experimental desorption data plotted versus log t to be
fit to the theoretical curve by a shift of the abscissa, corresponding to a change in the t scaling factor.
However~ Foster, McNabb, and Payne (4) have analyzed desorption data using
their (2) trapping model and have found more than just choosing an appropriate time scaling factor necessary to fit the data. Equation (V) is fundamentally different from Eqo (8) and the solutions of Eq. (5) must be obtained by numerical methods °
Figure 1 shows the flux of hydrogen atoms through a slab,
for several values of a, for the permeation boundary and initial conditions below: H:
0
H = C0 H=0
O
t=
0
X : a~ H=O ~
t>O
It can be seen that, as ~ increases, the shape of the transient flux deviates more and more from the shape expected for pure interstitial diffusion (a = 0).
The range of applicability
of this model may be obtained by using the method of McNabb and Foster (2) to determine the permeation time lag t i (5)° t i = ~ a2 where a is the sample thickness.
1 +
,
(VI)
i D If we use a 20% deviation of t i from its normal value of [ -~
as a criterion of deviation from normal diffusion behavior, ~ must be greater than 0.4. Figure 2 shows log H plotted versus IOO0/TK for hydrogen in the lattice and in voids°
a For
temperatures in which 0°4 H < H x, a will be nonnegligible and the normal diffusion behavior will be perturbed.
It can be seen from Armbruster's
(6) solubility data for hydrogen in iron
that this occurs at approximately 100°C for a void volume of l0 -3 cm3/cm 3 metal and a saturation pressure of 1 arm.
This void volume has been easily obtained in cold worked steels (7).
Also shown in Fig. 2 are data for hydrogen solubility in gold (8), copper (9), and an austenitic stainless steel (lO).
It is seen that trapping in voids will be important in gold and
copper until relatively high temperatures are reached. in both these systems (ll,12).
Anomalous effects have been observed
For the austenitic steels, however, void trapping will not be
important but atomic hydrogen traps (2) may be the source of abnormal diffusion behavior.
766
COMMENTS ON "EFFECTS OF M I C R O V O I D S UPON D I F F U S I O N OF H IN Fe AND STEEL"
Vol.
I. 000
Jo 0. I00
Jlo
- O [cq (HICo)~ \(3 (x/a) /L
(a : I00)
o.J
O. 0
0
1 O. 10
O. O1
~ 1. O0
10. O0
Dt/a 2 FIG. 1 Hydrogen flux (J) through a permeation sample as a function of time. Jo is the normal flux in the absence of any trapping.
1020
800
400
T°C 200
100
25
O
I
I
]
I
I
I
H/3%S S. S
.
~
-- - - -
i019 1018
~ lO17
x •
/
Vo : 10-3 cm31cm3 , Po = IATM
1016 I
/
1015
x
~
/
('
~,Vo : 10 -4 cm31cm 3 ~,,~( f < --
\
H
N Po : I A I M "
1014 1013
i
i
1
I
I
2 IO00/TK
i
I
3
i
I
4
FIG. 2 Lattice hydrogen concentration (H) and hydrogen in the gas phase (Hx) as a function of temperature. H is at a pressure of 1 arm.
8, No.
7
Vo ]~
8,
X'o.
7
COMMENTS ON "EFFECTS OF i~IICROVOIDS UPON DIFFUSION OF H rN Fe AND STEEl,"
767
In conclusion, gas phase hydrogen trapped in voids may be an Janportant contributor to anomalous behavior in a wide variety of metal/hydrogen systems, especially in systems where the hydrogen solubility is lowo
However, the equilibrium law which determines hydrogen pres-
sure in internal voids must be the same as that one which determines hydrogen solubility in the lattice.
Since the model presented by Allen-Booth and Hewitt (2) is actually an atomic
trapping model but leads to the same conclusions about the temperature dependence of the diffusion coefficient as does the void (molecular) trapping model, it will be necessary to apply both models to transient diffusion behavior to deter~aine the applicability of either. References io
D. Mo Allen-Booth and Jo Hewitt~ Acts. Met. 22, 171 (1974)o
2.
A. McNabb and P. K. Foster, TranSo Met. Soc. AIME 227, 618 (1963).
3.
J. Crank, The Mathematics of Diffusion, ist. Ed., Chapto 8, Oxford University Press (1956).
4.
P. K. Foster, A. McNabb, and C. M. Payne, Trans. Met. Soc. AIME 233, 1022 (1965).
5o
See for exsanple, Wo Jost, Diffusion in Solids, Liquids, and Gases, p. 44, Academic Press, New York (1960).
6.
Marion H. Armbruster, J. Am° Chem. Soco 65, 1043 (1943)o
7.
G.M.
8.
R. B. McLellan, Jo Phys. Chem. Solids 34, i137 (1973).
9.
W. Eichenauer and Ao Pebler, Z. Metallkunde 48, 373 (1957).
Evans and E. Co Rollason, J. Iron and Steel Inst. 207, 1591 (1969).
i0.
Wo J. Kass and Wo Jo Andrzejewski, The Permeation of Hydrogen and Deuterium in 309S Stainless Steel, Sandia Laboratories Development Report, SC-DR-720136, Sandia Laboratories, Albuquerque, New Mexico (April, 1972).
Iio
Go Ro Caskey, Jr. and W. L. l>illinger, Effects of Trapping on Hydrogen Permeation in Copper, SRL DP-MB-73-53, du Pont de Nemours (E.I.) & Co., Aiken, South Carolina (1973)o
12.
Unpublished data on hydrogen and deuterium diffusion in a Au 20~0 Ag alloy by Wo Jo Kasso