Zheng - Effects of Absorption and Desorption on Hydrogen Permeation - II Theoretical Modeling and...

8/3/2019 Zheng - Effects of Absorption and Desorption on Hydrogen Permeation - II Theoretical Modeling and Room Tempera

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EFFECTS OF ABSORPTION AND DESORPTION ON

HYDROGEN PERMEATIONI. THEORETICAL MODELING

AND ROOM TEMPERATURE VERIFICATION

T.-Y. ZHANG{ and Y.-P. ZHENG

Department of Mechanical Engineering, Hong Kong University of Science and Technology, ClearWater Bay, Kowloon, Hong Kong

(Received 28 October 1997; accepted 27 April 1998)

AbstractA new model for hydrogen permeation is proposed to evaluate hydrogen diusivity, considering

absorption and desorption processes. An analytical solution derived from this model can predict eects ofsample thickness, absorption and desorption processes on the permeation behavior. Drift velocity throughthe surface and drift velocity in the bulk are introduced herein and the ratio of drift velocity through thesurface to that in the bulk determines the appropriateness of the time-lag method. The proposed methodcan be applied for any value of the ratio of drift velocities, while the time-lag method is appropriate onlyas the ratio approaches innity. In order to verify the theoretical predictions, electrochemical permeationexperiments at room temperature were performed on fully-annealed-commercially-pure iron samples thathad dierent thicknesses and were coated by palladium on two sides. Two groups of samples, group Isamples with plasma cleaning and pre-sputtering and group II samples without it, were adopted to varythe eects of the absorption and desorption processes. The diusivity evaluated from the proposed modelreveals an almost identical value being independent of sample thickness and surface treatment, while thetime-lag method leads to a diusivity two orders smaller in magnitude. The experimental results show thatthe ratio of drift velocities for the group I samples is about 100 times higher than that for the group IIsamples, conrming that the dierence in diusivity between values obtained by the present model andthe time-lag method decreases by increasing the ratio of drift velocities. # 1998 Acta Metallurgica Inc.Published by Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

The permeation experiment is the most commonly

used method in measuring hydrogen diusivity

because it is dicult to measure hydrogen concen-

tration with high spatial accuracy. In 1920

Daynes [1] proposed the time-lag method, which is

derived from Fick's second law and has been widely

used for several decades. The time-lag method

ignores the absorption and desorption processes

and hence adopts concentrations as the entry and

exit boundary conditions. Thus, the permeation ux

and integrated ux can be derived from the solutionto Fick's second law. The integrated ux has an

asymptote that intersects the time axis at L2a6D,

where D is the diusivity and L the sample thick-

ness. Daynes called this time the lag [1]. The hydro-

gen diusivity can be evaluated from the time lag,

tL, as

D L2

6tLX 1

An alternative method is to measure the half time,

t1/2, which is the time required for the permeation

ux to attain a half height of the steady-state

level [2]. There is no essential dierence in nature

between the time-lag method and the half-time

method; hereafter, only the time-lag method will be

used for comparison.

The time-lag method is simple and has been a

very appropriate method, particularly in the past

when computers were not available for data acqui-

sition and tting. However, there are problems in

using the time-lag method to determine diusivity

and to understand the diusion mechanism. One

major problem is the large scatter in measured

hydrogen diusivities determined by the time-lag

method at low temperatures. Vo lkl and Alefeld [3]

summarized 46 sets of data for the hydrogen diu-

sivity in a-iron within the temperature range of 25

14758C. These data exhibited a large degree of

inconsistency at low temperatures, diering by 34

orders of magnitude. In addition to this, the diu-

sion activation energies also revealed a large scatter,

ranging from 4.52 to 69.1 kJ/mol. These inconsis-

tencies indicate that hydrogen diusivities deter-

mined by the time-lag method from the permeation

tests can be aected by spurious surface conditions

and/or the existence of traps [4]. The absorption

and desorption processes may lead to the ``thickness

eect'' in measuring hydrogen diusivity when the

time-lag method is used, wherein permeation testson membranes thinner than 1.0 mm (approx.) yield

progressively lower diusivities in pure iron [5, 6].

Acta mater. Vol. 46, No. 14, pp. 50235033, 1998# 1998 Acta Metallurgica Inc.

Published by Elsevier Science Ltd. All rights reservedPrinted in Great Britain

1359-6454/98 $19.00 + 0.00PII: S1359-6454(98)00176-1

{To whom all correspondence should be addressed.

5023


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Wach et al. [7, 8] attributed the thickness eect to

energy barriers at the entry and exit surfaces. They

added an additional layer on the sample surface

and adopted a mean value of 1 mm of the ad-ditional thickness to correct the apparent diusiv-

ities. The corrected diusivities, however, still

exhibit scatter one order in magnitude [7, 8].

Obviously, the thickness eect is caused by the

absorption and desorption processes in which

hydrogen atoms have to overcome energy barriers

to enter into or exit from the sample. Hence, the

surface eect has attracted great attention and a lot

of eort has been devoted to lowering the surface

energy barrier [712]. The problems of how to

avoid the surface eect in order to measure hydro-

gen diusivities in bulk materials and how to

measure the surface energy barrier using the per-

meation method have not been solved [13]. Another

major problem concerns hydrogen diusion along

grain boundaries and/or dislocations [1419]. Some

experimental results, on the one hand, show that

grain boundaries are fast diusion paths for

hydrogen [1416]. Contradictory results are

reported [1719], on the other hand, showing that

the apparent hydrogen diusivities are lower in

samples with ner grains. A similar contradiction

exists in studies of hydrogen diusion along dislo-

cations. It is clear that only when hydrogen diusiv-

ity in a bulk material is reliably measurable, can the

eects of microstructures such as grain boundaries

and dislocations be studied.Despite the fact that the time-lag method does

not consider the absorption and desorption pro-

cesses, these processes have been studied for many

years. In 1936, Wang [20] rst presented the disso-

ciative chemisorption permeation model. Wang's

model of the permeation process occurring at the

entry and exit surfaces is schematically shown in

Fig. 1 and described as follows. At the entry sur-

face, there are two forward and two reverse jumps.

(1) The gas molecules strike the material surface

and then split into atoms and adhere there. The

ux of adsorbing atoms Jads is second order in

1

y

, giving Jads

a

1

y

2P, where y is the frac-

tional surface coverage, a is a rate constant for

adsorption and P is the pressure of hydrogen gas.

(2) The reverse jump of the rst forward jump, i.e.

two adsorbed atoms combine to form a molecule

which evaporates into the gas phase. The desorbing

ux Jdes is second order in y since two adjacent

atoms are needed to recombine, giving Jdes dy2,where d is a rate constant for desorption. (3)

Following the rst forward jump is the passage of

the adsorbed atoms into the bulk of the material

and the associate ux is Jabs gy, where g is a pro-portionality constant. (4) The atoms inside of the

material can jump back into the surface and the

ux is proportional to the hydrogen concentration just below the surface and the fraction of unoccu-

pied surface sites, Jdsb b1 yC, where b is

another proportionality constant. The net ux at

the entry surface is then determined by the four

jumps and has the following form:

J1 a11 y12P1 d1y21, J1 g1y1 b11 y1C1entry surface, x 0X 2

Similarly, the ux at the exit surface is given by

J2

b2

1

y2

C2

g2y2, J2

d2y

22

a2

1

y2

2P2

exit surface, x L 3

where subscripts 1 and 2 refer to the entry and exit

surfaces, respectively. If the equilibrium case is con-

sidered where the ux nulls and P = P1=P2,

equation (2) or equation (3) leads to

yeq

1 yeqa

dP

r, Ceq

a

d

rg

b

P

p4

where subscript ``eq'' denotes equilibrium. Equation

(4) is well known as Sievert's law.

Wang's model was able to explain quantitatively

how the net ux changes with the external gas

pressure [20]. Later on, Wang's model was further

developed and modied [2126]. Andrew and

Haasz [26] highlighted the similarities and dier-

ences in the modied models and pointed out that

Wang's model is sucient to explain the observed

experimental results at the limiting case of high

pressure. These adsorption, absorption and deso-

rption analyses [2026] are, however, based on per-

meation uxes at steady state. Starting from

Wang's model, the present paper proposes a new

model that is simple and involves both the absorp-

tion and desorption processes at the surfaces and

the diusion process in the bulk. Using uxes rather

than concentrations as the boundary conditions,Fick's second law is re-solved and an analytical sol-

ution is obtained.

Fig. 1. The course of the entire permeation process includ-

ing adsorption and desorption processes based on Wang'smodel.

ZHANG and ZHENG: ABSORPTION/DESORPTION ON HYDROGEN PERMEATIONI5024


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2. THEORY

In the present work, the permeation process is

considered as a one-dimensional process. Each of

the absorption and desorption processes is simpli-

ed as a forward and a reverse jump without con-

sidering details of the adsorption and absorption

processes. From Wang's model, i.e. equation (2),

the continuity of ux at the entry surface, requires

k1 P1 k1 C DdC

dx, at x 0 5

where k1 and k1 are rate parameters, respectively,

for the forward and reverse jumps and are given by

k1 a1g11 y12g1 d1y1

, k1 b1d1y11 y1

g1 d1y1X 6

At equilibrium, the fractional surface coverage canbe calculated from the gas pressure, using

equation (4) and, consequently, the relationship

between k1 , k1 and the gas pressure is determined.

During permeation, however, values of k1 and k1

are not the same as the values at equilibrium, even

though the permeation reaches a steady state. At

equilibrium, there is no ux at all, while at the

steady state, the permeation ux reaches its maxi-

mum value. For simplicity, the fractional surface

coverage is assumed to be unchanged with the per-

meation time and then k1 and k1 are treated as

constants in the present work. In the same way, the

boundary condition at the exit surface is given by

k2 C k2 P2 DdC

dx, at x L 7

where k2 and k2 are obtained by exchanging the

subscript 1 to 2. In general, a sample has the same

energy barriers at its two sides, and during per-

meation tests, the entry surface is usually chosen

randomly, unless special treatment is applied to one

surface, e.g. palladium is coated on one surface.

Therefore, it is assumed in the present work that

k1 k2 and k1 k2 .Both hydrogen pressures P1 and P2 in the gas

phase at the entry and exit surfaces can be con-

stantly maintained during permeation tests and,usually, P2=0. Furthermore, the initial concen-

tration inside the material can be maintained as a

constant, which is usually zero also. Thus, the in-

itial and boundary conditions and Fick's second

law can be re-expressed as

dC

d t D d

2C

dx 2,

C 0,for t 0DdCadx kCx0 kp,

DdCadx kCxL 0,for t b 0 8

where t is the time, andk k1 k2 , kp k1 P1X 9

The solution to equation (8) is

Cpx q

Im1

Am eDl2mtDlm coslmx k sinlmx 10

where lm is the mth positive root of

tanlmL 2kDlmD2l2m k2

11

and

p kp2D kL , q

kpD kL2kD k2L ,

Am 2kplm

D2l2m

k2

L

2kD

X 12

The drift velocities are dened at steady state

through the surfaces and in the bulk, respectively,

as

Vs k, Vb DL

13

where subscripts ``s'' and ``b'' denote surface and

bulk, respectively. Then, the steady-state concen-

trations at the entry and exit surfaces are given, re-

spectively, by

Centry 1 VsaVb2

V

saV

b

C0, Cexit 1

2

VsaV

b

C0 14

where

C0 kp

k15

is termed the permeation concentration. As dis-

cussed above, C0 does not equal the equilibrium

concentration since the values of kp and k at steady

state do not equal the values at equilibrium.

Equation (14) shows that the concentration equals

C0 at the entry surface and null at the exit surface

when the drift velocity through the surface is much

larger than that in the bulk. In this case, the diu-

sion process in the bulk dominates and hence the

time-lag method is appropriate. On the other hand,

the time-lag method will be inappropriate if the

ratio of the drift velocity through the surface to

that in the bulk is not large enough, which is par-

ticularly true for thin samples with high diusivities.

In such cases, the concentration at the entry surface

will be lower than C0 and the concentration at the

exit surface will be higher than zero. Another

extreme is the ratio of drift velocities approaching

zero. In this case, the concentration at both surfaces

equals half of C0 and the diusion ux nulls.

The permeation ux and the integrated ux canbe calculated from equation (10) and are, respect-

ively, given by

ZHANG and ZHENG: ABSORPTION/DESORPTION ON HYDROGEN PERMEATIONI 5025


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J Dkp2D kL

Im1

DlmAm eDl2mt

k coslmL Dlm sinlmL 16

Q Dkp2D kL t

Im1

Am

lm

k coslmL Dlm sinlmL1 eDl2mtX 17

The integrated ux has also an asymptote

Q Dkp2D kL t

Im1

Am

lmk coslmL Dlm sinlmL

18which yields a time lag

tL 2D kL

Dkp

Im1

Am

lmDlm sinlmL k coslmLX

19As t approaches innity, the permeation ux will

reach its steady value, JI

, as

JI Dkp2D kL

kp

2 VsaVb C0 "V 20

where a new parameter, "V, the mean drift velocity,

is introduced and dened as

1"V

2V

s

1V

b

X 21

If Vs is much larger than Vb, the mean velocity is

approximately equal to Vb. In this case, Centry C0

and equation (20) is reduced to

JI C0DL

22

which is the expression of the steady permeationux in the time-lag method. If an additional thick-

ness dened as

DL 2DaVs 23is introduced here, equation (20) is re-expressed as

JI C0DDL L X 24

It should be noted that the additional thickness

changes with the temperature and microstructure of

the material. Clearly, the additional thickness intro-

duced in the present work diers from that intro-

duced by Wach et al. [7, 8].

The normalized permeation ux is a direct conse-

quence of equation (16) and has the following form:

J

JI1 22D kL

Im1

eDl2

m

t

k coslmL Dlm sinlmLD2l2m kL 2kD

X 25

As expected, the normalized permeation ux is inde-

pendent of kp, similar to the fact that the normal-

ized permeation ux is independent of the hydrogen

concentration at the entry surface in the time-lag

method. The diusivity, D, and the desorption rate,

k, will be evaluated from tting the measurednor-

malized permeation ux with equation (25) and

hence the absorption parameter, kp, will be calcu-

lated from the steady-state permeation ux. Note

that in the electrochemical permeation experiment,

permeation current density, i, is a product of the

permeation ux, J, and the Faraday constant, F.The experimental curve tting can be conducted

either on a single curve of a sample or on multi-

curves of a sample or on multi-curves of multi-

samples. If the tting is carried out on multi-curves

of a sample, those curves should be obtained from

the tests performed under the same conditions such

that D, k and kp should have the same values for

each of the curves. If the tting is carried out on

multi-curves of multi-samples, a physical justica-

tion is required that D, k and kp must have the same

values for each of the samples. The tting process is

described in detail in Appendix A. If the diusivity

is evaluated from the time-lag method, then it can

be expressed in terms of the diusivity and the ratio

of drift velocities, or the normalized diusivity is

determined by the ratio of drift velocities:

DC

D 1I

m1

122 VsaVbVsaVblmL2 VsaVb2coslmLlmL2lmL2 VsaVb2 2VsaVblmL2 VsaVb2

X 26

equation (26) shows that the normalized diusivity

approaches unity as the ratio of drift velocities

approaches innity.

3. EXPERIMENTAL PROCEDURE

In order to verify the theoretical predictions and

clarify the confusing situation in hydrogen diusion

in a-iron, a hydrogen permeation experiment on a-

iron was carried out. Fully-annealed- commercially-

pure iron (99.7 wt%) was used in the present study.

The composition of the material is listed in Table 1.

The permeation samples have a circular shape with

a radius of 1.0 cm, and hence a large permeation

area of 3.14 cm2 to avoid the edge eect and ensure

the appropriateness of the one-dimensional model.

Optical microscopy reveals an average grain size of

the fully-annealed iron sample of about 0.082 mm,

as shown in Fig. 2.



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Two kinds of surface treatments were adopted to

prepare samples in order to vary the eects of

absorption and desorption processes. First, all

samples were ground using grinding papers down to

No. 800 grit. Then electrochemical polishing was

carried out on both sides of the samples at room

temperature in a solution of 90% acetic

acid + 10% perchloride at a potential of 12.0 V for

30 min in order to remove surface or subsurface

damage caused by the mechanical preparation.

After the electrochemical polish, the samples were

divided into two groups for two dierent surfacetreatments. Before coating palladium on two sides,

surfaces of the group I samples were further treated

by plasma cleaning for 15 min and pre-sputtering

for 5 min, while there was no pretreatment for

samples of group II. Finally, double sides of all

samples were coated with 0.1 mm palladium using

sputter-deposition in an ultra-high vacuum. The

sputter-deposition and pre-sputtering were all con-

ducted in a thin lm sputtering system (DVI SJ/24

LL, Denton) at room temperature.

The double cell electrochemical permeation set-up

is shown in Fig. 3. A 0.2 N NaOH solution lled

both cells and the whole system was controlled by a

computerized potentiostat (EG&G273A). The

anodic current density was stable for 24 h under

0.3 V SCE anodic polarization. A low charging cur-

rent density of 2 mA/cm2 was used in the per-

meation tests to reduce the Joule heating eect [27].

The sensitivity of the complete experimental system

was about 0.1% of the measuring range, which

could be automatically adjusted. All the electroche-

mical permeation tests were carried out at room

temperature.

4. RESULTS

The repeatability of the experimental system hasbeen veried by running the permeation tests ve

times on a sample in group I with a thickness of

0.608 mm. From each of the ve permeation curves,

the diusivity, D, and the desorption rate, k, and

the absorption parameter, kp, were evaluated. The

results are listed in Table 2. As can be seen in

Table 2, the diusivity D varies from 3.47 105 to4.68 105 cm2/s, having a mean value of 4.07 105 cm2/s with a standard deviation of0.53 105 cm2/s. It is clearly shown in Table 2 thatthe rst test yields the lowest hydrogen diusivity.

This phenomenon may be caused by the existence

of hydrogen traps in the sample [4]. If the result of

the rst test is not taken into account, the mean

value of the diusivity D will be 4.22 105 cm2/swith a standard deviation of 0.33 105 cm2/s.Although the mean value increases just slightly, the

standard deviation drops by almost half. Therefore,

results of the second test for every sample will beused in the following in order to demonstrate dier-

ences between the present model and the time-lag

method.

Table 2 also indicates the repeatability in

measuring the desorption rate k and absorption

parameter kp. Similar to the diusivity, the rst

Table 1. The ingredients of the material

Element C Si Mn P SIng red ient (wt %) 0 .0 46 0. 140 0. 103 0. 00 65 0 .0 12

Fig. 2. Metallography of fully-annealed-commercially-pureiron samples.

Fig. 3. Double-cell electrochemical permeation set-up.

Table 2. Repeatability of permeation experiments on the sample with thickness of 0.608 mm ingroup I

nth test 1st 2nd 3rd 4th 5th mean

k (103

cm/s) 1.065 1.043 0.993 0.950 0.986 1.007D (105 cm2/s) 3.47 4.14 3.88 4.68 4.18 4.07kp (1013 mol/cm2 s) 6.41 6.10 6.23 5.64 5.85 6.05



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time test yields the largest deviation for both k

and kp. The mean values of k are 1.0120.05 103and 0.9920.04 103 cm/s, and kp are6.0520.31 1013 and 5.9620.26 1013 mol/cm2 s, respectively, for those tests both including

and excluding the rst one. Obviously, the standarddeviation is small when the rst time test is not

taken into account. The repetitive tests provide a

guideline to estimate the errors for the diusivity

and the desorption rate and the absorption par-

ameter.

Figure 4 shows the permeation curve for a

sample from group I with a thickness of 0.108 mm.

A diusivity D = 4.32 105 cm2/s and a deso-rption rate k = 1.01 103 cm/s are obtained fromtting the normalized permeation curve with

equation (25). For comparison, diusivity was also

evaluated using the time-lag method and it turned

out to be DC=3.26

106 cm2/s, which is about

one order of magnitude smaller than that evaluated

by the present method. After that, the value of DCwas used to plot the normalized permeation curve

based on the time-lag method. As can be seen in

Fig. 4, the curve plotted by the present model ts

the experimental data much better than the time-lag

one. Kiuchi and McLellan [10] analyzed 62 sets of

hydrogen diusivity data and believed that data

obtained by electrochemical methods using Pd-

coated membranes were reliable. The hydrogen

diusivity in pure iron is between 7 105 and8 105 cm2/s [10] at room temperature, which isabout twice as high as the present result. This

inconsistency may be caused by various purities of

iron used in dierent research groups. Figure 5

shows the diusivities for the group I samples,

where the solid circles denote the values resulting

from tting each of the permeation curves with

either the present model or the time-lag model, andthe solid lines are the predications of the present

model. As expected, the diusivity evaluated by the

present model remains almost unchanged and has

a mean value of 4.11 105 cm2/s with a standarddeviation of 0.13 105 cm2/s, while the diusivitycalculated by the time-lag method increases from

0.326 105 to 1.94 105 cm2/s as the samplethickness increases from 0.108 to 1.330 mm, which

demonstrates the ``thickness eect'' [7, 8, 28]. When

the ve permeation curves are put together

to t the diusivity, the value turns out to be

4.03 105 cm2/s, having a relative error of 2% tothe mean value. Table 3 also lists the tting value

of the desorption rate k for each permeation curve.

As expected, k is independent of the sample thick-

ness and its mean value is 9.8620.76 104 cm/s,while the value obtained by tting all ve curves is

9.87 104 cm/s, almost identical to the meanvalue.

Figure 5 indicates that the dierence in diusivity

from the two methods decreases as the sample

thickness increases. However, even for a 1.330 mm

thick sample, the value of the diusivity calculated

Fig. 5. Hydrogen diusivity as a function of the samplethickness for the group I samples, where the solid circlesdenote the experimental results and the lines represent the

theoretical predictions of the present model.

Table 3. Results of curve tting for samples in group I

L (mm) 0.108 0.267 0.608 0.995 1.330 mean

k (103

cm/s) 1.011 0.994 1.043 0.910 0.970 0.986D (105 cm2/s) 4.32 3.99 4.14 4.01 4.11 4.11DC (105 cm2/s) 0.326 0.687 1.24 1.67 1.94

Fig. 4. Experimental permeation curve and the curvestted by the present model and the time-lag model for thesample in group I with a thickness of 0.108 mm and the

sample in group II with a thickness of 0.429 mm.



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by the time-lag method is still half as small as that

evaluated by the present model. The appropriate-

ness of the time-lag method can be gauged by the

ratio of the drift velocity through the surface to

that in the bulk. Figure 6 shows the normalized dif-

fusivity, DCaD, for all samples in groups I and II as

a function of the ratio of drift velocities, VsaVbwhich is directly proportional to the sample thick-

ness for samples in each group. For the samples in

group I, the normalized diusivity increases from

0.08 to 0.47 as the ratio of drift velocities increases

from 0.25 to 3.14. The theoretical prediction of the

thickness eect is also shown in Fig. 6. For the nor-

malized diusivity larger than 0.85, the ratio of

drift velocities should be higher than 20.0, which

requires a sample thickness larger than 8.0 mm for

group I. The critical sample thickness of 1.0 mm

suggested by previous work [7, 8, 28] may be under-

estimated such that the eects of the absorption

and desorption processes could not be eliminated.

Since the samples in group II lacked pre-treat-

ment before coating with palladium, the surface

energy barrier would be higher than that in the

group I samples. Experimental results conrmed

this intuition. Figure 4 also shows a permeation

curve for a sample in group II with a thickness

of 0.429 mm and its tting curves from the two

models. Table 4 lists the evaluated results of the dif-

fusivity and the desorption rate k for the group II

samples. Comparing the results in Table 4 with

those in Table 3 shows that the scatter for the

group II samples is larger than that for the group I

samples and this may be caused by the unclean sur-

faces. The mean value of parameter k for the group

II samples turns out to be 8.6322.06 106 cm/s(the value obtained by tting all ve curves is

9.16 106

cm/s), two orders of magnitude smallerthan that for the group I samples. The desorption

rate k represents the rate of desorption and depends

on the surface condition. A small k means there is a

high energy barrier for hydrogen atoms to over-

come so that the absorption and desorption pro-

cesses will play an important role during

permeation. The unclean surfaces in the group II

samples lead to a smaller k and, consequently, a

smaller ratio of drift velocities. In this case, the

time-lag method is not appropriate for the group II

samples at all. Figure 7 shows the diusivity as a

function of the sample thickness, wherein the lines

denote the theoretical predictions and circles stand

for the values resulting from tting each of the per-

meation curves with either the present model or the

time-lag model. For the group II samples, neverthe-

less, the diusivity evaluated by the present model

has a mean value of 4.2720.73 105 cm2/s, whilethe diusivity obtained by tting all ve curves is

4.26 105 cm2/s, almost the same as the meanvalue. Furthermore, the relative dierence in

mean value of diusivity between the two groups

Fig. 6. Normalized diusivity, DCaD, vs the ratio of thedrift velocities, VsaVb, for both group I and II samples,where the circles and solid circles denote the experimentalresults, respectively, for group I and II samples, and theline represents the theoretical prediction of the present

model.

Fig. 7. Hydrogen diusivity as a function of sample thick-ness for the group II samples, where the circles denote theexperimental results and the lines represent the theoretical

predictions of the present model.

Table 4. Results of curve tting for samples in group II

L (mm) 0.429 0.864 1.184 1.474 mean

k (105

cm/s) 0.942 0.556 1.000 0.953 0.863D (105 cm2/s) 4.28 3.27 5.00 4.54 4.27DC (107 cm2/s) 1.41 1.54 3.92 5.68



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is just 4%. However, the time-lag model results

in the diusivities ranging from 1.41 107 to5.68

107 cm2/s when the sample thickness

changes from 0.429 to 1.474 mm, being two ordersof magnitude smaller than that evaluated from the

present model. The normalized diusivity vs the

ratio of drift velocities is also shown in Fig. 6, indi-

cating that the experimental results are well pre-

dicted by the present model. The results re-conrm

that one reason for a large scatter in measuring

hydrogen diusivity in iron using the time-lag

method lies on the eects of the absorption and

desorption processes on the permeation behavior.

Since the normalized diusivity depends only on

the ratio of drift velocities, a smaller value of k

requires a larger sample thickness to maintain an

unchanged ratio of drift velocities. Thus, the critical

thickness for the group II samples should be

1000 mm if 8 mm is the critical thickness for the

group I samples. There is no doubt that increasing

thickness will increase the size of the sample and

the time to reach a steady state. From a practical

point of view, the permeation experiment for a

sample with a thickness of 1000 mm would be im-

possible. In this case, therefore, the time-lag method

would not be appropriate at all.

Using equation (20), the absorption parameter kpis evaluated from the permeation rate at a steady

state. Figure 8 shows the permeation rate at a

steady state as a function of the reciprocal of

2 VsaVb for the group I and II samples, where

the circles denote the data evaluated from tting

each permeation curve with equation (20) and the

solid line represents the theoretical prediction. The

mean values of the absorption parameter kp are

5.9620.22 1013 and 1.1220.06 1011 mol/cm2 s, respectively, for the group I and II samples.

Since the ratios of drift velocities for the group II

samples are all much smaller than 2, ranging from

0.0094 to 0.031, the steady-state rate is almost inde-

pendent of the sample thickness, as shown by

equation (20) and Fig. 8.

Using the mean values of k and kp for the two-

group samples, the permeation concentrations are

calculated to be C0=6.04 1010 and1.30 106 mol/cm3, respectively, for the group Iand II samples. Under the same charging current

density, the permeation concentration for the group

II samples is about 34 orders of magnitude higher

than that for the group I samples. This fact

conrms the argument that the permeation concen-

tration does not equal the equilibrium concen-

tration because the values of k and kp during

permeation dier from those at equilibrium.

The additional thickness DL, representing the

eects of absorption and desorption processes, is

evaluated using equation (23) to be 0.83 mm for the

group I samples and 98.95 mm for the group II

samples, 100 times larger than that for the group I

samples.

Fig. 8. The permeation current density at a steady state as a function of 1a2 VsaVb, where the solidcircles and circles denote the experimental results and the line represents the theoretical predictions of

the present model.



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5. DISCUSSION

In the present experiment, both sides of the

samples were coated with 0.1 mm palladium in an

ultra-high vacuum. The real diusion system is a

three-layer medium, as shown in Fig. 9. The ux

at the entry surface could be divided into the

following steps. Hydrogen ions are reduced to

hydrogen atoms on the entry surface of the palla-

dium lm. Part of the reduced hydrogen atoms are

adsorbed and absorbed and the others are recom-

bined into hydrogen molecules. The continuity of

net ux at the entry surface of palladium lm

requires

kPdp Csks DPddC

dx

Pd1

27

where subscript ``s'' denotes the entry surface of

the palladium lm, DPd the hydrogen diusivity

in the palladium thin lm, kPdp the forward jump

ux at the palladium surface, and ks the reverse jump rate constant. Since the palladium lm is

so thin, the concentration gradient can be assumed

to be a constant. Thus, equation (27) is rearranged

as

k

Pd

p Csks DPd Cs

Ci

D 28where Ci is the concentration inside the lm adja-

cent to the interface between the lm and the sub-

strate and D the thickness of the coated palladium

lm. A similar equation to equation (27) is used to

present the ux continuity at the interface:

Ciki Cki D

dC

dxat x 0 29

where subscript ``i'' means the interface between the

palladium lm and the iron substrate. Combining

equations (28) and (29) results in

K CK D dCdx

, at x 0 30

where

K

kPdp ki

ks ki ks ki DaDPd

,

K ks k

i

ks ki ks ki DaDPdX 31

In the same way, if the hydrogen concentration

at the exit surface is assumed to be zero, the bound-

ary condition at the exit surface should be modied

for the palladium-coated samples as

CK D dCdx

, at x LX 32

Comparing equations (30) and (32), respectively,

with equations (5) and (7) shows both correspond-

ing equations have the same form, which means theboundary conditions at the entry and exit surfaces

can be described by equation (8) if the two par-

ameters of k and kp are interpreted as

k K, kp KX 33

It should be noted that a linear concentration distri-

bution in the palladium thin lm is used here to

ensure the concentration gradient is a constant.

This simplication is appropriate when the palla-

dium lm is very thin. Since the experimental

results can be well interpreted by the proposed

model, it is believed that 0.1 mm of the double-sided

coating is a suitable thickness. In this case, theabsorption and desorption processes include the dif-

fusion process in the palladium thin lms.

Equation (14) shows that the ratio of drift vel-

ocities also determines the hydrogen concentration

distribution at steady state. The concentration at the

entry surface ranges from 0.5C0 to C0 as the ratio of

drift velocities varies from zero to innity, and the

concentration at the exit surface changes from 0.5C0to zero. For the group I samples, C0=6.04 1010 mol/cm3, the ratio of drift velocities rises

from 0.26 to 3.20 when the thickness increases

from 0.108 to 1.330 mm. The corresponding concen-

tration ranges are from 0.56C0 to 0.81C0 for the

entry surface and 0.44C0 to 0.19C0 for the exit

surface, respectively. For the group II samples,

C0=1.30 106 mol/cm3, the ratio of drift velocitieshas values much smaller than unity, from 0.0094 to

0.0310, as the thickness increases from 0.429 to

1.474 mm. The small ratio of drift velocities leads to

the concentration at the entry surface being slightly

higher than 0.5C0 and the concentration at the exit

surface slightly lower than 0.5C0. Taking the highest

ratio of drift velocities, 0.0310, as an example, the

concentrations at the entry and exit surface are

0.51C0 and 0.49C0, respectively. As discussed above,

the hydrogen content at a steady state in the group II

samples is much higher than that in the group Isamples. This assumes charging concentrations are

the same, however, electrochemical charging is notor-

Fig. 9. Schematic illustration of a three-layer permeationsystem.



10/11

ious for variability. Under the same charging current

density, it must take a much longer time for hydrogen

to reach such a high content in the group II samples.

This phenomenon was also observed during per-meation tests. Comparing permeation curves between

the two groups, e.g. Fig. 4, indicates the time to reach

steady state is about 14 000 s for the group II

samples, while it is around 40 s, or three orders of

magnitude smaller, for the group I samples.

The absorption and desorption processes occur

not only at surfaces but also at interfaces. The

eects of absorption and desorption could be sig-

nicant in permeation through a multilayer system

with each layer having a small thickness. Takano et

al. reported that the measured diusivity of a multi-

layer system with each layer having a thickness of

micrometers is much lower than the true

diusivity [29]. The reason for this is that the

model [30] used by them ignored the eects of

absorption and desorption processes, i.e. concen-

tration boundary conditions rather than ux

boundary conditions were used at interfaces and

surfaces. In principle, the methodology developed in

the present work can be used to attack the per-

meation problem in the multilayer system.

6. CONCLUSIONS

1. A new model on hydrogen permeation is pro-

posed to evaluate hydrogen diusivity, consider-

ing absorption and desorption processes. An

analytical solution to Fick's second law and

satisfying the ux continuity rather than the con-

centration boundary conditions is derived from

this model. Drift velocity through the surface

and drift velocity in the bulk are introduced and

their ratio determines the validity of the time-lag

method.

2. The diusivity, evaluated from the proposed

model, for hydrogen diusion in fully-annealed-

commercially-pure iron at room temperature is

4 105 cm2/s and independent of sample thick-ness and surface conditions.

3. At room temperature and a charging current

density of 2 mA/cm2, the mean values of theparameter kp for the group I and II samples are,

respectively, 5.9620.22 1013 and 1.1220.06 1011 mol/cm2 s, and the mean value of the deso-

rption rate k is 9.86 104 cm/s for the group Isamples and 8.63 106 cm/s for the group IIsamples. This means a well cleaned surface yields

a larger k parameter and consequently a higher

drift velocity through the surface, but a lower kp.

4. At room temperature and a charging current

density of 2 mA/cm2, the permeation concen-

trations are C0=6.04 1010 and 1.30 106mol/cm3, respectively, for the group I and II

samples.

AcknowledgementsThis project was nancially supportedby the Hong Kong Research Grants Council under RGC

grant HKUST805/96E. The experimental tests were con-ducted at the Advanced Engineering Materials Facility,HKUST.

REFERENCES

1. Daynes, H. A., Proc. R. Soc. A, 1920, 97, 286.2. McBreen, J., Nanis, L. and Beck, W., J. electrochem.

Soc., 1966, 113, 2071.3. Vo lkl, J. and Alefeld, G., in Hydrogen in Metals,

Topics in Applied Physics, Vol. 28, ed. G. Alefeld andJ. Vo lkl. City and Press, 1978, p. 321.

4. Chu, W. Y., Hydrogen Damage and Delayed Failure(in Chinese), 1988, p. 81.

5. Adair, A. M. and Hook, R. E., Acta metall., 1962, 10,741.

6. Smilowaski, M., Hydrogen in Steel. Pergamon Press,Oxford, 1962.

7. Wach, S., Miodowink, A. P. and Mackowiak, J.,Corros. Sci., 1966, 6, 271.

8. Wach, S. and Miodowink, A. P., Corros. Sci., 1968, 8,271.

9. Hirth, J. P., Metall. Trans., 1980, 11A, 861.10. Kiuchi, K. and McLellan, R. B., Acta metall., 1983,

31, 961.11. Kirchheim, R. and McLellan, R. B., J. electrochem.

Soc., 1980, 127, 2419.12. Louthan, M. R. Jr and Derrick, R. G., Corros. Sci.,

1975, 15, 565.13. Fukai, Y., The MetalHydrogen System, 1993, p. 227.14. Calder, R. D., Elleman, T. S. and Verghese, K.,

J. nucl. Mater., 1973, 46, 46.15. Kimura, A. and Birnbaum, H. K., Acta metall., 1988,

36, 757.16. Tsuru, T. and Latanision, R. M., Scripta metall.,

1982, 16, 575.

17. Sidorenko, V. M. and Sidorak, I. I., Fiz. M. Khim.Mat., 1973, 9, 12.18. Chew, B. and Fabling, F. T., Metal Sci. J., 1972, 6,

140.19. Yao, J. and Cahoon, J. R., Acta metall., 1991, 39,

119.20. Wang, J., Proc. Camb. phil. Soc., 1936, 32, 657.21. Ali-Khan, I., Dietz, K. J., Waelbroeck, F. G. and

Winhold, P., J. nucl. Mater., 1978, 76/77, 337.22. Baskes, M. I., J. nucl. Mater., 1980, 92, 318.23. Pick, M. A. and Sonnenberg, K., J. nucl. Mater.,

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Follstaedt, D. M., J. appl. Phys., 1989, 65, 180.26. Andrew, P. L. and Haasz, A. A., J. appl. Phys., 1992,

72, 2749.

27. Otsuka, R. and Isaji, M., Scripta metall., 1981, 15,1153.

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APPENDIX A

Curve tting

Newton's method is used for curve tting. Consider thefunction

fMj1

ijtheo

D,k,tj ijexpD,k,tj2 A1



11/11

where ijtheo

D,k,tj is the normalized theoretical currentdensity expressed by equation (5), ijexpD,k,tj the normal-ized experimental current density and M the number ofthe experimental data. f= min requires&

f1 dfadD 0f2 dfadk 0 X A2

From Newton's theory, the unknown parameter can beevaluated by the following iteration formula:

Dn1 Dn 1

Jn

df1adk f1df2adk f2

DDn ,kkn

kn1 kn 1

Jn

f1 df1adDf2 df2adD

DDn ,kkn

VbbbbbbX

A3

where

Jn

df1adD df1adk

df2adD df2adk

DDn ,kknA4

is the Jacobi matrix, and subscript n is the iterationnumber.

If the permeation tests were performed for dierentsample thicknesses, more accurate results can be obtainedby tting the curves for dierent sample thicknesses. Atthis time, the function f in equation (A1) can be rewritten

as

fM1j1

ijtheo

D,k,L1,tj ijexpD,k,L1,tj2

M2j1

ijtheoD,k,L2,tj ijexpD,k,L2,tj2

Mnj1

ijtheoD,k,Ln,tj ijexpD,k,Ln,tj2 A5

where Mn is the maximum number of experimental datawhen the sample thickness is Ln. In order to keep thesame weight for each curve, M1 M2 Mn is usedfor curve tting. Then, following the same procedureabove, the unknown D and k can be evaluated.


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