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## Abstract

This article explores the effect of hydrostatic pressure on the diffusivity and concentration of hydrogen in 2205 duplex stainless steel. The Devnathan-Stachurski (DS) technique was employed to study the hydrogen permeation process housed in a high-pressure autoclave. Fick’s Second Law was modified, in order to describe reversible trapping of hydrogen in the steel. The concentration of hydrogen in the steel was measured by electrochemical charging techniques. A diminishment of the apparent diffusion coefficient, an increase of trapping rate constant and the concentration of trapped and free hydrogen, and a decrease of release rate were observed as the hydrostatic pressure increased.

**Key Words:** A. stainless steel; B. hydrogen permeation, galvanostatic, potentiostatic; C. kinetic parameters

## Introduction

Duplex stainless steels (DSSs), comprising both ferrite and austenite phases, which offer attractive performance by combining excellent chloride corrosion resistance with good mechanical properties, are widely used in the oil, petrochemical, and marine industries^{[1-4]}. Since the duplex stainless steels contain very low carbon (C 0.014 wt%), intergranular corrosion (IGC) due to the carbide precipitation in these steels is not a serious problem^{[5]}. However, duplex stainless steels used in marine facilities (e.g., platforms) suffer from hydrogen-induced cracking (HIC) on account of the large, negative over-potential resulting from cathodic protection. The excessive cathodic potential induces absorption and diffusion of hydrogen into the steel^{[6-8]}. The hydrogen gathered in the lattice, dislocations, and other defects in the steel may cause blistering, hydrogen embrittlement (HE), HIC, and structural failure of the steel^{[9-11]}. DSSs are inherently susceptible to hydrogen embrittlement (HE) in particular, as a result of hydrogen charging, temperature, and stress^{[12]}. Experimental data show^{[13]} that the hydrogen absorbed in the steel has a significant effect on the fatigue crack growth behavior and reduces the fatigue life.

Beyond that, in marine service, HE and HIC in DSSs are also produced H_{2}S^{[14]}, anaerobic microbial activity^{[15]} (e.g., sulfate-reducing bacteria, SRBs), and as the result of some pre-machining process^{[16, 17]}. In order to induce hydrogen-related corrosion phenomena (HE, HIC), a critical hydrogen concentration must be reached in the defects in the steel^{[18, 19]}. Therefore, the hydrogen diffusion coefficient and the concentration of hydrogen become key factors in the development of hydrogen-induced corrosion damage. Mechanical and chemical durability is extremely important in deep sea engineering materials. Although DSS has good performance, in general, in deep sea environment it is still at risk of HE and HIC^{[20]}.

In deep sea environments, as the hydrostatic pressure rise with increasing depth, the concentration of oxygen decreases and the temperature declines. According to the Arrhenius equation, the logarithm of the hydrogen permeability is proportional to inverse temperature when the hydrogen partial pressure is fixed^{[21-23]}. In the work of Chen et al.^{[24, 25]}, it was found that a threshold pressure difference between the both sides of the membrane for hydrogen permeation is exhibited and the hydrogen permeation flux versus the pressure difference is characterized by a liner relationship, regardless of the value of the pressure exponent. In this work, the specimen is charged with hydrogen by maintaining different pressures on the two sides of the sample with the difference being less than 1MPa. Wang et al.^{[26]} presented results showing that the total hydrogen content of a duplex stainless steel increased with rising pressure, wherein, the reversible hydrogen content decreased and irreversible hydrogen content increased. However, little research has been reported on the effect of hydrostatic pressure on hydrogen in metals, especially upon the diffusivity.

In addition, the trapping process^{[27, 28]} is also an extremely significant factor in determining the kinetics of hydrogen diffusion. Therefore, it is important to ascertain the effect of hydrostatic pressure on the diffusion coefficient, concentration and trapping process of hydrogen in duplex stainless steel, in order to predict and prevent failures of facilities serving in deep sea environments.

In the present work, using a Devnathan-Stachurski (DS) cell^{[29]} in a high pressure autoclave, we measured the hydrogen permeation rate and trapping rate constant as a function of hydrostatic pressure. We also employed an electrochemical cathodic charging method to study the effect of hydrostatic pressure on the hydrogen concentration in 2205 duplex stainless steel.

## Material and Method

### Material

The specimens were cut from a sheet of AISI 2205 duplex stainless steel supplied by Avesta Inc. The nominal chemical composition was (wt. %): C 0.014, Cr 22.39, Ni 5.68, N 0.17, Mo 3.13, Si 0.39, Mn 1.38, S 0.001, P 0.023 and Fe balance. Fig. 1 shows the metallographic structure of the specimen, in which the dark section is the ferrite phase and the bright section is the austenite phase. The proportion of ferrite phase is nearly 50%. The specimens used for hydrogen permeation studies were circular membranes of 2 cm diameter and less than 1 mm thick. Two sides of the samples were grounded with SiC grit paper from grade 150 to 2000 and polished with 1μm alumina, cleaned with deionized water, degreased in acetone, and dried in alcohol under hot air. The specimens applied for the determination of hydrogen concentration were cut into 10 mm × 10 mm squares × 1mm, sealed within epoxy resin with one face exposed to the solution and then polished and cleaned.

### Method

#### Hydrogen permeation measurements

The method used here for measuring the hydrogen permeation rate was an optimized Devanathan Stachurski cell ^{[18]}, as shown schematically in Fig. 2(a). In the double electrochemical cell system, hydrogen is gathered and absorbed on cathodic side (*x**=**0*) from the charging chamber, then diffuses to the opposite anodic side (*x**=**L*) which is held at a suitable anodic potential by means of potentiostatic polarization. The releasing hydrogen at *x* = *L* is immediately oxidized, due to the anodic potential imposed on the anodic side^{[29, 30]}.

The steel membrane (specimen) was located and sealed between the two cells and an area of 2.01 cm^{2} was exposed to each solution inside the cells. The anodic side used for hydrogen collection was filled with 0.2 M NaOH solution and an applied potential of 0.2 V_{SCE} imposed potentiostatically. The resulting current was measured using an electrochemical work station (Beijing Zhongfu Corrosion & Protection Co., Ltd PS268A). Hydrogen charging was carried out galvanostatically on the cathodic side in contact with 0.5 M H_{2}SO_{4} + 0.25 g/L thiourea solution. Both of the solutions in the releasing chamber and the charging chamber were deaerated using high purity nitrogen gas (99.999%). Before charging, an oxidation potential was applied, in order to reduce the hydrogen already existing in the sample and the resulting current is referred to as the “background current”. After the background current had decayed to under 0.2 μA/cm^{2}, a cathodic current density of 10 mA cm^{-2}was applied on the cathodic side using a galvanostat. Then the permeation curves were measured by using the electrochemical work station as the hydrogen diffused from the cathodic charging side, through the steel sample, to the extraction side. Hydrostatic pressures (0.1, 3, 5, 7 MPa) were imposed in the autoclave using high purity nitrogen.

#### Hydrogen concentration determination

In these experiments, hydrogen charging was carried out at 20 mA/cm^{2} for 12 h or 24 h in the autoclave with different hydrostatic pressures (0.1, 3, 5, 7 MPa). The apparatus is shown schematically in Fig. 2 (b). The solution in the charging chamber was deaerated 0.5 M H_{2}SO_{4} + 0.25 g/L thiourea, which was added to inhibit hydrogen atom recombination. After charging, the specimens were removed from the autoclave, washed with 0.2 M NaOH solution and were immediately installed in conventional electrochemical device immediately containing deaerated 0.2 M NaOH. A constant 0.2 V_{SCE }potential was applied on the charged samples for 10000 s, in order to oxidize the desorbing hydrogen and the current was recorded using a Princeton Applied Research multichannel electrochemical workstation (VMP3).

## Results

### Hydrogen permeation process

Figs. 3 (a) to (d) show the normalized initial hydrogen permeation curves as a function of increasing hydrostatic pressure. When hydrogen diffuses from the entry side (*x*=0) and is oxidized at exit side (*x*=L), the hydrogen flux is measured by the anodic current *I** _{a}*. As time goes by, the quantity of oxidized hydrogen increases and the anodic current rises toward a maximum, stable value,

*I*

*. The values of stable current density in each pressure are: 8.68 × 10*

_{∞}^{-4}mA/cm

^{2}at 0.1 MPa; 4.86 × 10

^{-3}mA/cm

^{2}at 3 MPa; 9.59 × 10

^{-3}mA/cm

^{2}at 5 MPa; and 1.42 × 10

^{-2}mA/cm

^{2}at 7 MPa. An apparent diffusion coefficient (hydrogen diffuse through both of lattice and traps),

*D*

_{app}, can be obtained, if we consider the system (steel and the passive film) to be a homogeneous element. Assuming that diffusion is unidirectional, Fick’s laws describe diffusion as Equation (1) and (2)

^{[31]}:

Ix,t = -Dapp∂C(x,t)∂x (1)

∂C∂t = Dapp∂2C∂x2 (2)

where *C* is the concentration of hydrogen, *D*_{app} is the apparent diffusion coefficient, *x* is the diffusion distance. In almost all cases, there are traps in the lattice defects of the specimen. In order to analyze the hydrogen permeation taking the reversible traps into account, Fick’s Second Law is modified. Assume that H atoms diffuse and that some are reversibly trapped, as follows:

H+Te ⇄ H.T (3)

where H is a hydrogen atom, H.T is a trapped hydrogen atom, and T_{e} represents an empty trap that is spatially fixed. Fick’s Second Law is now written as:

∂C∂t = Dapp∂2C∂x2 – k1CCT+ k-1CHT (4)

and the rate of change of trapped hydrogen can be expressed as:

dCHTdt =k1CCT – k-1CHT (5)

where

CT is the concentration of empty traps,

CHT is the concentration of trapped hydrogen, and

k1 and

k-1 are the trapping rate constants for the forward (trapping direction) and the reverse (de-trapping) direction of Reaction (3), respectively. Equation (4) must be solved subject to the initial and boundary conditions:

For

t = 0, 0 ≤ x ≤ L, C = 0, CHT = 0, CT = CT0 (Initial condition indicating no hydrogen exists in the membrane at zero time). By writing k1’=k1CT, we then write the kinetic equations as:

∂C∂t = Dapp∂2C∂x2 – k1’C + k-1CHT (6)

dCHTdt = k1’C – k-1CHT (7)

where

k1’ is a pseudo first-order rate constant. These equations must be solved for the following initial and boundary conditions:

For

t > 0:

x = 0, icF = -Dapp∂C∂xx=0 (Boundary condition for hydrogen input), which assumes that hydrogen is injected into the lattice galvanostatically.

For

t > 0:

x = L, C = 0 (Boundary condition for hydrogen release by oxidation).

In these initial and boundary conditions, *L* is the thickness of the membrane and ic is the cathodic current density for the input of hydrogen. This latter condition assumes that every hydrogen atom reaching the exit side of the membrane is immediately oxidized and hence that the concentration of H is zero.

Once *C(x,t)* is obtained, the current due to the oxidation of hydrogen is calculated as

iaF = -Dapp∂C∂xx=L (8)

where *i*_{a} is the anodic current density for the output of hydrogen. Possibly, the best way of solving this set of equations is by Laplace transformation^{[32]}. We first assume that the concentration of traps is so large that CT remains constant at CT0.

Laplace transformation of Equation (6) yields:

d2C̅dx2 – s+k1’DappC̅ + k1’CHT̅ + Ct=0Dapp = 0 (9)

where *s* is the Laplace frequency and the superscript bar signifies the Laplace transformed variable. Likewise, the Laplace transform of Equation (7) yields:

sCHT̅ = k1’C̅ – k-1CHT̅ (10)

Thus, we may derive and expression for the Laplace transform of the trapped hydrogen as:

CHT̅ = k1’s + k-1C̅ (11)

Substitution of this result into Equation (9) therefore yields:

d2C̅dx2 + k1’2Dapps + k-1 – s + k1’DappC̅ + Ct=0Dapp = 0 (12)

Noting that at *t* = 0, *C* = 0 (no hydrogen in the metal), this equation reduces to:

d2C̅dx2 + k1’2Dapps + k-1 – s + k1’DappC̅ = 0 (13)

The solution to Equation (13) is:

C̅ = Aeαx + Be-αx (14)

where

α = ±k1’2Dapps + k-1 – s + k1’Dapp1/2.

From the boundary conditions for

x = L, we write:

AeαL + Be-αL = 0 (15)

and from the boundary condition at *x* = 0:

dC̅dxx=0 = αA – B = -icFDapp (16)

Equations (15) and (16) are two simultaneous equations that are solved for *A* and *B*, yielding:

A = -icαFDappe-αLeαL + e-αL (17)

B = icαFDappeαLeαL + e-αL (18)

Thus, the Laplace transform of the hydrogen concentration in the steel becomes:

C̅ = -icαFDappe-αLeαL + e-αLeαx + icαFDappeαLeαL + e-αLe-αx (19)

From Equation (11), we then have the concentration of trapped hydrogen as:

CHT̅ = k1’s + k-1[-icαFDappe-αLeαL + e-αLeαx + icαFDappeαLeαL + e-αLe-αx (20)

Finally, the Laplace transform of the hydrogen oxidation current on the exit side of the membrane is written as

ias̅ = -FDapp∂C̅∂xx=L (21)

From Equation (14) we find that

ias̅ = -FDappαAeαL – Be-αL (22)

where *F* is Faraday’s constant. To obtain the concentrations and the hydrogen oxidation current as a function of time (i.e., in temporal space), we must take the inverse Laplace transforms of Equations (19), (20), and (22). In this work, we apply another method instead of inverse transforms to solve the equations, because the inverse transformations are very difficult. An alternative strategy is to transform the experimental current transient at the exit side of the membrane (Devanathan-Stachurski experiment) into Laplace space ^{[21]}. Thus, if *i(t)* is the oxidation current transient, the Laplace transform of the data is expressed as:

ias̅ = ∫0∞ite-stdt (23)

where the Laplace frequency (*s*) is chosen according to the sampling theorem:

2tt < s < 12∆t (24)

where *t** _{t }*is the length of the total time record and

1/∆t is the sampling frequency. Thus, if *t** _{t}* = 10000 s and

∆t= 1 s, the Laplace frequency used in evaluating Equation (23) is 0.0002 < *s* < 0.5 Hz. The values of ias̅ can be obtained by integrating the curves of *i(t)**e** ^{-st}* plotted in Fig. 4. Each value of Laplace frequency (

*s*) corresponds to one value of Laplace transform of oxidation current transient (ias̅). The logarithm values of ias̅ for different value of

*s*at various hydrostatic pressure are presented in Fig. 5, substituting the values of

*s*and ias̅ to Equation (22), a system of equations with respect to apparent hydrogen diffusion coefficient (

*D*

*) and trapping rate constants (*

_{app}*k*

_{1}*and*

^{’}*k*

*) is obtained.*

_{-1}

The correlation between the apparent diffusion coefficient and hydrostatic pressure is displayed in Fig. 6. The apparent hydrogen diffusion coefficient declines with increasing hydrostatic pressure. In Fig. 7, the correlation between the trapping rate constants and hydrostatic pressure is shown. Both the forward (trapping direction) and the reverse (de-trapping direction) rate constants increase with increasing hydrostatic pressure. The degree of increase for the forward trapping rate constant is greater than that for the reverse trapping rate constant.

### Hydrogen concentration determinations

Figs. 8 (a) and (b) show the hydrogen oxidation current transients after hydrogen charging process at various hydrostatic pressures for 12 h and 24 h, respectively. The top right corner image is the magnified view for the initial, few hundreds of seconds. Hydrogen atoms adsorbed in the metal are oxidized instantly when they diffuse to the surface of metal. The value of anodic current transient is proportional to the gradient of the hydrogen concentration in the metal surface. According to Głowacka’s work^{[33, 34]} there could be phase transformations in duplex stainless steels, such as the transformation of austenite into martensite, under our pre-charging conditions. However, in the pre-charging process every sample was subjected with the same current density and for the same time. Accordingly, during pre-charging we assume that the hydrogen-induced transformation would have occurred and may well affect the trap density. Thus, we assume that the phase transformations are the same in all samples and hence our observed results are due to the effect of hydrostatic pressure and not from the phase transformations.

The concentration of hydrogen in the bulk of sample (*C*_{H}) can be determined from the anodic current (*i*_{a}) and the corresponding charge (*Q*), which is obtained by integrating the *i*_{a}*-t *curves. The relationship between *C*_{H} and *Q* is:

CH = QzFv = ∫0∞iatdtzFv (25)

where *F* is Faraday’s constant, *v* the thickness of the sample, and *z* is the electron number (*z=1*). The hydrogen concentration for different charging conditions is calculated and is displayed in Fig. 9. The experimental data shows that the concentration of hydrogen increases as the hydrostatic pressure is raised and with the extension of time. However, the hydrogen dissolved in the specimens does not reach saturation under the conditions of the current experiments.

We transform the anodic current density transient of hydrogen in Fig. 8(a) to the concentration transient of hydrogen in Fig. 10(a) using the Equation (25). This figure shows that the concentration increases with experimental time and at ambient pressure the concentration increases much more slowly than at high hydrostatic pressures. We define the hydrogen release rate *R*_{R} as:

RR = ∂C∂t (26)

The curve of release rate with time is shown in Fig. 10(b). The data indicates the concentration of the hydrogen that is oxidized per second on the sample surface. The amount oxidized depends on the anodic current density and the total quantity of atomic hydrogen diffused to the surface of metal. The results show that the release rate declines sharply for the first tens of seconds; at longer times, the release rate is faster at higher hydrostatic pressure.

## Discussion

Fig. 11 shows a schematic of the hydrogen entry, diffusion, and exit/oxidation processed. A fraction of the hydrogen atoms that are discharged onto the cathodic side of the membrane combine to form hydrogen molecules and the remainder are adsorbed in the matrix. Due to the gradient of hydrogen concentration, hydrogen diffuses from the cathodic (entry) side of the membrane to the hydrogen anodic (exit) side. Because of the lattice structure of the steel, hydrogen is more likely to be absorbed into specimen in atomic form than in molecular form. The absorbed atomic hydrogen diffuses through the matrix with some being captured in various traps. Once reaching the exit (release) side of the membrane, the hydrogen atoms are oxidized very rapidly and the hydrogen atom concentration in the matrix is effectively reduced to zero.

In Fig. 6(a) the apparent hydrogen diffusion coefficient declines with increasing hydrostatic pressure. The lattice diffusion coefficient is a constant if the temperature (*T*), gas partial pressure (PH2) (i.e., fixed potential) and lattice constant (*a*_{0}) are fixed, as found in this work. Consequently, it is postulated that the trap occupancy is affected by the hydrostatic pressure and that results in an extension of the charging time and also resulting in the change in the hydrogen apparent diffusion coefficient.

It is common practice to express the diffusion coefficient in the form:

Dapp = D0e-∆G0,#/RT (26)

lnDapp = lnD0-∆G0,#RT (27)

∆G0,# = ∆V0,#dP (28)

(∂lnDapp∂P)T = -1RT∂∆G0,#∂P = -∆V0,#RT (29)

On the basis of Equation (27), we could also write:

lnD0.1D3 = ∆G30,# – ∆G0.10,#RT (30)

lnD0.1D5 = ∆G50,# – ∆G0.10,#RT (31)

lnD0.1D7 = ∆G70,# – ∆G0.10,#RT (32)

where

∆G0,#(kJ/mol) is the change in the standard Gibbs energy of activation and

∆V0,#(cm^{3}/mol) is the change in the standard volume of activation (“activation volume”) for the diffusion of hydrogen in the lattice, *P* (MPa) is the hydrostatic pressure, and *D*_{0} (m^{2}/s) is the lattice diffusion coefficient of atomic hydrogen, which is a constant as stated above, *D** _{i}* (m

^{2}/s) is the apparent diffusion coefficient at pressure

*i*

*, R*(8.314 J mol

^{-1}K

^{-1}) is the gas constant and

*T*(K) is the experimental temperature. As shown in Fig. 6(b) the data of ln

*D*

_{app}vs.

*P*can be fitted as ln

*D*

_{app}*=*-0.0195P

^{2}– 0.0555P – 23.197. Therefore,

(∂lnDapp∂P)T = -∆V0,#RT = -0.039P – 0.0555 (34)

∆V0,# = (0.039P + 0.0555)RT (35)

And the standard compressibility of activation becomes

∆κ0,# = -(d∆V0,#dP)/∆V0,# = -0.0390.039P+0.0555 (36)

∆Gi0,# can also be calculated by the data in Table 1. Then we can obtain the results listed in Table 2.

The data indicate that the standard activation Gibbs energy increases with increasing hydrostatic pressure, so it is more difficult for hydrogen to diffuse in metal at high hydrostatic pressure and this is also the reason why the apparent diffusion coefficient declines. This is presumably due to a reduction in the free volume of the metal. Interestingly, the volume of activation is out of the normal range (a few cm^{3} mol^{-1}) ^{[35, 36]}. However, this result assumes an elementary interstitial exchange reaction,

ISj-1+Hj →ISj-1-H-ISj#→Hj-1+ISj, where

IS is an interstitial position, Hj is a hydrogen atom occupying an interstitial at location, *j*, and ISj-1⋯H⋯ISj# is the (symmetrical) activated complex. Although Equation (34) is almost always used uncritically, in that an elementary process for diffusion is tacitly assumed, in fact the experimental observation is the pressure-dependence of the current on the exit side of the membrane, from which the apparent diffusivity is calculated. However, many elementary processes contribute to the observed current, including trapping. Thus, the volume and compressibility of activation data listed in Table 2 are labeled as being “apparent”. As shown in Fig. 11 for the “apparent” diffusion coefficient not only includes hydrogen atom diffusion through the metal but also reflects hydrogen ion discharge on the surface, possibly transport through a surface oxide film, hydrogen entry into the metal, and capture/release of hydrogen from the traps, so the calculated values are expected to be much different from those for the elementary reaction indicated above. We are now working on separating these processes, in order to resolve this issue.

In Fig. 7 both of the forward and reverse trapping rate constant increase with increasing pressure, demonstrating that increasing hydrostatic pressure accelerates the process of trapping and de-trapping. The increase of the forward trapping rate constant with pressure indicates that at high hydrostatic pressure more hydrogen atoms will be captured in the traps; this also accounts for the decrease in the apparent diffusion coefficient.

The calculated concentrations of hydrogen, as displayed in Fig. 9, are the apparent concentration of hydrogen (*C*_{H}) including both hydrogen in the lattice of the metal (*C*_{L}) and the hydrogen in the traps (CHT). The apparent concentration of hydrogen increases with rising hydrostatic pressure; this manifests that more hydrogen will dissolve into the metal when the hydrostatic pressure is elevated.

CH = CL + CHT (37)

The quantity of hydrogen in the lattice of the steel is correlated with the solubility of hydrogen in steel according to the temperature (*T*) and the lattice constant (*a*_{0}). In the present work, the temperature is fixed and lattice constantchanges only slightly, leading to a negligible change of concentration of hydrogen in the lattice of the steel (*C*_{L}). Therefore, the increase in the total quantity of hydrogen with increasing hydrostatic pressure is attributed to the increase in the concentration of hydrogen in the traps (CHT), as expressed in Equation (37). This is a consequence of the rate constants displayed in Fig. 7 and may be explained by an increase of the rate constant with increasing pressure as expressed by Equations (38) and (39)

(∂lnk1’∂P)T = -∆V1,app0,#RT (38)

(∂lnk-1∂P)T = -∆V-10,#RT (39)

where

∆V1,app0,#(cm^{3}/mol) is the apparent volume of activation of the trapping process and,

∆V-10,#(cm^{3}/mol) volume of activation of the de-trapping process. We then obtain

∆V1,app0,#=-444.48 cm3/mol and

∆V-10,#=-164.78 cm3/mol. These values appear to be high but, again, few data are available in the literature with which the values can be compared. The fact that the values are negative shows that the partial molal volume of the activated complex is smaller than that of the initial state of hydrogen in the trap or in the lattice. However, a direct comparison of the volumes of activation for trapping and de-trapping is obscured by the fact that k1’=k1CT is a pseudo-first order rate constant.

As a consequence, the concentration of hydrogen increases with rising hydrostatic pressure as stated above. Atomic hydrogen captured in higher activation energy traps is less readily released and may recombine to form molecular hydrogen within the traps, particularly if the traps are voids (e.g., condensed vacancies), hence, leading to an enhancement of the internal hydrogen pressure. When the internal hydrogen pressure reaches at a critical value, hydrogen embrittlement will occur and may even lead to material failure, possibly aided by hydrogen-induced grain boundary de-cohesion. When the release rate displayed in Fig. 10(b) is nearly stable (at long times), the final release rate is greater at higher hydrostatic pressure than at ambient hydrostatic pressure. This also could be attributed to the accumulation of hydrogen in the traps. The total hydrogen atoms oxidized on the metal surface are composed of hydrogen diffused from the lattice and from the reversible traps in the metal. The hydrogen in the traps can be regarded as a supplement to the lattice hydrogen. When the hydrogen oxidized at the metal surface, the hydrogen from traps will be released and will be add to the flux of lattice hydrogen to the surface. If the concentration of hydrogen in traps is higher than that in the lattice, the quantity of hydrogen oxidized on the metal surface will be greater than would be the case in the absence of trapped hydrogen and will be oxidized over a longer time. Accordingly, the release rate will be larger at higher pressure, as shown in Fig. 10(b).

Thus, from the above, the two prominent effects of trapping are the increase in the apparent hydrogen concentration and the decrease the apparent diffusivity^{[37]}. When steel is equilibrated against a constant external chemical potential of hydrogen, hydrogen will be absorbed to the solubility limit of the lattice, meanwhile extra hydrogen atom will be trapped by microdefects (dislocation, grain boundary, vacancy and so on) during diffusion process. When the external chemical potential is equal to the chemical potentials of the hydrogen existing in the lattice and in the trap sites, equilibrium steady-state of diffusion will be established, consequently the anodic current will reach the steady-state. Larger energy is required for a trapped hydrogen atom to escape from the trap than for hydrogen migrating through the lattice. Hence, the mean residence time of diffusing hydrogen is considerably longer in a trap site than in a lattice site. As mentioned above, the concentration of hydrogen captured in the traps is lager at higher hydrostatic pressure and, because the release of hydrogen from the traps is kinetically limited, the time taken for the oxidizing current at the exit side to reach a steady-state will also increase. As a result, the apparent diffusion coefficient declines with rising hydrostatic pressure, as shown in Fig. 6.

## Conclusions

A Devnathan-Stachurski cell located in a high pressure autoclave was employed in the present work to determine the hydrogen diffusivity at high hydrostatic pressure. We modified Fick’s Second Law to allow it to be applied to the case of the reversible trapping of hydrogen. The apparent diffusion coefficient of hydrogen in duplex stainless steel decreases but both of the forward and reverse trapping rate constant increase with rising hydrostatic pressure. The increase in the forward constant (for trapping) is higher than that for the reverse rate constant (for de-trapping). The standard Gibbs energy of activation for diffusion also increases with raising hydrostatic pressure, and this is regarded as being a contributing factor for the decline in the apparent diffusion coefficient.

The changes of hydrogen solubility in the lattice of the steel and of the lattice constant are negligible under the variable hydrostatic pressure employed in this work. Thus, the increase in the concentration of hydrogen due to increasing pressure is attributed to the augmentation of the hydrogen content by that in the traps, which also prolongs the time needed for equilibrium between external hydrogen chemical potential and the chemical potential in the lattice and in the traps of steel. Consequently, the apparent diffusion coefficient is diminished at higher hydrostatic pressure. The conclusion inferred from this work is that duplex stainless steels applied in deep-sea facilities are more likely to suffer from hydrogen embrittlement and material failure than in surface applications at ambient pressure.

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