Effects of self-induced mechanical stress in hydrogen sorption by metals, by EIS

Electrochimica Acta 44 (1999) 4415±4429

E€ects of self-induced mechanical stress in hydrogen p sorption by metals, by EIS P. Zoltowski* Institute of Physical Chemistry of the Polish Academy of Sciences, ul. Kasprzaka 44/52, 01-224, Warsaw, Poland Received 7 August 1998; received in revised form 28 January 1999

Abstract Electrochemical impedance spectroscopy (EIS) approach to the processes of hydrogen sorption by metals has been extended to account for the e€ects of mechanical stress induced by the presence of hydrogen in the metal perfect-crystal matrix. The EIS equations for the simplest mechanism of the hydrogen absorption have been modi®ed to account for the e€ects of self-induced stress. That has been done on the basis of the recently elaborated theory for the di€usion of hydrogen in metals in the transient breakthrough experiment, where the so-called `uphill e€ect' can be observed. In this paper it is shown that the general character of the EIS data is not in¯uenced by the stress. However, the apparent di€usion coecient, derived directly from the EIS data, is always larger than the di€usion coecient of hydrogen. The higher concentration of hydrogen in the metal near its surface, the larger is the di€erence. For some Pd alloys, the di€erence can attain one order of magnitude. The rate of hydrogen absorption process is in¯uenced by the stress only as the result of the change of hydrogen concentration in the metal. The e€ect of stress on the di€usion of hydrogen is discussed in comparison with the e€ect of hydrogen traps in a real metal matrix. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Hydrogen sorption by metals; Di€usion; Absorption; Stress; Electrochemical impedance spectroscopy

1. Introduction Some metals, e.g. Pd, can sorb large amounts of hydrogen. It enters the original metallic crystal matrix as interstitial component, H. At low concentration, hydrogen forms in the metal a solid solution, known as a-phase. This results in expansion of the crystal lattice proportionally to the concentration of hydrogen.

Presented as Keynote Lecture at the 4th International Symposium on Electrochemical Impedance Spectroscopy, 2±7 August 1998 in Angra dos Reis, Rio de Janeiro, Brazil * Fax: +48-3912-0238 and +48-22-632-5276. E-mail address: [email protected] (P. Zoltowski) p

Thus, gradient of hydrogen concentration causes selfinduced mechanical stress in the solid metal. At higher concentrations, b-phase appears, resulting in a larger, discontinuous expansion of the original lattice. That is an additional source of stress [1±4]. The di€erences in the chemical potential of hydrogen in various parts of the system are the driving force for its absorption and di€usion in the metal. The contribution of the stress to chemical potential of the components of solids was considered already one century ago by Gibbs [4]. The disregard of this contribution leads to an oversimpli®ed description of the transport processes of hydrogen in metals. A typical setup for studies of the sorption processes by metals consists of two chambers separated by a

0013-4686/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 4 6 8 6 ( 9 9 ) 0 0 1 5 7 - 7

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P. Zoltowski / Electrochimica Acta 44 (1999) 4415±4429

Nomenclature Cab Cd cH+ cH(z ) DE D sa ,D ta

DH E Edc EIS eq F F f fH Hab Hab(z ) Had IS i JH jF k L LH l Mp.i. m

hydrogen absorption capacitance electrical double layer capacitance of the electrodevsolution interface concentration of protons in the solution near the electrode surface concentration of Hab in the metal at distance z from its surface electrical circuit distributed element modelling the di€usion of hydrogen in the metal matrix apparent di€usion coecients, accounting for the e€ect of stress (s ) or hydrogen traps (t ) in the metal matrix on hydrogen di€usion di€usion coecient of hydrogen in the metal matrix electrode potential electrode direct-current potential electrochemical impedance spectroscopy as superscript indicates the equilibrium value of the given quantity electrical circuit element modelling the faradaic processes Faraday constant frequency activity coecient of hydrogen absorbed hydrogen absorbed hydrogen at distance z from the surface adsorbed hydrogen Impedance (Immittance) Spectra imaginary unit, Zÿl ¯ux of di€using hydrogen, equivalent to vd(z ) current density of faradaic processes index of kth reaction maximal distance of hydrogen di€usion in the metallic sample along the z-coordinate phenomenological coecient (see Eq. (25)) index of lth process variable transport operator for di€usion in permeable (p) or impermeable (i) conditions, respectively equals DH/L (see Eq. (10))

metallic membrane (¯at or tube-shaped sample). When the hydrogen is supplied to one chamber (entry), after a certain time lag a gradually increasing ¯ux of hydrogen is observed in the other chamber (exit) (e.g. [5,6]). However, the e€ect is di€erent for a Pd or Pd-alloy

R Rab Rs RHE s ss SID T t VH vd(z ) vk vt x Y Yec YF Y Z ZDE Zim Zre z Dl Dvk DcH dl dvk mH s s st o

gas constant resistance of hydrogen absorption reaction resistance of the electrolyte solution reversible hydrogen electrode imaginary angular frequency, io as superscript indicates the steady-state value of the given quantity stress-induced di€usion temperature time partial molar volume of hydrogen in the metal matrix rate of hydrogen di€usion in the thin metallic plate at distance z from the electrode surface, equivalent to JH rate of the reaction k equals ZsL 2/DH (see Eq. (10)) stoichiometric number in MeHx admittance (Y=Z ÿ1) admittance of electrical equivalent circuit of a process faradaic admittance elastic bulk modulus of metallic matrix impedance (Z=Y ÿ1) impedance of DE imaginary component of impedance real component of impedance dimensional coordinate normal to the thin ¯at plate specimen surface (0 E z E L ) amplitude of the ac component of the quantity l amplitude of the ac component of the rate vk excess cH arising from the e€ect of stress ac component of the quantity l ac component of the rate k molar chemical potential of hydrogen in the metal matrix mechanical stress induced by hydrogen in the metal matrix as superscript indicates that the stress is taken into account in the given quantity ®nite-di€usion parameter (see Eq. (19)) angular frequency, o=2pf

membrane when the hydrogen of a signi®cant pressure is present in the whole system prior to the experiment. In such a case, an abrupt increase of the hydrogen pressure in the entry chamber results mainly in an initial decrease of its pressure in the exit chamber.

P. Zoltowski / Electrochimica Acta 44 (1999) 4415±4429

Hydrogen is soaked by the membrane from the exit chamber. That is a negative ¯ux of the permeating hydrogen. This phenomenon of uphill di€usion was observed for the ®rst time by Lewis et al. in 1983 [7]. It results from the local chemical potential of hydrogen in the metal, dependent not only on the local activity of hydrogen, but also on the stress, of non-local character, induced in the solid metal matrix by the gradient of hydrogen concentration [6±14]. Until now, the e€ect of stress on the di€usion of hydrogen in metals has been noticed and analysed only for the outlined above transient breakthrough experiment [9±12]. The aim of this paper is to describe the e€ects of self-induced stress on the rates of hydrogen absorption and di€usion processes in metals close to the steady state and even equilibrium of the system. This is done using a small-signal approach in electrochemical environment of the metal, i.e. by electrochemical impedance spectroscopy (EIS). At the beginning, the usual EIS equations for the processes of hydrogen sorption will be given. Next, the theory of e€ect of hydrogeninduced stress in the transient breakthrough experiment will be presented. Then, this theory will be used for appropriate modi®cation of the former EIS equations. It is the ®rst time when the stress induced in a solid matrix is taken into account in the EIS formalism. 2. Mechanisms of hydrogen sorption Hydrogen can enter the metal from the gaseous molecular hydrogen phase. The hydrogen absorption has to be preceded by its dissociation into atoms. The latter process can take place on the metal surface: H2 42Had

…R1†

where Had is hydrogen atom adsorbed at the surface. In the electrochemical cell, the chemical potential of hydrogen is controlled by the electrode potential [5,10,15±19]. The ®rst step of the sorption process is the reduction of proton (acidic solutions) or water (alkaline solutions). Generally, two alternative mechanisms of hydrogen absorption should be taken into consideration: the direct and the indirect one [19±21]. For acidic solutions, they can be described by reactions (R2), and (R3) followed by (R4), respectively (the hydration of the proton has been neglected): H‡ ‡ e4Hab

reduction with absorption in one step

H‡ ‡ e4Had

reduction with adsorption …Volmer

reaction†

…R2†

…R3†

Had 4Hab

absorption of adsorbed hydrogen:

4417

…R4†

At relatively low values of the electrode potential, the evolution of molecular hydrogen can also take place in parallel to the sorption reactions (R2)±(R4). In this paper, only the direct mechanism of hydrogen absorption (R2) will be taken into account. The parallel adsorption (R3) as well as the evolution of molecular hydrogen will be neglected. It will be assumed that the metal±hydrogen sample is a single-phase continuous metallic crystal, i.e. that there is (i) only the a-phase, (ii) without grain boundaries or other imperfections. For Pd the upper limit of hydrogen concentration at room temperature is relatively low (PdH0.025) [1±3]. However, for some Pd alloys (e.g. Pd81Pt19 and Pd77Ag23) this limit is one order of magnitude and a half larger [6±14,22]. Onedimensional di€usion, normal to the surfaces of a large thin ¯at sample (0 E z E L, where L is the sample thickness e€ective for di€usion, see Fig. 1) will be discussed. Hence, Hab in (R2) should be substituted by Hab(z = 0), and the di€usion can be denoted as follows: Hab…zˆ0† 4Hab…z>0† :

…R5†

3. EIS approach without taking stresses into account Recently, the hydrogen sorption by Pd and its alloys was studied both theoretically [23,24] and experimentally by EIS in the so-called `permeable' [25] and `impermeable conditions' [26±30] (Fig. 1). In the permeable conditions (membrane separating two di€erent solutions), one has to assume that the di€usion coecient of hydrogen does not depend on its concentration in the metallic matrix. This assumption is not necessary in the impermeable conditions (foil immersed in a single solution) [24]. Hence, the dependence of the diffusion coecient of hydrogen on its concentration can be studied. Another advantage of impermeable conditions is that the experiment is performed at the conditions close to the equilibrium in the metallic matrix [24]. Moreover, one has to note that the steady state or equilibrium of a process also comprises the chemical potentials of its reactants. Therefore, in the case of the direct mechanism of hydrogen absorption (R2), this holds also for the concentration of protons in the solution near the electrode. In experiments in the impermeable conditions, a thin-layer sample on an inert substrate can also be used in place of a foil [24,31,32]. Three requirements of the EIS method with respect to the system under consideration should be quoted here [33,34]: (i) it should be possible to treat the system as one-port, i.e. a system where the input is applied and the output measured at the same couple of poles,

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P. Zoltowski / Electrochimica Acta 44 (1999) 4415±4429

Fig. 1. General scheme of the two types of EIS experiments on the hydrogen sorption by metals.

(ii) the steady state, and (iii) the linearity. The ®rst requirement should be ful®lled by an appropriate arrangement of the system and measuring equipment. The steady state and the linearity are usually ful®lled by controlling the dc potential of the working electrode, Edc, and using a small amplitude, DE, of the perturbing ac signal, dE, respectively. In such a case, the applied potential is E=Edc+dE, where dE=DE exp(st ), s = io, o denotes the angular frequency (o=2pf ), i the imaginary unit (i=Zÿ1), and t the time. In the experiments, ac signals of various frequencies in wide range (several decades) are successively applied, and the impedance (more generally, immittance, i.e. impedance, Z, or admittance, Y=Z ÿ1) spectrum (IS) of the electrode system is recorded for a given Edc. The goal is to describe the properties of the system by their immittance, in terms of a so-called

`measurement model' and the set of its frequency-independent parameters [35±37]. The electrode system is usually modelled by an electrical circuit presented in Fig. 2(a). It consists of the solution resistance, Rs, the double layer capacitance of the electrodevsolution interface, Cd, and the element F modelling the faradaic processes. From the point of view of the present study, the faradaic processes are the most important. Hence, none-purely capacitive behaviour of the solid electrodevsolution interface [34,38,39] is neglected. The nature of F depends on the process mechanism, the kinetics of particular processes and the experimental conditions. The equations for the immittance of electrode processes of hydrogen sorption by the direct mechanism of absorption into the metal in the impermeable conditions, as well as the indirect mechanism in the permeable and impermeable conditions are given in the

P. Zoltowski / Electrochimica Acta 44 (1999) 4415±4429

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Fig. 2. Electrical equivalent circuits: (a) for the whole electrode system: Rs, solution resistance; Cd electrode double layer capacitance, F, element modelling the faradaic processes. (b) for F, without taking the e€ects of the stress into account: Rab, direct absorption reaction resistance; DE, distributed element. (c) for F, including the e€ects of the stress: R sab, direct absorption resistance; DE s, distributed element

literature [23±25,29,30]. Below, these equations for the direct mechanism will be rewritten for both experimental conditions (Fig. 1), in the notation similar to that used by Chen et al. [24]. The rates of the elementary reactions of the sorption process (R2) and (R5) are de®ned in terms of their variables, in a general form: vab ˆ vab …E, cH…0† †  vd…z† ˆ vd…z†

@ cH…z† @z

…1†  …2†

where

dvk ˆ Dvk exp …st†:

…3†

All the dvks vary in time in response to dE. Similarly, all variables of elementary reaction rate, l (e.g. E and cH(z )), are the sums of their steady-state and ac components, l ss and dl, respectively l ˆ lss ‡ @ l

where

@ dcH…z† @z

…5†

@ dcH…z† @ 2 dcH…z† ˆ sdcH…z† ˆ DH @t @ z2

…6†

where DH is the di€usion coecient of hydrogen in the metal. Let us remember that the II Fick's equation assumes that the di€usion coecient does not depend on the concentration. The boundary conditions for the hydrogen di€usion for t e 0 are (see Fig. 1): . for permeable conditions

where vd(z ) denotes the rate of di€usion of hydrogen at the distance z from the electrode surface (vd(z ) is equivalent to the ¯ux of di€using hydrogen, JH [10± 12,23,24]), and cH(0) the concentration of Hab beneath the metal surface (z= 0). In EIS, the reaction rate, vk, is the sum of its steadystate and ac components, v ss k and dvk, respectively: vk ˆ vss k ‡ dvk ,

@ vd…z† ˆ ÿDH

dl ˆ Dl exp …st†:

…4†

EIS deals only with the ac components. Hence, one has to rewrite Eqs. (1) and (2) in their ac forms. We will start by the formulation of appropriate equation for dvd(z ) (see Eq. (2)). Hydrogen di€uses in the metal sample along the z coordinate (0 E z E L, where L is the electrode thickness e€ective for di€usion, see Fig. 1). According to the I and II Fick's equations, respectively (for dcH(z ) see Eq. (4)):

dcH…0† ˆ DcH…0† exp …st†,

dcH…L† ˆ 0

…7a†

. for impermeable conditions dvd…0† ˆ Dvd…0† exp …st†,

dvd…L† ˆ 0:

…7b†

The problem of ®nite di€usion, being the so-called `bounded' in permeable, and `restricted' in impermeable conditions (see Fig. 1), was analysed by many authors. The solution of Eq. (6) for the boundary conditions given in Eq. (7) is [23,24,34,40] dcH…0† ˆ dvd…0† Mp, i

…8†

Mp,i denotes the transport operator for permeable and impermeable conditions, respectively: Mp ˆ …mvt †ÿ1 tanh vt

…9a†

Mi ˆ …mvt †ÿ1 coth vt

…9b†

where m ˆ DH =L

and vt ˆ

p sL2 =DH :

…10†

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P. Zoltowski / Electrochimica Acta 44 (1999) 4415±4429

Now let us de®ne dvab (see Eq. (3)) in terms of its variables (Eq. (1)). This can be done by the ®rst-order Taylor expansion:  dvab ˆ

@ vab @E

 cH…0†

 dE ‡

@ vab @ cH…0†

 E

dcH…0† :

…11†

From the mass and charge balances, it follows: dvab ˆ dvd…0†

…12†



 @ vab   @ cH…0† E @ vab  ˆ ÿRab ˆ C ÿ1 st ˆ  ab : @ cH…0† E @ vab F @ E cH…0†

…19†

Rab (Eq. (18)) is the charge-transfer resistance, because the reaction (R2) involves charge transfer. The term !   @ vab @ vab F = @ E cH…0† @ ccH…0† E

djF ˆ ÿFdvab

…13†

where djF is the ac component of the faradaic current density, and F the Faraday constant. The faradaic admittance, YF, of electrode processes is YF ˆ djF =dE:

…14†

From Eqs. (8) and (11)±(14), the following equation for YF can be derived:   @ vab ÿF @ E cH…0†   YF ˆ : @ vab 1ÿ Mp, i @ cH…0† E

…15†

YF can be identi®ed with the admittance of the electrical circuit presented in Fig. 2(b): Yec ˆ

1 Rab ‡ ZDEp, i

…16†

where the symbols correspond to the circuit elements in the ®gure. Rab denotes the absorption resistance (R2), and DEp,i the distributed element that models the di€usion in permeable (p) or impermeable (i) conditions, respectively. Unlike in the case of the lumped elements, the immittance of a distributed element has more than one parameter [33,34,38]. The impedance of DEp,i is de®ned as follows (for Mp,i see Eq. (9)): ZDEp, i ˆ st Mp, i

…17†

where st is the ®nite-di€usion parameter (see Eq. (19)). ZDEp,i has three parameters: st, DH and L. By comparing Eqs. (15) and (16), the equivalent circuit parameters can be expressed in physico-chemical terms: "   #ÿ1 @ vab Rab ˆ ÿ F @ E cH…0†

…18†

in Eq. (19) can be called `absorption capacitance' (Cab), on the analogy to the frequently used term `adsorption capacitance'. Cab is the reciprocal of st. At equilibrium, Eq. (19) simpli®es to:  ÿ1 ÿ
ÿ1 dcH…0† seq ˆ ÿ F ˆ C eq t ab dE

…20†

C eq ab determines the slope of the absorption isotherm, cH(0)) = cH(0)(E ).

4. E€ects of stress in the transient breakthrough experiment In Section 3 it has been assumed that the rate of hydrogen di€usion in the metal matrix is a function of the gradient of its concentration (Eqs. (2) and (5)). Also the concentration of hydrogen in the metal directly beneath its surface has been assumed to be the variable of the absorption process (Eqs. (1) and (11)). However, in reality the chemical potential of hydrogen should be used as the variable, instead of the concentration. Until now, the e€ect of stress on the hydrogen di€usion in metals was noticed and analysed only for the transient breakthrough experiment. In this case, a large stepwise change of hydrogen pressure or electrode potential is applied at the entry side of the membrane (signal at z = 0), and the ¯ux of permeating hydrogen is measured at its exit side (response at z=L ) as a function of time. In such an experiment, the boundary conditions are changing in time at both sides of the membrane [4,10,12]. The chemical potential of interstitial hydrogen, mH, as a mobile component in a solid matrix depends on hydrogen concentration, cH, and mechanical stress, s [4,10,12]: mH …s, cH † ˆ mH …0, cH † ÿ VH s

…21†

where mH(0, cH) denotes the chemical potential of hydrogen in the stress-free state (s=0), VH the partial molar volume of hydrogen in the solid matrix, and

P. Zoltowski / Electrochimica Acta 44 (1999) 4415±4429

s(called `stress' in this paper) is the hydrostatic part of the stress tensor induced by the presence of hydrogen in the metal. The last quantity is de®ned as the trace of the tensor: s ˆ sxx ‡ syy ‡ szz :

…22†

In Eq. (21) s is an analogue of pressure [4,10,12]. VH is observed experimentally as expansion of the volume of the metallic matrix in the result of hydrogen sorption. It is assumed to be independent of the hydrogen concentration [22,41,42]. mH(0,cH) can be replaced by the conventional expression [10,12]: mH …0, cH † ˆ m0H ‡ RT ln fH cH

…23†

where R and T have their conventional meanings, and fH is the activity coecient of hydrogen. Transport equation for the ¯ux of hydrogen, JH, diffusing in the bulk of the metal under the one-dimensional gradient, along z coordinate, of the chemical potential de®ned by Eqs. (21) and (23), is [4,10,12]: @ mH ˆ ÿDH @z    @ ln fH @ cH VH cH @ s ÿ 1‡ @ ln cH @ z RT @ z

JH ˆ ÿLH

…24†

where LH is the phenomenological coecient of hydrogen di€usion in the metal matrix, de®ned (in the case of ideal solution, fH=1) as LH ˆ

DH cH : RT

…25†

In Eq. (24) the second term within the brackets represents the elastic part of the chemical potential gradient. It describes the stress-induced di€usion (SID) [11]. Balance equation (modi®ed II Fick's equation), under the assumptions that (i) the di€usion coecient of hydrogen depends neither on its concentration nor stress, and (ii) LH is de®ned by Eq. (25), is [4,10,12]: "  @ cH @ ln fH @ 2 cH VH cH @ 2 s ÿ ˆ DH 1 ‡ @t @ ln cH @ z2 RT @ z2 …26† # VH @ s @ cH ÿ : RT @ z @ z In order to solve Eq. (26), the relationship between 1 In reality, the elastic properties of metals can depend on the hydrogen content. For instance, it has been noticed that the presence of hydrogen in Pd reduces its elastic modulus up to ca. 20% [45].

4421

the gradients of cH and s is needed. For this purpose, the analogy with the e€ect of thermo-stress, arising from the thermal expansion of the elastic matrix, on the conduction of heat is usually used [4,10,12,43,44]. For the case of one-dimensional di€usion, in z direction, of hydrogen through an isotropic thin plate, the components of stress in the three normal directions are [10,12]: sxx ˆ syy

" … VH Y 1 L ˆÿ DcH ÿ DcH dz 3 L 0

12…z ÿ L=2† ÿ L3

…L 0

 #  L dz and szz ˆ 0 DcH z ÿ 2

…27†

where Y is the elastic bulk modulus (assumed to be independent of the interstitial hydrogen concentration1) of the metal matrix (in the literature, Y is sometime denoted by symbol B [46]; Y=E/(1ÿn ), where E and n denote the Young modulus and the Poisson ratio, respectively), and DcH=cHÿcH,0, where cH denotes the actual hydrogen concentration, and cH,0 its concentration in the initial stress-free state. Eq. (27) presents the required relation between the concentration and stress ®elds for the large thin ¯at plate [12]. On the right-hand-side of Eq. (27), the ®rst term within the brackets is of the local character, whereas the two integral terms are associated with the nonlocal character of stress. When the local concentration of hydrogen changes slowly due to the di€usion process, the associated stress is transmitted across the whole elastic specimen with the velocity of sound (in Pd, ca. 3100 m sÿ1 [45]). The local term expresses the contribution to stress arising in each volume element of the plate due to the concentration change, deviating from the stress-free conditions. Its ®nal in¯uence on the di€usion ¯ux will express a correction of the concentration gradient term, similar to that due to the non-ideally of the solution (see e.g. Eq. (31)). The ®rst integral term has the physical meaning of average concentration. If the hydrogen concentration within the plate thickness would be homogeneous, this will exactly compensate the local term. The second integral is important only when the stress is non-symmetrical with respect to the plane at z=L/2. It vanishes with symmetrical initial and boundary conditions at both sides of the plate (at z = 0 and z=L ). If di€usion starts from both sides of the plate (z = 0 and z=L ), at both surfaces at t = 0 [12] sxx ˆ syy ˆ ÿ

VH Y DcH 3

Later on (t>0) [12]

and

szz ˆ 0:

…28†

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P. Zoltowski / Electrochimica Acta 44 (1999) 4415±4429

"

sxx ˆ syy

VH Y 1 ˆÿ DcH ÿ L 3

…L 0

# DcH dz

and

…29†

szz ˆ 0 and the stress will decrease in time, because the integral term increases when the di€usion front proceeds. When the integral term would compensate exactly the concentration term, the stress will vanish, i.e. the di€usion equilibrium will be attained. The integral in Eq. (29) represents a gradient-less distribution of stress. Hence, it will not contribute to the driving force of diffusion [12]. s is the stress-tensor trace (Eq. (22)). In the case of non-symmetrical initial and boundary conditions, the gradient of stress is [10] " … @s @ sxx 2VH Y @ cH 12 L ˆ2 ˆÿ ÿ 3 @z L 0 3 @z @z   # L dz : DcH z ÿ 2

…30†

In contrast to stress (Eq. (27)), its gradient (Eq. (30)) includes only one non-local term. The origin of this term is the bending of the plate, arising from the non-symmetrical boundary conditions. This non-local term does not appear for a plate with symmetrical boundary conditions [10]. If @cH/@z is constant, Eq. (27) tells us that sxx and syy also vanish [4]. Introducing Eq. (30) into Eq. (24), one obtains [10] JH ˆ ÿDH "

@ ln fH 2V 2H Y cH ‡ 1‡ @ ln cH 3RT

8V 2H Y cH ‡ RTL3

!

@ cH @z

…L 0

 #  L dz : DcH z ÿ 2

…31†

The ®rst term within the brackets, proportional to the gradient of concentration, describes the local (Fickian) di€usion accounting for the stress. Thus, the stress always enhances the local di€usion, similarly as the activity coecient of hydrogen does, even in the case when VH < 0. The second term, proportional to the local concentration of hydrogen, is a non-local one, since it depends on the integral of the composition pro®le taken over the whole thickness of the plate (0E z E L ) [4]. The existence of this term is due to the distortion (bending) of the plate that creates a ¯ow of the interstitial hydrogen in each volume element of the

metal matrix as soon as a non-negligible hydrogen concentration exists. Introducing Eq. (30) into Eq. (26), one obtains [4,10,12] 2 ! @ cH @ ln fH 2V 2H Y cH @ 2 cH 4 ‡ ˆ DH 1 ‡ @t @ ln cH 3RT @ z2 2V 2H Y ‡ 3RT



@ cH @z

2

8V 2H Y ÿ RTL3

…L 0

3   ! L @ cH 5 : DcH zÿ dz 2 @z …32†

It is a second-order (partial) non-linear integro-di€erential equation. It has no general solution [10]. The ®rst term in the brackets at the right-hand-side of Eq. (32) is proportional to the second derivative of concentration, similarly as the only term in the II Fick's equation. It includes the local e€ect of stress on the Fickian di€usion. The second term, proportional to the square of the concentration gradient, results only from the stress. It will be probably negligible if the concentration gradient is small. On the other hand, it could be a source of non-linear phenomena. The third term, proportional to the concentration gradient and including the integral, is due to the asymmetry of the stress ®eld [10,12]. Eqs. (31) and (32) describe correctly the results of the transient breakthrough experiment in the almost whole period of time when the uphill di€usion is observed [10,12]. It is noteworthy that, according to Eq. (27), the concentration of hydrogen in the metal matrix is in a feedback relation with the self-induced stress. Hence, starting from this equation, the in¯uence of the stress is included in any quantity dependent on the hydrogen concentration. In other words, the hydrogen concentration is corrected for the e€ects of the stress. The problem of the stationarity and the pro®le of the hydrogen concentration over the thickness of the plate in the steady state should be discussed, because it will be of crucial importance in the next section. In contrast to Section 3 (see Fig. 1), it follows apparently from Eqs. (24), (26), (31) and (32) that the condition of steady state is not dc ss H/dz = const when the stress is taken into account. In fact, it has been experimentally proven that the concentration pro®le of hydrogen across the membrane in the ®nal, steady-state period of the breakthrough experiment is not linear [13,47]. This has been concluded for the steady state attained after a suciently long time. The criterion of the stationarity in this type of experiment is the time invariance of the ¯ux of the di€using hydrogen. This is observed as the linear increase of the hydrogen pressure in the exit chamber. However, this also means a

P. Zoltowski / Electrochimica Acta 44 (1999) 4415±4429

permanent change of the boundary conditions. Hence, this speci®c stationarity is not the true one in the whole system. A precise analysis of this problem, based on the condition of the minimum entropy production as criterion of the stationarity, puts in favour the opinion that the metal matrix is stress-free (s ss (z )=0) in the true steady state of the system. Therefore, it can be concluded that dc ss H/dz in the membrane is constant when the whole system attains the steady state [12].

5. Accounting for the e€ects of stress in EIS In the transient breakthrough experiment, the signal is applied at one side of the membrane, and the response is measured at its opposite side. Therefore, this is a two-port system. In the experiments on hydrogen sorption by EIS, the situation is di€erent. (i) The experiments are performed close to the (true) steady state, or even equilibrium, of the system. (ii) The perturbing signal (dE ) and also all responses are small enough to consider the system as linear. (iii) In both experimental conditions (see Fig. 1), the system can be considered as one-port, since the response (dj ) is measured at the same surface where the signal (dE ) is applied (z = 0). For the sake of simplicity, it will be assumed that the activity coecient of hydrogen ( fH) does not depend on concentration, and it is equal to one. To account for the e€ects of the stress on the hydrogen sorption processes, Eqs. (1), (2) and (11) should be modi®ed as follows: vsab ˆ vsab …E, cH…0† †  vsd…z† ˆ vsd…z†  dvsab ˆ

@ mH…z† @z

@ vsab @E

cH …0†

 dE ‡

@ vsab @ cH…0†

E

ˆ

m0H…z†

‡ RT ln

…css H…z†

‡ dcH † ÿ VH ds…z†

…37†

Therefore dmH…z† ˆ mH…z† ÿ mss H…z† ˆ RT ln 1 ‡

dcH…z† css H…z†

!

…38†

ÿ VH ds…z† :

By expansion of the logarithmic term of Eq. (38) in series and considering only the ®rst two terms, one obtains dmH…z† ˆ

RT dcH…z† ÿ VH ds…z† css H…z†

…39†

where dcH(z )=DcH(z ) exp(st ) (see Eq. (4)). Now we can reformulate Eq. (5) in terms of dmH(z ). According to Eqs. (3) and (25) s vsd…z† ˆ vss d…z† ‡ dvd…z†

vss d…z† ˆ ÿDH

where

ss css H…z† dmH…z†

RT

dz

…40†

and

dvsd…z† ˆ Dvsd…z† exp…st†: Therefore dvsd…z† ˆ ÿDH

…css H…z† ‡ dcH…z† † @ dmH…z† : RT @z

…41†

…34†

dvsd…z†  ÿDH

…35†

The superscripts s indicate that the respective quantities depend on stress. Since the system is linear, the small-signal oscillations of the hydrogen chemical potential in the metal matrix should be proportional to those of the hydrogen concentration. Hence, let us linearize the dependence of dmH on dcH, having in mind that s ss (z )=0, as concluded in the last paragraph of Section 4. According to Eqs. (4), (21) and (23) mH…z† ˆ mss H…z† ‡ dmH…z†

0 ss mss H…z† ˆ mH…z† ‡ RT ln cH…z† :

Taking into account that at any value of the electrode dc potential, Edc, dcH(z ) <
 dcH…0† :

where

…33† 



4423

…36†

css H…z† @ dmH…z† RT @z

…42†

where dmH(z ) is de®ned by Eq. (39). Consequently, the small-signal forms of Eqs. (24) and (26) are, respectively:  dvsd…z† ˆ ÿDH

ss @ dcH…z† VH cH…z† @ ds…z† ÿ @z RT @z



 @ dcH…z† @ 2 dcH…z† ˆ sdcH…z† ˆ DH @t @ z2  ss 2 VH cH…z† @ ds…z† VH @ ds…z† @ dcH…z† ÿ ÿ : RT @ z2 RT @ z @z

…43†

…44†

The two last equations replace Eqs. (5) and (6). The small-signal forms of Eqs. (27) and (30) are, respectively:

4424

P. Zoltowski / Electrochimica Acta 44 (1999) 4415±4429

"

dsxx ˆ dsyy ˆ ÿ

ÿ

VH Y 1 dcH…z† ÿ L 3

12…z ÿ L=2† L3

…L 0

 dcH…z†

…L 0

where A is a constant:

dcH…z† @ z

 # L zÿ dz 2

…45†

and dszz ˆ 0 "   # … @ ds…z† 2VH Y @ dcH…z† 12 L L ˆÿ ÿ 3 dcH…z† z ÿ @z : 3 @z @z L 0 2 …46† In these equations, the original DcH(z ) has been changed to dcH(z ), because the reference state (no stress) is dcH(z ) at t = 0, that is equal to zero (see Eq. (4)). By insertion of Eq. (46) into Eqs. (43) and (44) one obtains the small-signal forms of Eqs. (31) and (32) "  @ dcH…z† 2V 2H Y ss s dvd…z† ˆ ÿ DH 1 ‡ c 3RT H…z† @z   # … 8V 2H Y ss L L c dcH…z† z ÿ ‡ @z …47† 2 RTL3 H…z† 0 "

 @ 2 dcH…z† 2V 2H Y ss 1‡ cH…z† 3RT @ z2  2 2V 2H Y @ dcH…z† ‡ 3RT @z #  !  …L @ dcH…z† 8V 2H Y L @z ÿ dcH…z† z ÿ : 2 RTL3 @z 0

@ dcH…z† ˆ DH @t

…48†

Similarly as in Eqs. (6) and (44), @dcH(z )/@t=sdcH(z ) for dcH(z ) see Eq. (4)). In Eq. (47) the ®rst term within the brackets represents the local (Fickian) di€usion accounting for stress, and the second is a non-local one. Eq. (48) is a second-order (partial) non-linear integro-di€erential equation, similarly as Eq. (32). Let us simplify the structure of Eqs. (47) and (48): @ dcH…z† dvsd…z† ˆ ÿ …DH ‡ Acss H…z† † @ z  …L L ss 12 ÿ AcH…z† 3 dcH…z† z ÿ @z L 0 2

…49†

@ dcH…z† ˆ sdcH…z† @t

 2 @ 2 dcH…z† @ dcH…z† ˆ …DH ‡ Acss † ‡ A H…z† @ z2 @z   …L @ dcH…z† 12 L ÿ A 3 dcH…z† z ÿ @z @z L 0 2

…50†

Aˆ

2DH V2H Y : 3RT

…51†

Eqs. (49) and (50) di€er from Eqs. (5) and (6) by the presence of (i) the integral expression in both equations, and (ii) the square term in Eq. (50). Let us discuss both these additional terms. In Eqs. (49) and (50), the integrals are the only quantities related directly to dcH(z ). Since dcH(z ) and especially its amplitude (DcH(z )) are small, the value of the integral is also small. Hence, the non-local (integral) terms in Eqs. (49) and (50) may be neglected as being small in comparison with the others. However, let us analyse the integral from Eqs. (49) and (50) in details. Taking Eq. (4) for dcH, one obtains …L 0

    …L L L @ z ˆ exp…st† DcH…z† z ÿ dz: …52† dcH…z† z ÿ 2 2 0

Thus, only the amplitude of dc(z ) is a function of the space coordinate. It is attenuated when dcH(z ) is penetrating the plate along 0 E z E L with the velocity of sound. dcH(z ) oscillates symmetrically around c ss H(z ). These oscillations do not need a long-range di€usion of hydrogen. Therefore, the integral in Eqs. (49) and (50) equals zero in both experimental conditions (permeable and impermeable). In the case of the impermeable conditions, there is an additional argument to neglect the non-local terms in Eqs. (49) and (50). The thickness of the foil is 2L (see Fig. 1). The plane at z=L is the plane of symmetry of the foil, its both external surfaces being at z = 0. Due to this symmetry, there is neither concentration nor stress gradients at z=L, both in the steadystate and small-signal scales. Hence, no bending of the foil takes place, in spite of the evident existence of the small-signal oscillations of hydrogen concentration and stress at any other value of z. Thus, according to the comments given above in Eqs. (30)±(32), the non-local terms in Eqs. (49) and (50) vanish. According to the discussion of Eq. (32), the term proportional to the square of the concentration gradient is probably negligible when the concentration gradient is small in comparison with the preceding term [10]. It is specially the case in EIS experiments (Eq. (50)), because dcH(z ) itself is small. The assumption on linearity of the system in the small-signal scale is an additional argument for neglecting the square term in Eq. (50). Therefore, Eqs. (49) and (50) can be simpli®ed, respectively, as follows: dvsd…z† ˆ ÿDsa

@ dcH…z† @z

…53†

P. Zoltowski / Electrochimica Acta 44 (1999) 4415±4429

@ dcH…z† @ 2 dcH…z† ˆ sdcH…z† ˆ Dsa @t @ 2z

…54†

where  Dsa ˆ DH

2V 2H Y ss c 1‡ : 3RT H…0†

…55†

The only di€erence between Eqs. (53) and (54) and the original Eqs. (5) and (6) is that the di€usion coecient of hydrogen, DH, is changed to the apparent diffusion coecient, D sa . In order to solve Eq. (54), one should ®nd the proportionality factor between dcH(0) and dv sd(0), i.e. to rede®ne the original transport operator Mp,i from Section 3 (see Eqs. (8) and (9)) to a new one, accounting for the e€ect of stress on the rate of hydrogen diffusion. The boundary conditions are the same as in Section 3 (Eq. (7)). Hence, the solution of Eq. (54) is similar to that of Eq. 6 (see Eq. (8)): dcH…0† ˆ dvsp,i Msp,i

…56†

M sp,i denotes the transport operator for permeable and impermeable conditions, respectively, taking the e€ects of the stress into account. It is similar to the original Mp,i (Eq. (9)): Msp

…ms nst †ÿ1

nst

…57a†

Msi ˆ …ms nst †ÿ1 coth nst

…57b†

ˆ

tanh

where ms ˆ Dsa =L

and nst ˆ

q sL2 =Dsa :

Y sF can be identi®ed with the admittance of the electrical circuit presented in Fig. 2(c): Y sec ˆ



…58†

It is evident from Eq. (56) that dcH(0) is dependent on the stress. The original mass and charge balance equations (Eqs. (12) and (13)) remain essentially unchanged. Only the quantities should be supplemented by the superscript s: dvsab ˆ dvsd…0†

…59†

djsF ˆ ÿFdvsab

…60†

From Eqs. (35), (56), (59) and (60), the equation for the faradaic admittance of the electrode processes, accounting for the e€ects of stress (analogue of Eq. (15)) can be derived:  s  @ vab ÿF @E c  s  H…0† : Y sF ˆ …61† @ vab 1ÿ Msp, i @ cH…0† E

4425

1 : Rsab ‡ ZDEsp, i

…62†

Eq. (62) di€ers from Eq. (16) by the presence of the superscripts s. The equivalent circuit in Fig. 2(c) di€ers similarly from that in Fig. 2(b). The superscripts s indicate that the given quantities are modi®ed by the stress. The most important e€ect of the stress is in the de®nition of ZDEsp, i : ZDEsp, i ˆ sst Msp, i :

…63†

The following quantities in Eqs. (18)±(20) should be supplemented with the superscript s: Rab, vab, st, s eq t , Cab and C eq ab. Under this condition, Eqs. (18)±(20) and the comments on them near the end of Section 3 also hold for the respective quantities in Fig. 2(c) and Eq. (62).

6. Discussion According to Eq. (55), the stress induced by the presence of hydrogen in the metal matrix enhances its diffusion. The e€ective transport of hydrogen is characterized by D sa . It equals DH only when s c ss H(0) 3 0. Otherwise, D a is always larger than DH. The higher the concentration of hydrogen in the metal directly beneath its surface, the larger is the di€erence between D sa and DH. The ISs give information only on D sa . DH can be calculated from D sa by Eq. (55). The values of VH, Y and c ss H(0) cannot be obtained in EIS experiments. In particular, the concentration of hydrogen in the metal matrix can be measured by hydrogen extraction, using electrochemical or vacuum methods, from the foil specimen saturated with hydrogen at a given electrode potential (Edc). This equilibrium concentration, c eq H , is the same over the whole thickness of the electrode ss (0 E z E L ). Hence, c eq H =c H(0). Values of DH presented in the EIS literature [24±30] have to be considered as D sa . As an example, let us perform the correction for the e€ect of stress for several data collected for Pd±H samples in the impermeable conditions (i.e. formally close to the equilibrium) [27,29]. For VH and Y of palladium, we assume 1.77  10ÿ6 m3 moleÿ1 of H [22,42] and 1.844  1011 Pa [46], respectively. The data [27,29] and their corrections are given in Table 1. The relative corrections, ÿ(DHÿD sa )/D sa , are small, up to ca. ÿ2%, because c eq H are small. Probably these concentrations measured using the H2SO4 electrolyte [27] correspond to the equilibrium saturation.

4426

P. Zoltowski / Electrochimica Acta 44 (1999) 4415±4429

Table 1 s Data on the PdHx system measured by EIS (Edc, c eq H , DH, and D a are electrode potential, hydrogen equilibrium concentration in the metal matrix, di€usion coecient of hydrogen and apparent di€usion coecient, respectively) Samplea, electrolyte, temperature

Edc vs RHE (V )

a., s., 1 M H2SO4, 300 K a., abr., 0.1 M NaOH, Room temp.

0.165 0.115 0.160 0.120 0.100 0.080 0.060 0.040 0.020

x

c eq H

± ± 0 4.0  10ÿ5 0 8.0  10ÿ5 0 1.3  10ÿ4 0 2.0  10ÿ4 0 3.4  10ÿ4 0 6.0  10ÿ4 0 1.1  10ÿ3

3.9  101 1.5  102 04.6  100 09.1  100c 01.5  101c 02.3  101 03.9  101c 06.8  101c 01.3  102c

Original DH (D sa )  1011 (m2 sÿ1)

ÿ DH ÿ Dsa Dsa  100 (%)

4.9 3.8 0 5.8 0 5.0 0 4.2 0 3.2 0 2.7 0 2.1 0 1.3

0.6 2 0.07 0.1 0.2 0.4 0.6 1 2

Ref.

27 29b

a

a., annealed; s., smooth; abr., Pd samples abraded with the emery paper SiC ]1200 directly prior to the experiments. Values of DH and x taken from Figs. 4 and 6, respectively, in [29]. c Value calculated from x in the preceding column.

b

The data obtained using the NaOH electrolyte [29] are smaller by up to 1.5 orders of magnitude. Hence, they cannot correspond to the saturation. Furthermore, the scale of potentials in [29] can be erroneous, because the potential of the a t b-phase transition of the Pd± H system is quoted by the authors as ca. +80 mV/ RHE [25,29,30,48], instead of +50 mV/RHE [1± 3,49,50]. The maximal value of x for a-PdHx at room temperature is ca. 0.025 [1±3], i.e. cH32.9  103 mole mÿ3. In this case the correction of the di€usion coecient derived directly from the EIS data can be estimated as ÿ31%. For some Pd alloys, where the upper limit of hydrogen concentration in the a-phase is much higher (e.g. for the Pd81Pt19 alloy at room temperature, the value of x in a-MeHx can equal up to ca. 0.4 [7], i.e. cH34.9  104 mole mÿ3), this correction can be up to ca. ÿ90%. In other words, DH can be lower by one order of magnitude than D sa . In Fig. 3, typical examples of the ISs for the electrode system when the electrode processes are hindered mainly by the ®nite di€usion are presented in the impedance complex plane. ISs have been computed for the model given in Fig. 2(a), where F is alternatively the circuit of Fig. 2(b) or 2(c). The similarity of these subcircuits should be underlined. Rs has been assumed as equal to 1 O m2. In order to minimize the e€ects of these parameters, very small values for Cd and Rab or R sab were used. Hence, the plots are in¯uenced only by Rs and the parameters of ZDEp,i or ZDEsp,i (see Eqs. (9), (17), (57) and (63)): L, and st and DH or s st , and D sa , respectively. It has been assumed that L = 5  10ÿ5 m, and st or s st =1  10ÿ6 O m3 sÿ1.

Curves a ' and b ' (empty points) present ISs simulated for DH=1  10ÿ11 m2 sÿ1 under the assumption of no e€ect of stress (D sa =DH), for restricted and bounded di€usion, respectively. For simulation of curves a0 and b0 (solid points), the same value of DH and the presence of the e€ect of stress as for aPdH0.025 has been assumed. Namely, D sa =DH(100+31)/10031.3  10ÿ11 m2 sÿ1. Therefore, the di€erence between the curves a ' and a0, and b ' and b0 illustrates the e€ect of stress on ISs in the case of restricted and bounded di€usion, respectively. When the EIS experiment in impermeable conditions is performed using the thin-layer electrode (see Section 3), i.e. a layer of the hydrogen-sorbing metal deposited on a substrate neutral for hydrogen absorption, there is no symmetry plane at z=L. Nevertheless, there is no bending of the plate, because of the rigidity of the thick substrate. Hence similarly as in the case of the foil specimen, the integral non-local terms in Eqs. (49) and (50) would probably vanish. It should be emphasized that the EIS experiments in the impermeable conditions o€er a chance for studying the dependence of the di€usion coecient of hydrogen on the hydrogen concentration in the metal matrix. The real metal matrix is not perfect. There are microstructural imperfections (grain boundaries, dislocations, etc.), which can act as traps for hydrogen. Their presence a€ects the directly observed di€usion coecient of hydrogen. Hence, the last should be treated also as only an apparent di€usion coecient, D ta [17,51,52]. The correction is to some extent similar to that derived in this paper for the e€ect of stress, D sa . However, in contrast to the stress, the traps hinder the

P. Zoltowski / Electrochimica Acta 44 (1999) 4415±4429

4427

Fig. 3. Complex-plane impedance diagram of spectra for the electrode system when the faradaic process is hindered only by ®nite di€usion. The data were simulated for the model of Fig. 2(a) with the model of Fig. 2(b) or 2(c) inserted for F. The following values (assumed for unit surface area of 1 m2) have been applied: Rs=1 O m2, Cd=1  10ÿ10 Fmÿ2 and Rab and R sab=1  10ÿ10 O m2 (to eliminate the e€ects of the last two elements). For the parameters of ZDEp.i and ZDEsp.i (see Eq. (10), (17), (58) and (63)) the following values have been applied: st=sst =1  10ÿ6 O m3 sÿ1, DH=1  10ÿ11 m2 sÿ1, and L = 5  10ÿ5 m. Curves a' and b' are without the e€ect of stress (Fig. 2(b)), for bounded and restricted di€usion, respectively. Curves a0 and b0 are for D sa =1.3  10ÿ11 m2 sÿ1 (Fig. 2(c)), i.e. when the di€usion is enhanced by stress as in the case of the maximal saturation of a-Pd by hydrogen (room temperature). Numbers at the curves indicate the frequency, in Hz.

di€usion of hydrogen in the metal matrix. According to Eq. (55), D sa e DH, and D sa approaches DH when the hydrogen concentration decreases. The e€ect of traps is opposite, i.e. Dta E DH, and Dta approaches DH when the hydrogen concentration increases (e.g. see Eq. (86) in [18]). The presence of traps results also in excess values of the apparent c ss H [52]. In this paper, the possible e€ects of the term proportional to the square of the gradient of hydrogen concentration in the balance equation (Eq. (48) or (50)) have been neglected as being insigni®cant. This term can cause some non-linear e€ects [12], and EIS gives opportunity for their analysis. It should be emphasized that the di€erence between DH and D sa represents only the local e€ect of stress on the di€usion of hydrogen in the metal matrix. The non-local, i.e. uphill e€ect, is inaccessible in a one-port system.

7. Conclusions The e€ects of self-induced stress should be taken into account in the EIS analysis of the rates of hydro-

gen sorption processes in metals. These e€ects do not alter the general character of the EIS data of the system, neither in the bounded nor restricted di€usion conditions of hydrogen. However, as the result of the e€ect of stress on the local di€usion of hydrogen in the metal matrix, the di€usion coecient derived directly from the EIS experimental data is not the true (DH), but the apparent one (D sa ). DH E D sa . The higher the hydrogen concentration in the metal matrix, the larger is the di€erence. For a-phase Pd±H system at room temperature, ÿ(DHÿD sa /)D sa can be up to ca. 30%. For some a-phase Pd alloys, where the upper limit of hydrogen concentration is much higher, ÿ(DHÿD sa /)D sa can attain even one order of magnitude. To calculate DH from D sa , information on the equilibrium value of the hydrogen concentration in the metal at the given electrode potential, the partial molar volume of hydrogen in the metal, and the elastic modulus of the metal matrix is needed. The EIS experiments in the restricted di€usion conditions give opportunity to study the dependence of the di€usion coecient of hydrogen on its concentration in the metal matrix. The e€ect of the self-induced stress on the rate of hydrogen di€usion is opposite to the e€ect of the hydrogen traps.

4428

P. Zoltowski / Electrochimica Acta 44 (1999) 4415±4429

The e€ect of the self-induced stress on the rate of the surface processes of hydrogen absorption is only an indirect one. It arises from the change of the concentration of hydrogen directly beneath the metal surface, modi®ed by the e€ect of the stress on the hydrogen di€usion. By EIS, only the local e€ect of stress on the di€usion of hydrogen can be studied. The non-local, uphill e€ect is inaccessible.

[17] [18] [19] [20] [21] [22]

Acknowledgements This work was supported by the State Committee for Scienti®c Research through grant 2P303 075 07. The author expresses his gratitude to Professor B. Baranowski from Warsaw for the inspiration to undertake this work and important discussions.

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