Journal of Electroanalytical Chemistry 501 (2001) 89–99www.elsevier.nl/locate/jelechem

Diffusion of hydrogen in self-stressed metals — transfer functionspectroscopy approach

Piotr Zoltowski *Institute of Physical Chemistry of Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland

Received 22 March 2000; received in revised form 14 October 2000; accepted 14 November 2000

Abstract

The transfer function spectroscopy approach to the diffusion of hydrogen in a self-stressed isotropic elastic metal matrix isproposed. The system should be close to equilibrium. It is perturbed by a sine-wave input signal applied at one surface of thethin-plate specimen. The magnitude of this signal, of varied frequency, is small enough to treat the system as linear. The responsesignal is measured at the opposite surface of the specimen. The transfer function is the ratio of steady-state response to inputsignals. The hydrogen concentration and hydrogen flux are the input and output signals, respectively. The diffusion equations arederived, and they are solved analytically. The resulting transfer function is discussed in terms of hydrogen permeation through aspecimen of properties similar to palladium and Pd81Pt19 alloy, in a wide range of hydrogen concentrations in the metal matrix.It is demonstrated that at relatively high frequencies the transfer function is highly sensitive to the non-Fickian diffusion, resultingfrom the non-local effect of self-stress. In contrast, at infinitesimally low frequency, i.e. at steady-state, both local and non-localeffects compensate. Hence, the self-stress is absent. Under the proposed experimental conditions the transfer function spectroscopyis more appropriate for studying the diffusion of hydrogen in self-stressed metals than the commonly used transient break-throughmethod. It should allow the study of the diffusion coefficient of hydrogen in metals, and, moreover, of the elastic modulus ofmetal–hydrogen solids, both these quantities as function of hydrogen concentration in isotropic matrixes. © 2001 Elsevier ScienceB.V. All rights reserved.

Keywords: Hydrogen diffusion; Stress; Palladium and its alloys; Transfer function spectroscopy; Electrochemical impedance spectroscopy

1. Introduction

It is well established that gradients of concentrationof interstitial hydrogen in solid metals result in self-in-duced mechanical stress. Stress influences the chemicalpotentials of components of the solid. Hence, the self-stress influences the transport of hydrogen in metals[1–9].

There are two effects of stress, local and non-local,on the transport of hydrogen in an elastic solid [1–9].The former enhances the transport in comparison tothat described by the classical formulation of Fick’sequations [1–10]. Owing to this effect, the apparentdiffusion coefficient of hydrogen in a metal may beeven one order of magnitude larger than the true one[10]. The non-local effect results from the fact that the

stress induced by a local gradient of the hydrogenchemical potential is transmitted with the velocity ofsound (i.e. immediately, in comparison with the rate ofdiffusion processes) over the whole volume of the elas-tic solid. The higher the hydrogen concentration in thematrix, the higher are the two effects [1–10].

Until now, the impact of self-stress on the transportof hydrogen in metals has been studied mainly bytransient break-through experiments (TBT). A largethin plate of a metal sorbing large amounts of hydrogen(usually palladium alloys) is used as the specimen. Thisplate is placed as a membrane separating two chambersof controlled activity of hydrogen. If the initial activityis non-zero, its sudden increase in one (input) chamberresults in an immediate transient sucking of hydrogeninto the specimen from the opposite (output) chamber,prior to the appearance of an increasingly positive flux.This phenomenon is called ‘uphill diffusion’. The ap-propriate diffusion equations are fairly complex. More-

* Fax: +48-22-6325276.E-mail address: pizolt@ichf.edu.pl (P. Zoltowski).

0022-0728/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved.PII: S0022-0728(00)00507-6

P. Zoltowski / Journal of Electroanalytical Chemistry 501 (2001) 89–9990

over, the boundary conditions of diffusion evolve dur-ing the experiment [1–9]. In spite of that, a goodagreement between the transport equation and the ex-perimental results has been obtained for initial times,when the boundary conditions are approximately con-stant [5,6].

Recently, electrochemical impedance (more generally,immittance) spectroscopy (EIS) has been proposed tostudy the impact of self-stress on the absorption ofhydrogen into a metal electrode [10,11] and its diffusioninside [10]. Immittance is the ratio of electric potentialinput and current output signals (or vice versa) of aone-port system, i.e. at the same electrode surface.Thus, the non-local effect of stress on hydrogen diffu-sion cannot be noticed. Consequently, only the appar-ent diffusion coefficient of hydrogen, i.e. the (true)diffusion coefficient increased by the local effect ofstress, can be estimated by EIS [10].

In the past, several types of generalized immittancespectroscopy, i.e. limited neither to electric signals norto one-port systems, have been applied to various sys-tems [12]. In these methods, the ratio of output signalto input signal is the transfer function (TF) of thesystem studied. So, any generalized immittance spec-troscopy can be named ‘transfer function spectroscopy’(TFS). Electro-gravimetric [12–14], magneto-electro-chemical [12,15] and opto-electrochemical spectroscopyof one-port systems are among the examples [12]. TFSstudies on the transport of hydrogen in metals involv-ing both sides of the specimen were also performed,hence treating the system as a two-port one. Experi-ments were carried out for palladium, iron and theiralloys, and other materials, in gas as well as in elec-trolyte solution environments [16–28]. Recently, a TFSstudy on hydrogen diffusion through a metal specimenhas been reported, where the input and output signalswere the hydrogen pressure sine-wave and ac current ofhydrogen ionization, respectively [28]. Also a large re-view and theoretical analysis of electrochemical acmethods, including TFS, for the investigation of sorp-tion of hydrogen in metals has been published [29–31].However, in none of these studies was the impact ofstress on diffusion of hydrogen in metals taken intoaccount, nor were the experimental conditions neces-sary to observe its non-local effect even attained.

The aim of this work is to describe the transport ofhydrogen in an elastic solid matrix for a two-portsystem, in terms of TFS. The goal is to analyze thelocal and non-local effects of self-stress on the diffusionof hydrogen. At first, the TBT approach to this trans-port is reviewed. Next, the proposed TFS approach ispresented and discussed.

Throughout this paper, the metal specimen is as-sumed to be a continuous single-phase (isotropic) elas-tic solid. It consists of a large thin flat plate. It isapplied as a membrane, of thickness L, separating two

compartments of controlled activities of hydrogen.Only one-dimensional diffusion of absorbed hydrogen,Hab, from one large surface of the specimen (z=0) to aparallel surface at a distance L is discussed:

Hab(z=0)l6d

Hab(05z5L) (1)

where 6d denotes the rate of this process. 6d is equiva-lent to the flux of diffusing hydrogen, JH.

Our attention is focused on the diffusion process. Forthe sake of simplicity, all other possible processes areneglected, except the ab/desorption process of atomichydrogen into/out of the specimen (HlHab). The lat-ter is assumed to be at equilibrium, irrespective of thepresence of stress.

Essentially, it is not important whether the surround-ings (input and output) of the specimen are a gas orsolution of an electrolyte. The absorption of atomichydrogen must be preceded by dissociation of molecu-lar hydrogen (H2l2H), or by reduction of proton orwater (e.g. H++e−lH).

The majority of quantities in TFS are vectors (simi-larly as in EIS). In the notation used in this paper theiramplitudes (D) are complex, i.e. their modulus andargument describe the magnitude and phase angle, re-spectively. For instance, the ratio of magnitudes ofsecondary and primary concentration signals describesthe attenuation resulting from the diffusion process,and the argument of the secondary signal describes thepertinent delay, i.e. the phase shift.

2. Transient break-through (TBT) approach

TBT experiments are performed for the system beingoriginally at equilibrium at some non-zero concentra-tion of hydrogen. At the beginning of the measurement,a large step change of hydrogen chemical potential(pressure or electrode potential) is suddenly applied atthe input side of the specimen (z=0), and the resultingflux of permeating hydrogen is measured at its outputside (z=L) as a function of time. Initially, the sign ofthe response is opposite to that of the signal, because atearly times the non-local effect of self-stress on diffu-sion dominates [5,6].

The chemical potential of interstitial hydrogen, mH,as a mobile component in a solid matrix depends onhydrogen concentration, cH, and mechanical stress, s

[5,6,9]:

mH=mH(0, cH)−VHs (2)

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

(below called ‘stress’) is the trace of the stress tensorinduced by the presence of hydrogen in the metal(s=sxx+syy+szz). So, s is the hydrostatic part of

P. Zoltowski / Journal of Electroanalytical Chemistry 501 (2001) 89–99 91

this tensor, and is an analogue of hydrostatic pressure.VH is assumed to be independent of the hydrogenconcentration.

In Eq. (2), mH(0, cH) can be replaced by the conven-tional expression:

mH(0, cH)=mH0 +RT ln fHcH (3)

where R, T and fH denote the gas constant, temperatureand activity coefficient of hydrogen, respectively.

The transport equation (modified 1st Fick equation)for the flux of hydrogen, JH, diffusing along the zcoordinate in the bulk of the metal, caused by thegradient of the chemical potential is [5,6,9]:

JH= −LH

#mH

#z= −DH

��1+#ln fH

#ln cH

�#cH

#z−

VHcH

RT#s

#zn(4)

where LH and DH denote the phenomenological coeffi-cient (in terms of irreversible thermodynamics) anddiffusion coefficient of hydrogen in the metal matrix,respectively. The latter is assumed to be independent ofcH. For an ideal solution ( fH=1):

LH=DHcH/(RT) (5)

The second term within the brackets in Eq. (4)represents the elastic part of the chemical potentialgradient. It describes the stress-induced diffusion (SID)[4].

The balance equation (modified 2nd Fick equation),under the assumptions that (i) the diffusion coefficientof hydrogen depends neither on its concentration noron stress, and (ii) LH is defined by Eq. (5), takes theform [5,6]:

#cH

#t=DH

��1+#ln fH

#ln cH

�#2cH

#z2 −VHcH

RT#2s

#z2−VH

RT#s

#z#cH

#zn(6)

In order to solve Eqs. (4) and (6), a relationshipbetween the gradients of cH and s is needed. For thispurpose, the analogy with the effect of thermo-stressarising from the thermal expansion of the elastic matrixon the conduction of heat is employed. That results ina feedback relationship of the gradients of hydrogenconcentration and stress in the metal matrix [5,6]:

#s

#z= −

23

VHY( �#cH

#z−

12L3

& L

0

DcH�

z−L2�

dzn

(7)

where Y( denotes the bulk elastic modulus of the solidspecimen (Y( =E( /(1−n), where E( and n are the Youngmodulus and the Poisson ratio, respectively; Y( is as-sumed to be independent of the hydrogen concentrationin the metal matrix), and DcH=cH−cH,0, where cH

denotes the actual hydrogen concentration, and cH,0 itsconcentration in the initial stress-free state.

Accordingly, Eqs. (4) and (6) change respectively to[5,6,9]:

JH= −DH��

1+#ln fH

#ln cH

+2VH

2 Y( cH

3RT�#cH

#z

−8VH

2 Y( cH

L3RT& L

0

DcH�

z−L2�

dzn

(8)

#cH

#t=DH

!�1+#ln fH

#ln cH

+2VH

2 Y( cH

3RT�#2cH

#z2 +2VH

2 Y(3RT

�#cH

#z�2

−�8VH

2 Y(L3RT

& L

0

DcH�

z−L2�

dzn#cH

#z"

(9)

In Eq. (8), the first term within the brackets describesthe local (Fickian) diffusion accounting for the stress.Thus, stress always enhances the local diffusion (even ifVHB0), just as the activity coefficient of hydrogendoes. The second term is the non-local one, because itdepends on the integral of the composition profiletaken over the whole thickness of the plate (05z5L).It has to be emphasized that the flux resulting from thesecond term arises in each elementary volume of thespecimen, irrespective of the local concentration gradi-ent [6].

Eq. (9) is a partial second order non-linear integro-differential equation. The first term within the braces isproportional to the second derivative of concentration,similarly as the analogous term in the 2nd Fick equa-tion. It includes the local effect of stress on diffusion.The second, non-linear term, proportional to the squareof the concentration gradient, results only from thestress. It will be negligible if the concentration gradientis small. However, as emphasized by Baranowski [6], ‘itcould be a source of interesting non-linear phenomena,including oscillations and more complex dissipativestructures’. The third term, proportional to the concen-tration gradient and including the integral, is due to theasymmetry of hydrogen distribution with respect to thez=L/2 plane [5,6].

The above diffusion equations are fairly complex.Moreover, at both surfaces of the specimen theboundary conditions for diffusion of hydrogen evolvein time [5,6].

Recently, a success in the theoretical description ofthe TBT experiments for initial times has been reported[5,6]. The flux of diffusing hydrogen (Eq. (8)) wasformally decoupled in local and non-local parts, andthe integral in the non-local part was solved undersimplifying assumptions: (i) the diffusion is of infinitecharacter (L=�), (ii) z in the term within parenthesesunder the integral can be neglected as much smallerthan L, and (iii) the boundary conditions of diffusionare constant. This approach is substantiated as long asthe Fickian transport resulting from the input signalhas not penetrated deep into the specimen. The result-ing solution has given an apparent consistency of theplots of theoretical and experimental fluxes of hydrogenat the output side of the Pd81Pt19 specimen in the firstperiod of time (up to more than 10 min in the appliedexperimental conditions) when the magnitude of the

P. Zoltowski / Journal of Electroanalytical Chemistry 501 (2001) 89–9992

output signal, opposite to the input one, increasesmonotonically.

3. Transfer function spectroscopy (TFS) approach

3.1. Principles

High and low frequencies in the frequency domaincorrespond in the time domain to short and long times,respectively. On the basis of the results of TBT one mayexpect that the non-local effect of stress on the flux ofpermeating hydrogen will dominate at relatively highfrequencies, whereas the Fickian diffusion will domi-nate at low frequencies.

The essential requirements for the proposed TFSexperiments on hydrogen diffusion in metals are: (i) thesystem should be a two-port one, i.e. the output signalshould be measured at the side of the specimen oppo-site to that of the input signal (similarly as in TBT, andin contrast to EIS [29,32–38]), (ii) the input signalshould be small enough to treat the system as linear (asin EIS), and (iii) the chemical potential of hydrogenshould be close to equilibrium across the whole sectionof the specimen (05z5L) during the whole experi-ment (in contrast to TBT).

In Fig. 1 the system proposed for TFS measurementsis presented in comparison to those used in EIS.Namely, the specimen is a membrane separating twochambers (similarly as in EIS permeable conditions,Fig. 1A). The concentration of hydrogen in the speci-

men is close to equilibrium (similarly as in EIS imper-meable conditions, Fig. 1B). Hence, the systemproposed for TFS resembles that used in TBT experi-ments prior to the application of the input signal. Theinput signal is a sine-wave concentration of hydrogensuperimposed at one (input) surface on the constanthydrogen concentration. The output signal is the oscil-lating flux of hydrogen at the opposite (output) surface,where the constant concentration of hydrogen is main-tained during the experiment.

The proposed TF is the ratio of the oscillating fluxoutput signal to the oscillating concentration input one.It is very similar to the TF proposed and applied inmeasurements of hydrogen diffusion in metals byKedzierzawski et al. [19]. It is similar also to the TFsapplied by Sekine [20], and recently by Bruzzoni et al.[27,28]. However, in all these small-signal measure-ments, as well as in all reviews [18,25,29], the concen-tration of hydrogen at the output surface of thespecimen is kept equal to zero when the flux of perme-ating hydrogen is measured there. This is just as it wasin the original large-signal transient experiments ofDevanathan and Stachurski on diffusion of hydrogen[39]. That excludes the possibility of observation of thenon-local effect of stress on diffusion. On the contrary,in the proposed method the concentration of hydrogenat the output surface is the equilibrium concentration,as it is in the whole system. Only the small-signaloscillating component of hydrogen concentrationequals zero, and the small-signal flux of permeatinghydrogen is the sole component of the flux at theoutput surface.

Fig. 1. Schemes of the system in typical EIS experiments (A and B) and that proposed for the TFS experiments (C).

P. Zoltowski / Journal of Electroanalytical Chemistry 501 (2001) 89–99 93

The simplest technical means to perform the pro-posed measurements are offered by commercial electro-chemical equipment. They should be performed underelectric potential control because of the sensitivity ofthe MeHn systems to the a�b-phase transition.

The potential applied to the input side (subscript ‘i’)of the specimen, being in this case a bi-electrode, isE(i)=Edc(i)+dE(i), where dE(i) denotes an ac small-sig-nal input signal. The output side (subscript ‘o’) of thespecimen is also under potentiostatic control, E(o)=Edc(o) Edc(i). Hence, dE(o)=0, and the small-signaloutput current, dj(o), is the output signal. The resultingac diffusion is bounded (similarly as in the EIS experi-ment presented in Fig. 1A).

The ab/desorption process is assumed to be at equi-librium (see Section 1). Therefore, the small-signalchanges of chemical potential of hydrogen and its con-centration directly under the electrode input surface areproportional to the oscillations of the input potential.

The proposed TF of the system, H(v), is defined asfollows:

H=dj(o)

dE(i)

=Dj(o) exp(st)DE(i) exp(st)

=Dj(o)

DE(i)

(10)

H is a two-port-system counterpart of immittance inEIS.

A pretreatment of the system at Edc(o)$Edc(i) is re-quired prior to the measurements, to allow the systemto be close to equilibrium at the given dc potential. Thisequilibrium extends essentially from the place where theinput signal is applied to that where the output signal ismeasured. Moreover, an ac-pretreatment at every fre-quency of the input signal is also required, since in TFSthe small-signal periodic steady-state of the system isrequired.

3.2. Kinetics

The input signal is defined similarly as in EIS [10]:

dE(i)=DE(i) exp(st) (11)

where D denotes the amplitude, s= iv, i is the imagi-nary unit (i=−1), and v the angular frequency(v=2pf, f being frequency). The frequency of dE(i) isvaried over a wide range. In response, any variable, l,of process k and its rate, d6k, deviate from their equi-librium values, l eq and 6k

eq, similarly as does dE. In oursimplified case, the only variable is the concentration ofhydrogen in the metal matrix (05z5L):

dcH=DcH exp(st) (12)

and diffusion of hydrogen inside the specimen (Eq. (1))is the only process analyzed. Its rate is:

d6d=D6d exp(st) (13)

Let us notice in advance that from Eq. (12) it followsthat:

#dcH

(t=sdcH (14)

For simplicity, it will be assumed that the solution ofhydrogen in the metal is ideal. This means that theactivity coefficient of hydrogen ( fH) is equal one, irre-spective of the hydrogen concentration. As alreadymentioned, the ab/desorption process (H++elHab) isassumed to be at equilibrium. Hence, the dependence ofthe oscillating concentration of hydrogen at the inputsurface, dcH(0), on dE(i) for a linear system can bewritten as follows [10]:

dcH(0)=FcH

eq

RTdE(i) (15)

where F and cHeq denote the Faraday constant and the

equilibrium concentration of hydrogen, respectively.Formally, the above equation is valid only for relativelysmall concentrations of hydrogen. The assumption onlinearity of the system allows one to apply a lineardependence of small-signal oscillations of the chemicalpotential of hydrogen on oscillations of hydrogen con-centration and stress [10]:

dmH=RTcH

eq dcH−VHds (16)

Accordingly, the dependence of the oscillations of thehydrogen flux, i.e. rate of diffusion, on the oscillationsof hydrogen chemical potential is [10]:

dJH$−DH

cHeq

RT#dmH

#z(17)

Consequently, the small-signal forms of Eqs. (4), (6)and (7) are:

dJH= −DH�#dcH

#z−

VH(cHeq+dcH)RT

#ds

#zn

(18)

#dcH

#t=DH

�#2dcH

#z2 −VH(cH

eq+dcH)RT

#2ds

#z2

−VH

RT#ds

#z#dcH

#zn

(19)

#ds

#z= −

23

VHY( �#dcH

#z−

12L3

& L

0

dcH�

z−L2�

dzn

(20)

When Eq. (20) is substituted into Eqs. (18) and (19),the following equations of diffusion (modified 1st and2nd Fick equations) are obtained:

dJH= −DH!�

1+2VH

2 Y(3RT

(cHeq+dcH)

n(dcH

(z

−8VH

2 Y(RTL3(cH

eq+dcH)& L

0

dcH�

z−L2�

dz"

(21)

P. Zoltowski / Journal of Electroanalytical Chemistry 501 (2001) 89–9994

#dcH

#t=DH

!�1+

2VH2 Y(

3RT(cH

eq+dcH)n#2dcH

#z2

+2VH

2 Y(3RT

�#dcH

#z�2

−�8VH

2 Y(RTL3

& L

0

dcH�

z−L2�

dzn#dcH

#z"

(22)

Eqs. (21) and (22) are similar to the respective TBTequations (Eqs. (8) and (9)). In Eq. (21) the first termwithin the braces describes the Fickian diffusion ac-counting for the local effect of stress. The second term,proportional to the integral, is the non-local one. Theflux resulting from this term arises in each elementaryvolume of the specimen.

Eq. (22) is a partial second order non-linear integro-differential equation. Its solution should be fairly non-linear. For instance, it is directly visible that the second,square term within the braces will result in a secondharmonics (see also the comment on Eq. (9)). Similarly,the first and third terms within the braces are non-lin-ear. Namely, in every case dcH as well as the integral,(dcH/(z and (2dcH/(z2 are proportional to dcH(0) (seee.g. Eqs. (37), (39) and (41)). Therefore, also the firstand third terms must result in more complex phenom-ena than the fundamental harmonics.

However, for small-signal measurements the follow-ing relation can be applied:

cHeq+dcH$cH

eq (23)

The above approximation is crucial for obtaining aharmonic solution of Eq. (22). Namely, it allows theflux equation of diffusion (Eq. (21)) to be simplified asfollows:

dJH= −DH!�

1+2VH

2 Y(3RT

cHeq�(dcH

(z

−8VH

2 Y(RTL3cH

eq& L

0

dcH�

z−L2�

dz"

(24)

In turn, from Eq. (24) the following balance equationof diffusion results:

#dcH

#t=DH

�1+

2VH2 Y(

3RTcH

eq�#2dcH

#z2 (25)

Eqs. (24) and (25) are analogues of the 1st and 2ndFick equations for small-signal diffusion of hydrogen ina metal matrix, accounting also for the effects of self-stress. They are linear. Hence, they do not involvehigher harmonics or other non-linear effects. In otherwords, they filter out all possible non-linear effects.

Eqs. (24) and (25) will be used to deriving the finalequation for the transfer function defined originally byEq. (10). They can be rewritten as follows (see Eq.(14)):

dJH= −Da

ddcH

dz+DH

12AL3 cH

eq& L

0

dcH�

z−L2�

dz (26)

Da

d2dcH

dz2 −sdcH=0 (27)

where A is a dimensionless constant proportional to thelocal effect of stress, and Da is the apparent diffusioncoefficient of hydrogen, i.e. the (true) diffusion coeffi-cient increased by the local effect of stress [10]:

A=2VH2 Y( /(3RT) (28)

Da=DH(1+AcHeq) (29)

In contrast to EIS [10], the integral in Eq. (26),resulting from the non-local effect of stress on the fluxof hydrogen diffusing in the metal matrix is not equalto zero. This integral (apparently, the gradient of theamount of hydrogen over the z coordinate) is a measureof asymmetry in dcH distribution with respect to theplane z=L/2, weighted by the distance from this plane.The higher the dcH frequency, the larger is the modulusof this integral. It causes a linear oscillatory bending ofthe specimen, in response to the oscillations of the inputconcentration of hydrogen.

As pointed out earlier, the recent success in theinterpretation of the TBT experimental results has beengained by decoupling the hydrogen flux (Eq. (8)) intolocal and non-local components, and solving the inte-gral for early times under the assumption that diffusionis of infinite character. However, for longer times thisapproach cannot be followed, because of the evolutionof the boundary conditions of diffusion at both surfaces[5,6]. In contrast to TBT, in TFS the boundary condi-tions are invariant (periodic steady-state), and the sys-tem is linear. This allows one to proceed further thanwas possible in TBT, to extend the analysis down to thelowest frequencies, corresponding to arbitrary longtimes.

As in the TBT analysis [5,6], we will start by decou-pling the flux (Eq. (26)) into the local and non-localcomponents (subscripts ‘loc’ and ‘nloc’, respectively):

dJH=dJH,loc+dJH,nloc (30)

where:

dJH,loc= −Da

ddcH

dz(31)

dJH,nloc=DH

12AL3 cH

eq& L

0

dcH�

z−L2�

dz (32)

In contrast to TBT, the above decoupling is not onlyformal. It is physically substantiated for each value offrequency. Moreover, in TBT JH,nloc depends on thespace coordinate, because it is proportional to the localconcentration of hydrogen (see the second term withinthe brackets in Eq. (8)) [6]. On the contrary, in TFSdJH,nloc" f(z), i.e. dJH,nloc is the same within the wholespecimen, because it is proportional to cH

eq.Similarly as in Ref. [6], let us underline that the

non-local flux can be written as

P. Zoltowski / Journal of Electroanalytical Chemistry 501 (2001) 89–99 95

dJH,nloc=LcHeq (33)

where L has the physical meaning of mobility of hydro-gen in the metal matrix, i.e. the rate of displacement ofhydrogen. However, the reason of this displacement islimited solely to the self-stress. In TFS, the mobility isa complex quantity, dependent on the frequency of thesignal.

The general solution of Eq. (27) for neutral particles(like atomic hydrogen) diffusing in one-dimensional (z)space in an isotropic matrix is as follows (e.g. Ref. [40]):

dcH=B1 sinh(s/Daz)+B2 cosh(s/Daz) (34)

where the constants B1 and B2 should be selectedproperly in order to satisfy appropriate boundary con-ditions. According to the principles of the proposed TFmeasurements, the boundary conditions are:

dcH�z=0=dcH(0) and dcH�z=L=0 (35)

From Eqs. (34) and (35):

B2=dcH(0) and B1= −dcH(0) coth(s/DaL) (36)

Therefore, Eq. (34) can be written as:

dcH=dcH(0)[cosh(qz)−coth(qL) sinh(qz)] (37)

where

q=s/Da (38)

The hydrogen concentration defined by Eq. (37)should be introduced into the integral of Eq. (32). Theresult of integration is:& L

0

dcH�

z−L2�

dz= −dcH(0)

q2

!qL [1+cosh(qL)]2 sinh(qL)

−1"

(39)

Hence, Eq. (32) changes to:

dJH,nloc= −DHcHeqdcH(0)

12Aq2L3

!qL [1+cosh(qL)]2 sinh(qL)

−1"(40)

In turn, from Eq. (37) it follows also that:

ddcH

dz)z=L

= −dcH(0)

qsinh(qL)

(41)

Hence, the local component of flux, and the total flux(Eqs. (31) and (30), respectively) at the output surface(z=L) are:

dJH,loc�z=L=DadcH(0)

qsinh(qL)

(42)

dJH�z=L= −DHdcH(0)!

−q(1+AcH

eq)sinh(qL)

+12AcH

eq

q2L3

�qL [1+cosh(qL)]2 sinh(qL)

−1�"

(43)

The charge balance equation for z=L is:

dj(o)= −F(dJH�z=L) (44)

According to Eqs. (10), (15), (43) and (44), the pro-posed transfer function is as follows:

H=dj(o)

dE(i)

= −DHcHeq F2

RT!

q(1+AcH

eq)sinh(qL)

−12AcH

eq

q2L3

�qL [1+cosh(qL)]2sinh(qL)

−1�"

(45)

The first and second terms within the braces in Eq.(45) result from the local and non-local components ofthe flux of diffusing hydrogen, respectively. Effectively,the local component is proportional to cH

eq, and thenon-local to (cH

eq)2.Let us remember that the above equation for TF

should be valid for every value of the frequency. Thelinearity of the system and a magnitude of the signal ofthe hydrogen concentration, dcH, small enough to con-form with Eq. (23) are the only simplifying assumptionsused in its derivation.

4. Discussion

The proposed TF (Eq. (45)) will be analyzed as plotsof its spectra simulated for a metal–hydrogen system ofproperties similar to those of palladium- and Pd81Pt19–H a-phase systems (a-MeHn). The following parametervalues are assumed: VH=1.77×10−6 m3 mol−1 [41–43], Y( =1.844×1011 Pa [44], DH=1×10−11 m2 s−1

[1,5,7,45–48], L=5×10−5 m (similar to the thicknessof Pd specimens used in EIS [32–38]), and T=298.2 K.cH

eq is varied from 5×104 down to 5×102 mol m−3.The upper value is close to the largest concentration ofhydrogen observed in a-phase Pd81Pt19–H at roomtemperature (n in MeHn equal to ca. 0.4) [1], and thelower is smaller than the maximum concentration ina-phase Pd–H under the same conditions [45–47].

Plots of the TF spectra are presented in the frequencyrange from 10 to 1×10−4 Hz, both in polar andrectangular coordinates, i.e. in the complex plane. Forbest legibility, the phase angle is given in 2p units, i.e.as a fraction of the period.

In Fig. 2, spectra of the TF are presented in polar(Bode) coordinates, for three values of cH

eq in the speci-men. These differ by one order of magnitude (curves a,b and c). Also two additional spectra are presented forthe highest cH

eq, with the same parameters as for curve aexcept that for a%, it was assumed that: A"0 anddJH,nloc=0, (i.e. discrepant with the definition of thelast (Eq. (40)) and for a%% A=0. Thus, in the former (a%)only the non-local effect of stress on diffusion is sup-pressed, while for the latter a%% both effects are sup-

P. Zoltowski / Journal of Electroanalytical Chemistry 501 (2001) 89–9996

Fig. 2. Spectra of the transfer function (Eq. (45) in polar coordinates.(A), logarithm of modulus, and (B), phase angle, both vs. logarithmof frequency. The plots were computed at the following values ofparameters of the system: VH=1.77×10−6 m3 mol−1, Y( =1.844×1011 Pa, DH=1×10−11 m2 s−1, L=5×10−5 m, T=298.2 K, andcH

eq equal to 5×104 mol m−3 (curve a, solid), 5×103 mol m−3

(curve b, long dash), and 5×102 mol m−3 (curve c, short dash),respectively. Curves a% (dot–dash) and a%% (dots) differ from curve aby the suppression of either only the non-local or both effects ofstress, respectively.

results from the fact that in this case the diffusioncoefficient is increased by the local effect of stress(Da\DH). When the frequency is increased, the lack ofthe non-local effect results in a rapid decrease of themodulus (curves a% and a%%). That is characteristic forFickian diffusion. In contrast, the presence of the non-local effect of stress hinders the strong attenuation ofthe transfer function when the frequency is increased.

The phase angle of all curves (Fig. 2B) approachesthe value of −p at the lowest frequencies, i.e. close tothe steady-state of the diffusion process. It is a simpleconsequence of the fact that the output current is in theopposite phase to the output flux (Eq. (44)). When thefrequency is increased, in the absence of the non-localeffect of stress the phase angle decreases rapidly (curvesa% and a%%). However, when this is present (curves a–c),at frequencies over ca. 1×10−1 Hz, the phase angleshifts asymptotically to a value smaller by one eighththan one 2p unit. The last value (−p/4 radians) ischaracteristic for typical Fickian diffusion at frequen-cies high enough to treat it as infinite in character.

In rectangular coordinates, the strong dependence ofTF on cH

eq (Eq. (45)) results in different sizes of particu-lar curves. Hence, TF will be normalized with respect tothe TF at infinitesimally low frequency, i.e. when thesmall-signal diffusion of hydrogen is at steady-state.

From Eqs. (40) and (42) it can be derived that thelimiting values of the two components of the flux ofhydrogen at the output surface, when the frequencyapproaches zero, are, respectively:

limf�0

dJH,loc�z=L=DHdcH(0)

L(1+AcH

eq) (46)

limf�0

dJH,nloc�z=L= −DHdcH(0)

LAcH

eq (47)

Also, from Eqs. (33) and (47) it follows that thelimiting value of mobility of hydrogen in the metalmatrix (L) at infinitesimally low frequency is equal to−DHdcH(0)A/L. However, one should bear in mindthat this mobility results only from the non-local effectof stress.

From Eqs. (30), (45)–(47) it follows that:

limf�0

dJH�z=L=DHdcH(0)

L[(1+AcH

eq)−AcHeq]=

DHdcH(0)

L(48)

limf�0

H=limf�0

dj(o)

dE(i)

= −DHF2

LRTcH

eq (49)

According to Eq. (48) the local and non-local effectsof self-stress on the hydrogen flux at the output surfacecompensate. Therefore, at infinitesimally low frequencythe stress is absent.

Let us point out that there is apparently a closesimilarity between the system under TFS measurementsat infinitesimally low frequency and under TBT mea-

pressed. Consequently, the comparison of the curves aand a% illustrates the non-local effect of stress. Thedifference between curves a% and a%% results only fromthe fact that in the former case Da\DH (as for curvea), and in the latter Da=DH (see Eq. (29)).

Two issues have to be emphasized with respect tocurves a% and a%%. Firstly, both effects of stress ondiffusion result from a non-zero value of VH (see Eq.(2)). Hence, the assumption of A=0 is physicallyequivalent to VH=0 (see Eq. (28)). Secondly, the non-local effect of stress results from the same physicalfactors as the local one (see Eq. (21)). Thus, the separa-tion of the two effects is unphysical, and the elimina-tion of just the non-local one is possible only insimulations.

The modulus of H is strongly dependent both on cHeq

and on stress (Fig. 2A). It decreases with the decreaseof cH

eq (curves from a to c). When the non-local effect ofstress is eliminated, at low frequencies the modulus islarger than originally (compare curves a% and a). It

P. Zoltowski / Journal of Electroanalytical Chemistry 501 (2001) 89–99 97

surements after the elapse of a very long time. Thereare opinions based on TBT experimental observations[8,49,50] that the concentration gradient of hydrogenover the specimen section is not constant even when astationary output flux of hydrogen is attained. How-ever, Baranowski [6] has noted recently that the station-arity attained in TBT is not the true one, because thepressure of hydrogen in the output chamber increasespermanently. Based on the principle of minimal entropyproduction as the thermodynamic criterion of station-arity, Baranowski concluded that at true stationaritythe stress should be absent [6]. This conclusion isproved by Eq. (48).

Finally, let us define the normalized transfer func-tion, Hn. According to Eqs. (45) and (49):

Hn=H

limf�0

H=L

!q

(1+AcHeq)

sinh(qL)

−12AcH

eq

q2L3

�qL [1+cosh(qL)]2 sinh(qL)

−1�"

(50)

After normalization, the transfer function is nolonger directly proportional to cH

eq or to DH. Hence, inrectangular coordinates the normalization changes onlythe size of the curve, with other features being pre-served. In particular, both terms within the braces inEq. (50) are implicit functions of all system parameters,and the second, non-local term is directly proportionalto cH

eq. Normalization does not apply to curve a%, as thisis unphysical.

In Fig. 3A–C, spectra of the normalized TF (Eq.(50)) are presented in rectangular coordinates, in threescales. When both effects of stress are suppressed (curvea%%), the curve shape is similar to that of a logarithmicspiral (Fig. 3B,C). At the highest frequency the curvestarts from the close vicinity of the origin of coordi-nates, crosses all quadrants of the plane clockwise andterminates at the lowest frequency in the 3rd quadrantclose to the value −1 of the real axis. This is character-istic for Fickian finite diffusion of hydrogen, irrespec-tive of its concentration at the output surface [27–31].

Stress changes this shape at relatively high frequen-cies. Namely, at the highest frequency (10 Hz in thefigure) the curves a–c start in the 1st quadrant, by aspecific section which extrapolates at still higher fre-quencies to the origin of coordinates by a straight lineof slope equal −1 (Fig. 3A–C). This is characteristicfor infinite diffusion, and it corresponds to the respec-tive, high-frequency region of the phase angle (Fig. 2B).The higher cH

eq, the larger is the size of this section. Atthese frequencies the second, non-local term within thebraces in Eqs. (45) and (50), proportional to cH

eq, pre-vails. The high-frequency section of curves a–c corre-sponds to the output flux opposite to the input signal atearly times in TBT experiments. At lower frequenciesthe curve shape changes gradually to an arc similar tothat typical for Fickian diffusion (as curve a%%). Itterminates in the 3rd quadrant close to the value −1 ofthe real axis, just as does curve a%% (Fig. 3A–C). It isworth noticing that the smaller cH

eq, the lower is thefrequency and more characteristic is the shape of thecurve in the region where the non-Fickian and Fickiansections join each other. That can be especially impor-tant in the case of metals of low solubility of hydrogen.

The proposed transfer function is highly sensitive tothe effects of self-stress on diffusion of hydrogen inmetals. Obviously, details of the plots and the locationof their characteristic features with respect to frequencyare dependent on system parameters. However, forother sets of values of VH, Y( , DH and cH

eq the essential

Fig. 3. Spectra of the normalized transfer function (Eq. (50)) inrectangular coordinates. (A), full scale; (B), magnification of the boxin (A); (C), magnification of the box in (B). Legend as for Fig. 2.Decade frequencies are indicated by particular symbols: 10 Hz (), 1Hz (�), 1×10−1 Hz ( ), 1×10−2 Hz (2), 1×10−3 Hz (�) and1×10−4 Hz (�).

P. Zoltowski / Journal of Electroanalytical Chemistry 501 (2001) 89–9998

features of the plots will be preserved. The importantchange of modulus and phase angle by the non-localeffect of stress at relatively high frequencies gives achance for observation of the phenomenon of uphilldiffusion in metal–hydrogen systems of small concen-tration of hydrogen, e.g. Fe-base alloys.

The transfer function (Eq. (45)) can be applied as amodel for real Me–H systems. Namely, it can be fittedto experimental data collected in a chosen range offrequency, and the best-fit values of the modelparameters can be evaluated. In this way, values ofDH and VH

2 ×Y( ×cHeq (see Eqs. (28) and (29)) can be

estimated. Consequently, since VH is well-known andcH

eq can be measured directly (e.g. by extraction ofhydrogen from the specimen), Y( can be estimated ex-perimentally. Moreover, dependences of DH and Y( oncH

eq can be studied.One has to remember that in this paper the process

of ab/desorption of hydrogen has been assumed to beat equilibrium, and all other possible side processeshave been neglected. Examples of these can be dissoci-ation of molecular hydrogen (only in gas environ-ment), adsorption of hydrogen at the specimensurfaces, charging of the electrical double layer at themetal � solution interface, and ohmic drop of the elec-trode potential in the solution (the two last only in asolution environment) [31].

The thickness of specimens useful for TFS (andEIS) measurements of hydrogen diffusion in palladiumand its alloys is about one order of magnitude smallerthan of those applied in TBT experiments (0.25–0.5mm) [5,6,10,32–38]. Hence, if hindrances of surfaceprocesses are important, their disregard will result insimilarly higher errors. The usual method for mini-mization of surface hindrances is the activation ofspecimen surfaces by palladium black, regardless ofthe type of the metal the diffusion of hydrogen isstudied in. This is considered effective in TBT experi-ments [1–7]. Also abrasion with emery paper was usedto activate the surface of palladium for experiments onhydrogen diffusion by EIS [32–34]. However, suchtreatment destroys the crystalline structure of surfacelayers of the specimen, and creates a multitude ofhydrogen traps there [25,51]. This delays the satura-tion of the specimen with the interstitial hydrogen andhinders the diffusion of hydrogen (see Table 1 and thepertinent comment in Ref. [10]) [10,25,50].

In EIS the specimen surface activation by palladiumblack results in an important frequency-dispersion ofthe impedance of the interfacial processes and in anincrease of the surface capacitance, both faradaic (hy-drogen adsorption) and interfacial double layer. Nev-ertheless, in contrast to TBT, in EIS the surface anddiffusion hindrances can be separated, because the for-mer influence the immittance at higher frequenciesthan the latter. On the other hand, the non-Fickian

diffusion resulting from stress, as observed at higherfrequencies than the Fickian, can overlap with theeffects of surface processes. Therefore, the separationof the hindrances of non-Fickian diffusion and thoseof surface processes would need special care.

The possible hindrances resulting from non-diffusionprocesses should be carefully analyzed. The consider-ation of these processes at the input side will result inmore complex dependence of dcH(0) on dE(i) than isgiven by Eq. (15). The analysis of this effect is arelatively simple task. This can be performed in asimilar way as in EIS [10,32–38]. The consideration ofhindrances at the output side will result in more com-plex boundary conditions at z=L (Eq. (35)). Hence,Eq. (37) will be more complex. However, it has to beemphasized that at the output surface there are, forinstance, no interfacial charging processes (either dou-ble layer or faradaic), because dE(o)=0 [31].

5. Conclusions

(a) Transfer function spectroscopy is a promisingtechnique for studying the process of diffusion of hy-drogen in a self-stressed elastic isotropic metal matrix.It is more powerful than the commonly used transientbreak-through technique.

(b) The superiority of the new method resultsmainly from the fact that it is a small-signal technique,which allows the system to be treated as linear. Theproposed approach allows the description of the pro-cess of diffusion of hydrogen in a self-stressed matrixby analytical equations over the total scale of fre-quency, which corresponds to the total time scale.

(c) The proposed transfer function differs from simi-lar flux transfer functions presented in the literature bythat the measurements are performed on the systemwhen it is close to the equilibrium.

(d) The new method should allow the evaluation ofthe diffusion coefficient of hydrogen in metals, and,moreover, of the bulk elastic modulus of these metals.Also the dependence of the above parameters on theequilibrium concentration of interstitial hydrogen inthe metal matrix can be studied.

Acknowledgements

The author expresses his gratitude to Dr P.Kedzierzawski, of Warsaw, for important discussions.Also Professor B. Baranowski, of Warsaw, is acknowl-edged for the inspiration to undertake this work.

P. Zoltowski / Journal of Electroanalytical Chemistry 501 (2001) 89–99 99

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