Concentration transfer function of hydrogen diffusion in self-stressed metals

Journal of Electroanalytical Chemistry 512 (2001) 64–73

Concentration transfer function of hydrogen diffusion inself-stressed metals

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

Received 29 January 2001; received in revised form 30 May 2001; accepted 24 June 2001

Abstract

The effects of self-stress on the concentration transfer function of hydrogen diffusion in a continuous elastic solid-metal matrixwere analyzed. A large thin-plate metal specimen saturated with hydrogen is the essential part of the system. The equilibrium isperturbed by a small-magnitude sine-wave input signal of hydrogen concentration applied at one surface of the specimen. Inresponse, oscillations of hydrogen concentration appear at the opposite surface. The concentration transfer function is defined asthe ratio of the stationary response to input signals. The derived diffusion equations are non-linear. They are linearized, and thensolved analytically. The resulting transfer function is discussed in terms of hydrogen permeation through a specimen of propertiessimilar to palladium and Pd81Pt19 alloy, in wide ranges of hydrogen concentrations in the metal matrix and of frequencies of thesignal. At relatively high frequencies, the system is highly sensitive to non-Fickian diffusion, resulting from the non-local effectof self-stress. Transfer function spectroscopy seems to be a more powerful tool for studying the hydrogen transport in self-stressedmetals than the commonly used transient break-through method. In particular, it should allow the dependence of the hydrogendiffusion coefficient and of the elastic modulus of metal–hydrogen solids on the hydrogen concentration to be studied. © 2001Elsevier Science B.V. All rights reserved.

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

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1. Introduction

Interstitial hydrogen causes an expansion of the crys-tal lattice of a solid matrix. Thus, stress is induced bythe gradient of the hydrogen concentration in elasticsolids. Stress is one of the factors determining thechemical potential of components of solid systems.Therefore, self-stress resulting from the gradient of thehydrogen concentration affects the transport of hydro-gen in metals [1–11].

The stress induced by a local gradient of the hydro-gen concentration is transmitted within the whole vol-ume of the elastic solid with the velocity of sound, i.e.immediately in comparison with the rate of Fickiantransport. This results in two effects on diffusion: localand non-local. The former enhances the Fickian diffu-

sion, while the latter causes a non-Fickian diffusion.The higher the hydrogen concentration in the solid, thelarger are the two effects [5–11].

Until now, the influence of self-stress on the trans-port of hydrogen in metals has been studied by tran-sient break-through (TBT) experiments. A large thinplate of a metal sorbing large amounts of hydrogen(usually palladium [7] or its alloys [1–6,8–11]) is usedas the specimen. It is placed as a membrane separatingtwo chambers. Before the beginning of the experiment,the system is at equilibrium at some non-zero concen-tration of hydrogen. The experiment starts by a stepchange of the hydrogen pressure in one chamber. Thiscauses an evolution of the pressure in the oppositechamber, as a result of the flux of permeating hydro-gen. At early times the flux is opposite to the signal.This phenomenon, called ‘uphill diffusion’, is caused bythe non-local effect of self-stress, which results in atransient decrease of the chemical potential of hydrogenin the specimen close to the output surface [1–11].

* Tel.: +48-22-632-3221; fax: +48-22-632-5276.E-mail address: [email protected] (P. Zoltowski).

0022-0728/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved.PII: S 0022 -0728 (01 )00584 -8

P. Zoltowski / Journal of Electroanalytical Chemistry 512 (2001) 64–73 65

The chemical potential of interstitial hydrogen, �H,as a mobile component in a solid matrix depends on itsconcentration, cH, and mechanical stress, � [6,8,9]

�H=�H(0, cH)−VH� (1)

where �H(0, cH) denotes the hydrogen chemical poten-tial in the stress-free state (�=0), VH is the partialmolar volume of hydrogen in the matrix, and � (belowcalled ‘stress’) is the trace of the stress tensor induced inthe matrix by the presence of hydrogen (�=�xx+�yy+�zz). So, � is the hydrostatic part of this tensor,and is an analogue of the hydrostatic pressure. VH isassumed to be independent of hydrogen concentration.It has to be emphasized that the non-zero value of VH

is the origin of the stress.The transport equation (modified 1st Fick equation)

for the flux of hydrogen, JH, diffusing in the bulk of themetal along the z coordinate, caused by the gradient ofthe chemical potential is [6,8,9]

JH= −LH

��H

�z= −DH

��1+

�ln fH

�ln cH

��cH

�z−

VHcH

RT��

�zn(2)

where LH, DH, R, T and fH denote the phenomenologi-cal coefficient (in terms of irreversible thermodynamics,see Eq. (3)), diffusion coefficient of hydrogen in themetal matrix, gas constant, temperature and activitycoefficient of hydrogen, respectively. DH is assumed tobe independent of cH. For an ideal solution ( fH=1)

LH=DHcH/(RT) (3)

In order to solve Eq. (2), a relationship between thegradients of cH and � is necessary. For this purpose, theanalogy with the effect of thermo-stress arising fromthe thermal expansion of the elastic matrix on theconduction of heat is employed. That results in afeedback relationship between these two gradients inthe matrix [6,8,9]

��

�z= −

23

VHY� ��cH

�z−

12L3

� L

0

�cH�

z−L2�

dzn

(4)

where Y� denotes the bulk elastic modulus of the solid(Y� =E� /(1−�), where E� and � are the Young modulusand the Poisson ratio, respectively), and �cH=cH−cH,0, where cH is the real hydrogen concentration andcH,0 its concentration in the initial stress-free state.

Accordingly, Eq. (2) changes to [6,8,9]

JH= −DH��

1+�ln fH

�ln cH

+2VH

2 Y�3RT

cH��cH

�z

−8VH

2 Y� cH

RTL3

� L

0

�cH�

z−L2�

dzn

(5)

From Eq. (5) the following balance equation(modified 2nd Fick equation) results [6,8,9]:

�cH

�t=DH

��1+

�ln fH

�ln cH

+2VH

2 Y�3RT

cH��2cH

�z2

+2VH

2 Y�3RT

��cH

�z�2

−�8VH

2 Y�RTL3

� L

0

�cH�

z−L2�

dzn�cH

�z�

(6)

In Eq. (5), the first term within the brackets describesthe local (Fickian) diffusion accounting for the stress.Thus, stress always enhances the local diffusion, even ifVH�0, as does the activity coefficient of hydrogen. Thesecond term is the non-local one, because it depends onthe integral of the composition profile taken over thewhole thickness of the specimen (0�z�L). It has tobe emphasized that the flux resulting from the secondterm arises in each elementary volume of the specimen,irrespective of the local concentration gradient [6,8].

Eq. (6) is a partial second order non-linear integro-differential equation. The first term within the bracesrepresents the Fickian diffusion enhanced by the localeffect of stress. Both the next terms result only from thestress. If the concentration gradient is small, the termproportional to the square of this gradient will benegligible. However, as emphasized by Baranowski [8],‘‘it could be a source of interesting non-linear phenom-ena, including oscillations and more complex dissipa-tive structures’’. The last term, proportional to theproduct of the integral and the concentration gradient,is due to the asymmetry of hydrogen distribution withrespect to the z=L/2 plane [6,8].

The above diffusion equations are fairly complex.Moreover, at both surfaces of the specimen theboundary conditions are time dependent. Nevertheless,as reported recently [6,8], Eq. (5) describes correctly theresults of TBT experiments for a Pd81Pt19 specimen atearly times (up to more than 10 min under the appliedexperimental conditions). Namely, in the range of timeswhen the magnitude of the output signal, opposite tothe input one, changes monotonically, a consistency ofthe plots of theoretical and experimental fluxes of hy-drogen at the output side has been noticed [6,8].

Electrochemical impedance spectroscopy (EIS) hasalso been proposed to study the effect of stress on theabsorption of hydrogen into a metal electrode [12,13]and its diffusion in the bulk metal [12]. However, inaccordance with the definition of impedance, in thistechnique the system is one-port, i.e. the input andoutput signals are measured at the same electrode sur-face. Thus, the non-local effect of stress on hydrogendiffusion cannot be seen. Consequently, only the appar-ent diffusion coefficient of hydrogen, i.e. the (true)diffusion coefficient increased by the local effect ofstress, can be estimated [12].

Recently, a transfer function (TF) approach to thediffusion of hydrogen in stressed metals has been pro-posed [14]. In this case, the specimen is a large thin

P. Zoltowski / Journal of Electroanalytical Chemistry 512 (2001) 64–7366

plate applied as the membrane separating two gaschambers or electrochemical cells, as presented in Fig.1C. This system is close to equilibrium at a non-zerochemical potential of hydrogen, as in TBT before ap-plying the step signal. The equilibrium is perturbed bya sine-wave input signal, of small magnitude and ofvaried frequency, applied at one surface. At the oppo-site surface the concentration of hydrogen is forced tobe constant. The resulting oscillating flux of diffusedhydrogen, of the same frequency as the input one, isconsidered as the response signal. The ‘flux TF’ (TFf) isthe ratio of the periodically stationary small-signal flux(or electric current, opposite in phase) at the outputsurface to the small-signal concentration (or electricpotential) at the input surface. As output surface ispermeable for hydrogen diffused in response to theinput signal, this diffusion can be named ‘bounded’, inanalogy with the diffusion in one of the two types ofEIS conditions (Fig. 1A) [15].

Let us specify explicitly two prerequisites for observ-ing the effects of self-stress on hydrogen transport byany flux method, i.e. by TBT or TF. Firstly, the systemshould be a two-port one. Secondly, the concentrationof hydrogen at the output side should be non-zero. Itshould be emphasized that, in contrast to Ref. [14], inall earlier approaches to the hydrogen transport by TFf

[16–25] the second prerequisite was not fulfilled, as theprofile of hydrogen concentration in the specimencross-section was similar to that in Fig. 1A.

In this paper, the diffusion of hydrogen in self-stressed metals is described in terms of another transferfunction. The system differs from that recently pro-posed for TFf [14] only in the character of the outputsurface. Namely, it is impermeable to the diffusedhydrogen; i.e. there is no mass transfer throughout (seeTFc in Fig. 1C). In this case, the character of thediffusion can be called ‘restricted’, in analogy withanother set of EIS conditions (Fig. 1B) [15]. The sta-tionary oscillations of the hydrogen concentration atthe output surface, of the same frequency as those ofthe input, are considered as the response signal. Theratio of this response to the concentration input signalis named ‘concentration transfer function’ (TFc).

The proposed TFc is equivalent to ‘the concentrationtransfer function’ discussed by Montella (see Eq. (41)and Fig. 3 in Ref. [22]). It is also equivalent to theresults of ‘the enforced oscillation method’ of Boes andZuchner [17], and to those of ‘the potentiometric sinu-soidal technique’ reviewed by Pounds [21]. However,according to our knowledge, for the first time theeffects of stress on diffusion will be taken into accountin the case when the concentration of the diffusedhydrogen is the response of the system.

2. Derivation of the concentration transfer function

As usually in the analysis of the influence of stress onthe diffusion of hydrogen in metals [6,8,9,12,14], the

Fig. 1. Schemes of the system in typical EIS experiments (A and B) and in two types of TF experiments (C).


metal matrix specimen is assumed to be a continuous(single-phase) elastic solid. It is a large thin flat plate.Only one-dimensional diffusion of absorbed hydrogen,Hab, from one surface of the specimen (z=0) to theopposite one at a distance L, is discussed

Hab(z=0)��d

Hab(0�z�L) (7)

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

For the sake of simplicity, all other possible processesare neglected, except the absorption/desorption processof atomic hydrogen (H�Hab) at z=0. This process isassumed to be at equilibrium, irrespective of the pres-ence of stress. In order to simplify the equations, it isalso assumed that the activity coefficient of hydrogen( fH) is equal to one.

The small-magnitude sine-wave concentration signalapplied to the input surface (z=0) of the specimen,�c(0), is defined as follows:

�c(0)=�c(0) exp(st) (8)

where � denotes the amplitude, t the time, and s= i� ;i is the imaginary unit (i=�−1), and � the angularfrequency (�=2�f, f being frequency). This signal issuperimposed on the equilibrium concentration of hy-drogen, cH

eq, in the system. Under these conditions, theTBT equations (Eqs. (2) and (4)) have to be changed asfollows [14]:

�JH= −DH��cH

�z−

VH(cHeq+�cH)RT

��

�zn

(9)

��

�z= −

23

VHY� ��cH

�z−

12L3

� L

0

�cH�

z−L2�

dzn

(10)

Consequently, the following equations of diffusion(modified 1st and 2nd Fick equations) are obtained[14]:

�JH= −DH��

1+2VH

2 Y�3RT

(cHeq+�cH)

n��cH

�z

−8VH

2 Y�RTL3(cH

eq+�cH)� L

0

�cH�

z−L2�

dz�

(11)

��cH

�t=DH

��1+

2VH2 Y�

3RT(cH

eq+�cH)n�2�cH

�z2

+2VH

2 Y�3RT

��cH

�z�2

−�8VH

2 Y�RTL3

� L

0

�cH�

z−L2�

dzn��cH

�z�

(12)

Eqs. (11) and (12) are similar to the respective TBTequations (Eqs. (5) and (6)). However, they have aspecific feature, resulting from the periodicity of thesignal. For instance, the rise of the concentration gradi-ent to the square (see the second term within the bracesin Eq. (12)) results in a doubling of the frequency, i.e.

in the second harmonics. Similarly, the first and thirdterms in Eq. (12) as well as both terms within the bracesin Eq. (11) are non-linear. Namely, as will be seen later(see e.g. Eqs. (33) and (36)), �cH as well as the integraland ��cH/�z are proportional to �cH(0) even in the casewhen Eqs. (11) and (12) are linearized. Hence, all termsin braces in the right-hand side of [11,12] result in morecomplex phenomena than those described by the funda-mental harmonics. It has to be concluded that thediffusion discussed here is intrinsically non-linear. Inthe simplest case of non-linearity of the system, i.e.when it results only in higher harmonics, one has toassume that

�cH(z�0)=�m

[�cH(z�0) exp(mst)] (13)

where m denotes an integer (m=1, 2,…). This equationdescribes the response of the system to the perturbationby �cH(0). When the maximal value of m�1, it is aharmonic series. In Eq. (13) it is assumed tacitly thatthe source of the primary perturbation, �cH(0), is power-ful enough to eliminate all possible non-linear phenom-ena at the input surface.

Let us notice in advance that from [13] it followsthat:

��cH(z�0)

�t=�

m

[ms�cH(z�0) exp(mst)] (14)

In order to obtain a linear solution for this non-lin-ear process of diffusion described by Eqs. (11) and (12),the following relation can be applied [14]:

cHeq+ �cH�cH

eq (15)

The above relation is a mathematical formulation ofthe requirement that the concentration signal should besmall in comparison to cH

eq. This is crucial for obtaininga linear solution of Eq. (12). Namely, it allows the fluxequation of diffusion (Eq. (11)) to be simplified asfollows [14]:

�JH= −DH��

1+2VH

2 Y�3RT

cHeq��cH

�z

−8VH

2 Y�RTL3 cH

eq � L

0

�cH�

z−L2�

dz�

(16)

In turn, from Eq. (16) the following balance equationof diffusion results [14]:

��cH

�t=DH

�1+

2VH2 Y�

3RTcH

eq��2�cH

�z2 (17)

As a consequence of Eq. (15), the series defined byEqs. (13) and (14) reduce to their first terms, for m=1.

Eqs. (16) and (17) are analogues of the 1st and 2ndFick equations for linearized small-signal diffusion ofhydrogen in a solid matrix, accounting for the effects ofstress. They do not involve higher harmonics or other


non-linear phenomena. In other words, the applicationof Eq. (15) filters out every possible non-linearity [14].

Below, Eqs. (16) and (17) will be used for deriving adetailed equation for the concentration transfer func-tion, defined as follows:

Hc=�cHz=L

�cH(0)

=�cH(L) exp(st)�cH(0) exp(st)

=�cH(L)

�cH(0)

(18)

Eqs. (16) and (17) can be rewritten as follows (seealso Eq. (14) for m=1) [14]:

�JH= −Da

d�cH

dz+DH

12AL3 cH

eq � L

0

�cH�

z−L2�

dz (19)

Da

d2�cH

dz2 −s�cH=0 (20)

where A is a dimensionless constant resulting from thestress, and Da is the apparent diffusion coefficient ofhydrogen, i.e. the (true) diffusion coefficient increasedby the local effect of stress

A=2VH

2 Y�3RT

(21)

Da=DH(1+AcHeq) (22)

In contrast to EIS [12], the integral in Eq. (19)resulting from the non-local effect of stress on the fluxof hydrogen diffusing in the metal matrix is not equalto zero. This integral is a measure of the asymmetry in�cH distribution with respect to the plane z=L/2,weighted by the distance from this plane. It causes anoscillatory bending of the specimen, in response to theoscillations of hydrogen concentration [14].

The recent success in the interpretation of the TBTexperiments, mentioned in Section 1, resulted fromdecoupling of the hydrogen flux (Eq. (5)) into local andnon-local components, and solving the integral of thelatter for early times under the assumption of infinitediffusion. However, for longer times this approachcould not be followed, because of the evolution of theboundary conditions of diffusion at both surfaces [6,8].TF techniques allow the analysis to be extended downto the lowest frequencies, corresponding to arbitrarylong times, because the boundary conditions are invari-ant (periodic stationarity) [14].

As in the TBT analysis, we can start by decouplingthe linearized flux (Eq. (19)) into the local and non-lo-cal components (subscripts ‘loc’ and ‘nloc’, respectively)[14]

�JH=�JH,loc+�JH,nloc (23)

where

�JH,loc= −Da

d�cH

dz(24)

�JH,nloc=DH

12AL3 cH

eq� L

0

�cH�

z−L2�

dz (25)

The above decoupling is physically substantiated forevery value of frequency. Moreover, in contrast toTBT, �JH,nloc� f(z), i.e. �JH,nloc is the same within thewhole specimen, because it is proportional to cH

eq (Eq.(25)) [14].

The general solution of Eq. (20) for neutral particles(like atomic hydrogen) diffusing in one-dimensional (z)space in a continuous matrix is as follows (e.g. [26])[14]:

�cH=B1 sinh(qz)+B2 cosh(qz) (26)

where

q=�s/Da (27)

The constants B1 and B2 should be selected properlyin order to satisfy appropriate boundary conditions.According to the principles of the TFc measurementsdiscussed, the boundary conditions are

�cHz=0=�cH(0) and �JHz=L=0 (28)

Let us point out that recently the above boundaryconditions have been applied in the analysis of surfacehindrances of hydrogen insertion into metals [13].FromEqs. (19), (22), (26) and (28) it follows that

d�cH

dzz=L

=12AcH

eq

(1+AcHeq)L3

� L

0

�cH�

z−L2�

dz (29)

B2=�cH(0) (30)

According to Eq. (30), Eq. (26) can be rewritten as

�cH=B1 sinh(qz)+�cH(0) cosh(qz) (31)

Hence, from Eq. (31) it follows that

d�cH

dzz=L

=q [B1 cosh(qL)+�cH(0) sinh(qL)] (32)

The integral in Eq. (29) is evaluated for �cH definedby Eq. (31)� L

0

�cH�

z−L2�

dz

=B1�L

2q[cosh(qL)+1]−

1q2 sinh(qL)

�+�cH(0)

�L2q

sinh(qL)−1q2[cosh(qL)−1]

�(33)

Hence, from Eqs. (29), (32), and (33) it results that

B1= −�cH(0)

�aqL2

−q3�sinh(qL)+a [1−cosh(qL)]�aqL2

−q3�cosh(qL)+a�qL

2−sinh(qL)

n(34)

where

a=12AcH

eq

(1+AcHeq)L3 (35)


Therefore, Eq. (26) can be rewritten as follows:

For Eq. (38) two particular cases have to be ana-lyzed; namely, what will happen (i) when the systemattains thermodynamic equilibrium; and (ii) whenVH=0, i.e. there is no reason for the stress.

Firstly, from Eqs. (18) and (37), or Eq. (38) it can bederived that

limf�0

Hc=limf�0(�cHz=L)

�cH(0)

=1 (39)

Therefore, at infinitesimally low frequency there is nostress, in agreement with the principles of equilibrium.

Secondly, for VH=0, according to Eqs. (21), (22)and (35) Da=DH and a=0. In such a case it followsfrom Eq. (38) that

Hc(no stress)=�

cosh�� s

DH

L�n−1

(40)

Eq. (40) defines the same TF as Eq. (41) in Ref. [22],derived for the concentration TF without taking thestress into consideration. In this case the boundarycondition at the output surface was

d�cH

dzz=L

=0

[22], which is equivalent to �JH,locz=L=0 (see Eq.(24)).

3. Simulated spectra of the concentration transferfunction

In this section, Hc according to Eq. (38) is analyzedas plots of its spectra in a large range of frequencies.These spectra have been simulated for a metal–hydro-gen system of properties similar to those of the �-phasepalladium- and Pd81Pt19–H systems (�-MeHn). The

following parameter values were assumed: VH=1.77×10−6 m3 mol−1 [27–29], Y� =1.844×1011 Pa [30],DH=1×10−11 m2 s−1 [4,6,10,11,31–44], L=5×10−5

m (similar to the thickness of Pd specimens used in EIS[38–44]), and T=298.2 K. cH

eq is varied from 5×104

down to 5×102 mol m−3. The upper value (n in MeHn

equal to ca. 0.4) is about twice as large as the largestconcentration of hydrogen observed at room tempera-ture for the Pd81Pt19–H system [4,45,46], and the lowervalue is five times smaller than the maximum concen-tration in the �-phase Pd–H system under the sameconditions [33,36,37,47]. The above values are the sameas those applied previously in the analysis of theanalogous flux transfer function [14].

The spectra of Hc are presented both in polar andrectangular coordinates, i.e. in the complex plane, inthe frequency range from 1 to 1×10−4 Hz. For greaterclarity the phase angle is given in 2� units (i.e. periods).

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

eq in the specimen.These differ by one order of magnitude (curves a–c).Also, two additional spectra for the highest cH

eq (as forcurve a) are presented. They have been computed underthe following assumptions: A�0 and �JH,nloc=0 (curvea�), and A=0 (curve a�). Thus, for the former (a�) onlythe non-local effect of stress on diffusion is suppressed,while for the latter (a�) both effects are suppressed.Consequently, the differences between curves a and a�,and a� and a� illustrate the non-local and the localeffects of stress, respectively. The latter results onlyfrom the fact that for curves a� and a� Da�DH (as forcurve a) and Da=DH, respectively (see Eq. (22)).

With respect to the curve a� an important issue has tobe emphasized. Both effects of stress on diffusion arethe consequence of a non-zero value of VH (see Eq. (1)).The assumption of A=0 is physically equivalent to

�cH=�cH(0)��

�

�

�

cosh(qz)−

�aqL2



2−sinh(qL)

n sinh(qz)��

�

�

(36)

Consequently

�cHz=L=�cH(0)��

�

�

�

cosh(qL)−

�aqL2



2−sinh(qL)

n sinh(qL)��

�

�

(37)

Finally, from Eqs. (18) and (37) it follows that

Hc=cosh(qL)−

�aqL2



2−sinh(qL)

n sinh(qL) (38)


Fig. 2. Spectra of the concentration transfer function (Eq. (38)) inpolar coordinates. (A) Modulus; (B) logarithm of modulus; and (C)phase angle; all vs. logarithm of frequency. The plots were computedat the following values of parameters 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, and cH

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). Curves a� (dot-dash) and a� (dots) differ fromcurve a by the suppression of either the non-local or both effects ofstress, respectively.

limits the modulus changes monotonically, with a singleinflection point. The non-local effect of stress manifestsitself by the appearance of a shoulder on the curve,separating two inflection points (curves a–c). Thelarger cH

e is, the larger is this shoulder.In Fig. 2B it can be noticed that at relatively high

frequencies the effect of the non-Fickian diffusion re-sults in a nearly constant slope of the logarithm of themodulus, independent of the frequency logarithm(curves a–c).

In Fig. 2C the phase angle of Hc spectra is presented.When the frequency approaches its lowest value, thisquantity approaches zero for all curves. When thefrequency increases, the Fickian diffusion results in amonotonic decrease of the phase angle (curves a� anda�), with a steepness dependent on the effective diffu-sion coefficient. In contrast, the non-local effect of

Fig. 3. Spectra of the concentration transfer function (Eq. (38)) inrectangular coordinates. (A) Full scale; (B) zoom of the box in (A);(C) zoom of the box in (B). Legend as for Fig. 2. Decade frequenciesare indicated by particular symbols: 1 Hz (�), 1×10−1 Hz (�),1×10−2 Hz (�), 1×10−3 Hz (�) and 1×10−4 Hz (�).

VH=0 (Eq. (21)), and of A�0 to �JH,nloc�0 (Eq.(25)). Therefore, the separation of the two effects per-formed in the case of curve a� is unphysical. Theelimination of just the non-local effect is possible onlyin simulations. However, under this restriction, bothcurves a� and a� describe the Fickian diffusion. Theydiffer only by the operating diffusion coefficient.

In Fig. 2A the modulus of Hc spectra are presented.When the frequency approaches its lowest and highestvalues, the modulus approaches one and zero, respec-tively. In the case of Fickian diffusion, between these


stress results in a non-monotonic change of the phaseangle, up to the highest frequencies when a constantvalue of −3/4 of the period is approached (curves a–c).

In Fig. 3A–C, Hc spectra are presented in rectangu-lar coordinates. In the case of Fickian diffusion (curvesa� and a�), the curve shape is similar to that of alogarithmic spiral. At the highest frequency the curvesstart from the vicinity of the origin of coordinates (Fig.3C), cross all quadrants of the plane clockwise (Fig.3A–C) and terminate at the lowest frequency in the 1stquadrant close to the value one of the real axis (Fig.3A). Curves a� and a� differ only by the distribution offrequency, depending on the effective diffusion coeffi-cient. All of the above is characteristic for the Fickiandiffusion of hydrogen [22].

Stress changes this shape, the most at relatively highfrequencies. At the highest frequency (1 Hz in thefigure) the curves a–c start close to the border betweenthe 3rd and 4th quadrants, by a specific section whichextrapolates at still higher frequencies to the origin ofcoordinates by a straight line of infinite slope (Fig.3A–C). This section corresponds to the respective high-frequency region of the phase angle (Fig. 2C). It resultsonly from the non-local effect of stress. Let us noticethat the infinite slope of the curves, parallel to theimaginary axis, is characteristic for a conservative pro-cess, as e.g. the charging/discharging of a capacitor. Inour case it results from an implicit assumption of idealelasticity of the matrix. The higher cH

eq is, the larger isthe size of this section.

When the frequency is decreased, curves a–c ap-proach gradually to those characteristic for Fickiandiffusion (curves a� and a�) (Fig. 3A). Nevertheless, thedifference in shape between curve a from one side andcurves a� and a� from the other, due to the non-localeffect of stress, is important at least within two decadesof frequency.

It is worth noticing that the smaller cHeq is, the lower

is the frequency and more characteristic is the shape ofthe curve in the region where the non-Fickian sectionmeets the other section, which is influenced by Fickiandiffusion.

4. Discussion

The proposed concentration TF is highly sensitive tothe effects of stress on the diffusion of hydrogen inmetals, similar to the recently proposed flux TF [14].Obviously, details of the plots and the location of theircharacteristic features with respect to frequency aredependent on the system parameters. However, forother sets of values of VH, Y� , DH and cH

eq the essentialfeatures of the plots should be the same. In particular,the essential influence of the non-local effect of stresson the modulus and phase angle at relatively high

frequencies gives a chance for observation of this effectin metal–hydrogen systems of low hydrogen concentra-tion (e.g. Fe–H).

Eq. (38) can be applied as a model for real Me–Hsystems. Namely, it can be fitted to experimental spec-tra, and the best-fit values of the model parameters canbe evaluated. In this way, values of DH and VH

2 ×Y� ×cH

eq (see Eqs. (21), (22), (27) and (35)) can be estimated.Consequently, since VH is well known and cH

eq can bemeasured directly (e.g. by extraction of hydrogen fromthe specimen), Y� can be estimated experimentally.Moreover, the dependence of DH and Y� on cH

eq can bestudied.

One has to remember that in this paper all possiblenon-diffusional processes have been neglected, exceptthat of ab/desorption of hydrogen, which has beenassumed to be at equilibrium. The thickness of speci-mens useful in TF (and EIS) measurements of hydrogendiffusion in palladium and its alloys is about one orderof magnitude smaller than the thickness applied in TBTexperiments (0.25–0.5 mm) [1–11]. Hence, if hin-drances of the surface processes are important, theirdisregard will result in larger errors than in TBT.

Essentially, it is not important whether the surround-ings (input and/or output) of the specimen are themolecular hydrogen or a protic solution. In the firstcase, the equilibrium relation between the hydrogenconcentration in the metal matrix and the gas pressureis described by Sieverts’ equation [6,8,36]. For smallsignals this equation can be linearized. Thus, the oscil-lations of pressure of molecular hydrogen can be theinput and/or output signals [19,20,23,24]. However, theabsorption of atomic hydrogen must be accompaniedby the dissociation of molecular hydrogen (H2�2Had).The activation of specimen surfaces by deposition ofthe palladium black, regardless of the type of the metal,is the usual method for minimization of hindrances ofthe last process [1–11,24]. However, the Pd-black in-creases the quantity of hydrogen adsorbed at the speci-men surface. So, if the ratio of the last quantity to thisin the specimen bulk is relatively large, the signal willbe damped at the surface. Moreover, as the Pd-black isusually deposited at relatively high current densities[1–11], hydrogen traps [25,48,49] can be created di-rectly under the specimen’s surface [50].

In the case of a liquid environment, the linearizedNernst equation can be applied to describe the equi-librium relation between �cH(0) and the small signal ofelectrode potential [12,14]. In the simplest case [38,39],the process of hydrogen absorption must be accompa-nied by simultaneous reduction of proton (H++e−

�Hab). In the case of lack of equilibrium of thisprocess at the input surface, its hindrances can besubtracted from the overall ones just as in EIS[15,51,52]. At the output surface the situation is more


complex, due to the fact that this side of the specimenis under open-circuit conditions. Firstly, in response tooscillations of �cH(L) the process of charging/discharg-ing of the electrical double layer at the metal solutioninterface, accompanied by hydrogen adsorption/desorp-tion will be operating. Secondly, at relatively largeconcentrations of hydrogen in the Me–H system aleakage of hydrogen from the specimen throughout itsoutput surface, followed by an escape of molecularhydrogen, may pose a problem. The above processeswill result in an essential complication of the boundarycondition of diffusion at the output surface (see Eq.(28) for z=L, and the comments on Eq. (40)). All theabove problems are specific for the concentration TFitself [51,52], irrespective of the effects of stress ondiffusion. However, the region of frequencies character-istic for Fickian diffusion is extended by the non-Fick-ian one to higher values. Thus, the frequency region,where the non-local effect of stress can be observed, canoverlap with this characteristic for the surfaceprocesses.

The boundary condition about the lack of mass fluxthrough the specimen output surface is an importantdisadvantage of the concentration TF, in comparison tothe flux TFs. In long-lasting experiments such a systemshould be sensitive to parasitic processes. This may bethe reason why, for a long time, the concentration TFhas found no experimental application, despite the factthat it is well known and still largely discussed[17,21,22,51,52].

Moreover, in spite of the really small magnitude ofthe input signal (solely the first harmonics), other har-monics may be present in the primary response signalof the system. Filtering of higher harmonics is easy toperform. However, the so-called ‘zero harmonics’, ac-companying the second harmonics, may cause an in-crease of the essentially constant component ofconcentration (cH

eq) at the output side of the specimen[53].

Finally, let us point out that the effects of stress ondiffusion should operate in all processes of sorption ofinterstitial components into elastic matrixes. Hence,effects similar to those on the transport of hydrogen inmetals should influence also the intercalation processes,as noticed recently by Vakarin et al. [54].

5. Conclusions

1. The proposed approach to the process of diffusionof hydrogen in a self-stressed continuous elasticmetal matrix by the concentration transfer functionallows it to be described by analytical equations.Concentration transfer function spectroscopy is apromising technique for studying this process in realsystems.

2. The superiority of the method discussed over thecommonly used transient break-through method re-sults mainly because: (i) the boundary conditions ofdiffusion are invariant (periodic stationarity), and(ii) the system can be treated as linear. The equa-tions derived for this method are valid over thewhole scale of frequency.

3. The proposed method should allow the evaluationof the diffusion coefficient of hydrogen in metals,and of the bulk elastic modulus of the metal–hydro-gen systems. Also the dependence of both theseparameters on the equilibrium concentration of in-terstitial hydrogen in the metal matrix can be stud-ied, in the same way as by the recently proposed fluxtransfer function.

Acknowledgements

The author expresses his gratitude to Dr P.Kedzierzawski for important discussions.

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Concentration transfer function of hydrogen diffusion in self-stressed metals

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