CC BY
4Oermak * J., Kufudakis **A., Lewis ***F. A.
*Institute of Physics Na Slovance 2,182 21 Praha 8, Czech Republic **Nuclear Research Centre "Demokritos" Aghia Paraskevi, Attiki, Greece ***School of Chemistry The Queen's University of Belfast Belfast BT9 5AG, Northern Ireland, UK
20 _
Performance and interpretation of measurements of hydrogen diffusivity in metals have been improved steadily in regard to emphasizing the role of initial and boundary conditions at interfaces, the state of perfection of the crystal lattice of the solvent metal, the concentration of hydrogen and the strain-induced component of the diffusion flux. This paper presents an attempt to estimate the relative influence of the afore-mentioned factors on the diffusivity of hydrogen in a-Pd at 50 °C in three diffusion-elastic processes, over courses of electrolytic charging with respective subsequent equilibration and electrolytic discharging processes. Comparison of available data has suggested some non-negligible influences of regularly neglected systematic errors.
1. INTRODUCTION
The diffusion-elastic effect, as noted in its simplest form in 1909 by Stoney [1] can be regarded for study of processes of absorption, equilibration and descorption, for different initial and boundary conditions, in the sense of a classification [2] as a direct instantaneous response of a standard linear solid to a time- and co-ordinate-dependent stress field developed during diffusion of impurities/interstitials/in a crystal matrix. Stoney's experimental technique has since been generalised and experimentally developed [3,4] in terms of the simple geometry of uniaxial bending of a strip during one-dimensional diffusion [3,4]. Such a generalised diffusion-elastic effect has thus proved to be an effective and sensitive experimental tool for the investigation of, for example, the diffusion of hydrogen in metallic structures under very different initial and boundary conditions including absorption, equilibration within, and desorption of hydrogen from samples of the strip. Values of diffusion coefficient of the matrix material of the strip could then be derived from the time dependence of the bending deformation related directly to the diffusion process of the hydrogen interstitials through the thickness of the strip. The technique was absolute in the sense that it required no calibration. For determination of the hydrogen diffusion coefficient it was only necessary to record the deflection of the free end of the strip as a function of time and to know the other geometrical characteristics of the experimental system. No adjusting parameters need to enter the subsequent calculations.
It is important here to state explicitly that the associated bending distortions could also be expected to induce consequent Gorsky effect migration processes [5-8]. Thus from a detailed comparison of the diffusion-elastic (DE) and Gorsky effect (GE) in [9] it is impor-
tant to restate here at least two differences between the operation of the two effects:
a) The DE effect can be classified as a primary elastic effect while GE is correspondingly classified an anelastic effect
b) The sensitivity of the diffusion elastic effect to changes of concentration distribution of interstitials should be expected to be at least two orders of magnitude higher that the sensitivity of the GE [9]. The second statement (b), has allowed the GE to
have been effectively neglected in measurements of dif-fusivity obtained by the DE technique in conditions of current experimental precision, since the GE can never properly develop to its full theoretical magnitude during the course of the DE measurements. This fact further reduces conditions of description according to Ze-ner's classification [2], and so to any contribution of the GE to DE, and has fully justified neglecting the Gorsky effect in this connection in all regard to the following considerations below. On the other hand Gorsky effect influences have been fully taken into account in specialised measurements with tubular membranes of palladium alloys as, for example, in the studies of refs. [10, 11 and 12].
1.1. Scope of the present paper
On turning towards somewhat more specific attentions to the DE effect, it should be noted that the DE technique has been successfully applied to measurements of diffusion parameters of hydrogen in nickel [9,13], iron [14], platinum [15] and a series of palladium-platinum alloys [16]. Its reliability has been justified by overall agreement of results obtained in these studies [13-16] and results alternatively derived by other methods. In addition, the DE technique has been successful in obtaining results under difficult combinations of exper-
imental conditions, where more conventional methods would either have been very difficult to apply or would have proved inefficient as, for example, has been the case in measurements of hydrogen diffusion coefficient in platinum at 30 °C [15] which were the only ones available at that particular time.
All the earlier results that have been mentioned above, have referred to a single-phase solid solution of hydrogen in a homogeneous isotropic metallic matrix. However, in order to verify the principal assumptions of the model that have been previously outlined in full in [3] and [13], a more complicated system of a bimetal double-layer consisting of two parallel metallic strips of nickel and copper has also been experimentally investigated [4,17] and a reasonable agreement has been obtained with available related data, for hydrogen diffu-sivity in the cases of both the involved metals.
In all of the cases quoted here above, experimentally measured bending deflection against time curves have been analysed and compared to appropriate theoretical relation by a least-squares procedure, in terms of a model briefly summarised in ref. [13] where unified boundary conditions had included assumptions of perfect permeabilities of both entrance and exit surfaces together with other prescribed conditions of constant surface concentrations of hydrogen both at beginnings and also over the courses of the entire diffusion process.
In the case of studies with a-Pd-H specimens, close examination of bending deflection against time curves (BDT curves) for various electrolytic regimes, had early indicated a need for essential differences in assumptions as compared to those previously applied to the other measured metals [8, 9, 13-15]. In the case of Pd-H specimens [18], the piceine-coated surface of the palladium specimens had to be considered to be effectively impermeable to hydrogen and also that both preanod-ized and palladized surfaces had to be highly efficient both for the initial absorptions and retentions of elec-trolytically discharged hydrogen at 50 °C and also with considerations of the relatively low prescribed current densities of 20 Am-2. It should also be emphasised here, that the piceine coating of one outer surface had assumed a primary purpose of preventing this surface of experimental samples, from actually contacting the electrolyte. An effective combined impermeability to hydrogen of this piceine coating, in the measurements with palladium may also be associated with the generally lower values of hydrogen chemical potentials recorded over the whole course of the palladium studies.
A general survey of the variety of such influences observed in the DE measurements with palladium, including consequences of aob phase transition possibilities, has been presented qualitatively in refs [18 and 24], and has served as a reliable basis for the present more quantitative analysis.
The scope of the present paper, corresponds to a further presentation of results of the DE measurements of diffusivity of hydrogen in a-Pd-H at 50 °C obtained in conditions of the operation of three different processes of hydrogen diffusion namely: hydrogen absorption, hydrogen content equilibration and hydrogen desorption respectively, in the electrochemical regime [24]. Possible influences of lattice defects and hydrogen content concentration are also to be discussed. The particular influences of the choice of initial and boundary conditions on the descriptions and results of experiments, are also to be analysed.
2.EXPERIMENTAL
The essential arrangements have been described in previous papers as, for example, in refs. [3, 4, 13]. Long Pd strips, dimensions 60x10x0.3 mm3 of purity 3N as supplied by Safina, had first been annealed for two hours at 1000 °C in vacuum. Specimens were next clamped at one end and a mirror was attached to an arm extended from the other free end positioned to reflect a light beam from a projection lamp on to photographic paper held on a rotating drum.
After the coating of one face of specimens with pi-ceine hydrocarbon wax, the other surface could then be activated by electro-deposition of a thin layer of palladium black from a solution of 2% PdCl2 in 0.1 N HCl. Finally after immersion in an electrolyte solution of 1 N H2SO4 at 50 oC + 0.1 oC, hydrogen could be introduced or removed electrolytically through an estimated effective active surface area of 50x10 mm2 at current densities of 20 Am-2 or 200 Am-2 alternatively. A platinum sheet was employed as a counter electrode.
Examples of well defined experimental bending deflection-time (BDT) curves obtained for the three different experimental processes are presented in Figs. 1-4. With reference to these BDT curves, a sequence of procedures had to be applied [13], based on physicochemical reasoning concerning surface conditions applying to transmission of hydrogen through the electrolyte-Pd electrode interface to be discussed further below in Section 7.2.
Hydrogen concentrations cH(n) = nH/npd (where nH and nPd are the volume densities of atomic ratios of hydrogen and palladium atoms respectively) were restricted to below a limit of cH(n) = 0.01, in order to ensure limitations of cH(n) values to the a-phase structure composition range [13]. Restrictions to this limit were in accordance with experimental evidence of absences of alterations of the shape of BDT curves even over repeated cycles of the three successive processes of ab-sorption-equilibration-desorption (a-e-d, Fig. 1) with the same sample [18] and correspondingly also indicated absences of any corresponding measurable irreversible structural changes to specimens, arising from possible occurrences of a o b phase transitions.
3. POSSIBLE INFLUENCES OF LATTICE DEFECTS AND HYDROGEN CONCENTRATION
Major problems from measurements of diffusivities in metallic lattices, include those of closely related dependences of diffusivities on: (a) densities of various types of crystal defects and their interactions with the hydrogen atoms, and (b), the bulk lattice hydrogen concentration and its distribution during the diffusion process. A detailed analysis in ref. [9] has assessed the density of dislocations to be the most frequent and important defect factor concerning diffusivity of hydrogen in fcc nickel. In view of its crystallographic similarity to nickel, it has therefore seemed reasonable to apply a same analysis in the case of fcc Pd. Two interaction energies for hydrogen atoms with dislocations with values of 0.25 eV and -0.15 eV have therefore been similarly treated with reference to considerations of Wert [20], for hydrogen concentration values ranging from cH(n) = 0.8.10-6 up to cH(n) = 0.8.10-3. The main results in
the case of nickel that were plotted in Fig. 1 of ref [9]. have indicated that for structural states in a range from good monocrystals with dislocation densities J of mono values ~10-8 m-2 up to well annealed polycrystal values with J poly ~10-12 m-2, an effective value for hydrogen diffusivity is not changed within such quoted defect concentration limits and has seemed in accord with the unexpectedly narrow scatter of available data presented by different authors for values of hydrogen diffusivity in nickel and palladium [19].
Assuming, therefore, an interaction energy of hydrogen with dislocations in Pd of the order of -0.25 eV, again in keeping with general suggestions of Wert [20], any measurable dependence of significant concurrent introduction of dislocations on diffusivity, need not have been expected within an investigated range of hydrogen concentrations of 0.005 <cH(n) < 0.009.
These expectations also would seem consistent with findings of Bucur [21] and with considerations of a comprehensive review by Kirchheim [22] - especially with reference to Fig. 30 on page 292 of ref. [22]. The geometrical treatment of an imperfect lattice containing dislocations given in ref. [9], seems further to be supportive of this line of argument [23,24] by analogous consideration of results for deuterium trapping in Pd that have been obtained by small angle neutron scattering [25].
4. THE ROLE OF INITIAL AND BOUNDARY CONDITIONS
Acceptance of the two statements (a) and (b) of Section 3 now allows more concentration of attention to a search for IBC forms which correspond to adequate fits of the experimental curves in Figs. 1-4 without any additional requirements. Only simple IBC curves of this type, expressing for example surface concentrations of hydrogen atoms, might seem appropriate, from a considered point of view, to serve as reasonable approximations for the overall process, especially if suitable auxiliary factors may have to be considered, such as the surface treatment of electrodes, including any alterations of current densities, chemical composition of the electrolyte, alterations of temperature, choices of catalysts or conditions of stirring.
Thus, for example, the DE measurements on nickel have required [9,13,23], slight surface etching, additions of thiourea to the electrolyte and requirements of efficient stirring, in order to maintain effectively constant hydrogen concentrations at both entrance and exit surfaces at all times. Moreover in view of all these associations, it now seems quite probable from a standpoint of natural equilibrations of surface processes, that secondary chemical reactions and electrolyte ageing could also produce some measure of inconstancy of these surface concentrations over the whole periods of measurements. Such probabilities may be accepted within a given set of measurements and may be correlated with margins of errors in calculated values of diffusion coefficients. Considerations of plausible models may include input data estimations of values of hydrogen concentrations c and c,, in terms of the times that should be allowed
o 1'
for fully informative pictures of real concentration distributions. Indeed, derived quantities have suggested that surface concentration variations of the order of only 10% would not seem overstatements of necessary required corrections, in similarity to those required for
analogous considerations of similar problems which have arisen in the measurement of diffusivity of hydrogen in nickel [24].
5. FACTORS INVOLVED IN MATHEMATICAL MODELLING
Considerations of mathematical modelling problems should likely be of increased complexity with increasing numbers of the various parameters that may be necessary adequately to characterise the diffusion system. For example, the model suggested by Makhlouf et al [26] has involved four parameters, three of which (y , ^, v ) control the rates of capture and release of hydrogen by traps and the fourth hm corresponds to a surface impedance factor operating at x = 0. In this case hm was defined as a surface limited mass-transfer coefficient analogous to a convective heat-transfer coefficient and the final interpretation has included involvements of roughness, texture, grain size, heat treatment, and catalytic effects, together with chemical interactions including interactions between surface and fluid. Nevertheless the Makhlouf approach [26] is realistic in regard to the choices of factors influencing experimental results, while a more recent review by Lewis [27] has also enumerated similar varieties of condition thwt may govern any type of measurement in the case of the Pd-H system.
From these latter considerations it has to be concluded that the influence of any individual factor on dif-fusivity would only represent a part of a complex picture and that interactions of two or more factors are likely to play important roles. Indeed a comparison of results of different authors would seem only to be possible in cases of "identical" samples under very precise measurement conditions and even then systematic errors might still exceed the possibilities of obtaining good reproducibility of recorded values [29].
6. RESULTS AND DISCUSSION
The essential problem of solving Fick's equation for the DE model in the present case has been treated by a trial and error method with suitable choices of initial
'o (7 )
/ 0 / Q / y B
^ / « ■O / A m / (8 ) '
to (6) Time C /
Fig. 1. Diagrammatic representation of 3 types of bending deflection-time (BDT) curves corresponding to conditions of: (i) cathodisation of paiiadium-biack-coated surfaces over domain A, initiated at t0 (A); (ii) open circuit conditions (domain B) following interruption of cathodisation at t0 (B); (iii) anodisation of paiiadium-biack-coated surface commencing at t0 (C).
and boundary conditions [29, 30] and has been based on physicochemical considerations, coincidences of experimental and theoretical BDT curve shapes and satisfactory correlations with other available literary diffusiv-ity data.
The investigated processes of initial hydrogen absorption, internal equilibration of hydrogen concentrations and subsequent complete desorptions of hydrogen from the palladium strip were each carried out successively as represented in Fig. 1.
The following tables 1a,b represent a more detailed description of three chosen hydrogen diffusion processes (absorption, equilibration, desorption) in one-dimensional experimental arrangement for the diffusion-elastic measurements:
Table 1a: Will characterise the process and corresponding boundary conditions.
Table 1b: Indicates expressions for concentration distributions and bending moments. Free end deflections of the sample are always proportional to the bending moment.
Figs. 2,3 and 4 will represent corresponding BDT curves and are each representative of both tables.
Tables 1a and 1b summarise schematically, suitable correspondence of IBC together with expressions for concentration distributions c(x,t) and the corresponding bending moments M (t).
The boundary conditions indicated in Table 1a always refer to the right hand surface where x = 1. The left hand surface at x = 0, covered with piceine, has here been considered to be effectively hydrogen impermeable. All the solutions of ci(x,t) presented in Table 1b are either found directly in ref. [29] for the IBC indicated or may be derived from expressions quoted there. All the formulae for Mi(t) were derived from the corresponding solutions of ci(t) on the basis of the diffusion-elastic model [3,13]. Several constants of the system have been summarised into K1, K2 and K3. This abbreviation represents only the shape of curves corresponding to coordinates and are given unambiguously only by the sum S in Mi(t). Terms of the series include only the three parameters - D, t and l, of which t and l were readily measurable and values of D have been calculated directly without any assumptions as to constants. There has been no margin for arbitrary adjustments of auxiliary parameters since the procedure has been taken to be absolute [13-18].
6.1. Hydrogen Absorption
The right hand surface conditions for absorption has been considered as operative with the following stipulations:
(a) linear increases with time of the surface hydrogen concentration of the strip i. e. cright = ko + klt, where k and k are constants.
o l
(b) a constant flux of hydrogen through the surface. In this case resultant BDT curves clearly correspond to conditions of constant F0.
6.2. Equilibration conditions
For the process of equilibration of internal hydrogen concentration conditions of perfect impermeability of both surfaces have been assumed to hold throughout. At the moments of interruption of the electrolysis current, open circuit conditions have been taken to begin immediately. Equilibration process have then commenced from initial concentration distributions, attained at the conclusion of the preceding process of
absorption. It is important to appreciate here, that after sufficient periods of the absorption process, where the sequential concentration against time curves have been found to have the same shape, their position on the concentration axis, then simply shifts in proportion to the elapsed time. This follows from the form of c1(x,t)
- with the extended form given in Table 1b, where the third term in brackets can effectively disappear for long times, t - in our conditions approximately, after 20 min
- and cl(x,t) may then take the simplified form
c, (xr) r0iLi(ui, i1 - ((2^2))2) ,
as representing the stationary curves in the form of simple quadratic parabolas, with heights above the zero level, proportional to time.
At the beginnings of the equilibration stage, a dependency on the initial concentration also has a parabolic form, and the expression c2par(x,t) has been found to hold for the time development of internal hydrogen concentrations with corresponding moments expressed by: M2par (t) with an extended forms given in Table 1b.
It may be stated here that concentration distributions represented by nonparabolic monotonous smooth curves, may also be approximated to by a series of parabolas, so that the indicated forms of M2par (t) still hold.
Further considerations have led to an unexpected conclusion. Thus if one assumes an initial linear distribution of concentrations before the equilibration stage, then the time development follows the form of expression c2lin(x,t) in Table 1b and the resulting moment M2lin(t) will have exactly the same shape as M2par(t) but with rather different constants in front of the summation symbol. This means that for practical treatments of the BDT curves for the equilibration process, the initial starting concentration dependence may arbitrarily, then either be linear or parabolic or even be approximated to by one or several parabolas. This is a conclusion of extreme practical importance, suggesting that a wide range of shapes for the initial concentration distribution for equilibration would each yield the same or practically the same results on the basis of BDT curves. However, it can also be shown that no such tolerance exists for either the absorption or desorption processes.
This reasoning leads to a clear experimental rule: in that if after a process of absorption, the change of regime assumes a subsequent conditions of two impermeable surfaces, and it is preferable just to use this combination of measurements for calculation of the diffusivity. This conclusion could also allow an inclusion of Gorsky effect considerations as special cases within which the initial concentration distribution is only linear[5, 10-12, 30, 35, 36].
6.3. Hydrogen desorption conditions
The form of the experimental BDT curves for the desorption process c3(x,t) and (M3(t) in Table 1b) has led to only one combination of possible assumptions concerning the IBC. The main assumption is that at the moment of switching on anodisation, there has been an "instantaneous" decrease of the right side surface concentration to zero with subsequent continuation of this zero constancy throughout anodis-ation. It is a next requirement, that the form of the process of hydrogen desorption from Pd-H will be assymmetrical with respect to the form of the absorption process. Thus the flux of hydrogen removed
Table 1a.
Correspondences of Boundary Conditions.
Fig. Process (electrolytic conditions) Boundary conditions for t>0 for the right surface x=1 Total hydrogen content
2. Charging (cathodisation) -D(dc/dx) = F0 constant flux of hydrogen Increasing from zero level
3. Equilibration of concentrations (open circuit) dc/dx = 0 impermeable; no flux Effectively constant
4. Desorption (anodisation) c(l,t) = 0 kept on zero concentration Decreasing from constant level to zero
Table 1b.
Detailed description of three chosen hydrogen diffusion processes (absorption, equilibration, desorption)
characterised by BDT curves in Figs. 1-4.
.a
(Ô
a a
j_i_i_
t(s)
Fig. 2. Hydrogen concentration distribution c1(x1t) and bending moments M1(t), proportional to the deflection during hydrogen absorption by electrolytic cathodisation
r / I Dt ix1 -I2 L I Lin2n2r i ftiiX i C, i -X,f I — —— | — -I------->-—6XD,--5- ,COS- ,
Li , r or h i m* . r , i
M, (t) = K}1
\___4_ y 1
24 + l)
exp
f D (2n +1 t ^
for the initial zero concentration c,(x,0) = 0
150 .. , 300 t(s) —»
3E-8
Fig. 3. Hydrogen concentration distribution c2(x,t) and bending moment M2(t) during hydrogen content equilibration
() C°! 2 n2 ?(2n - 1);
■exp¡
D ( n - l)nV! ( 2n - l)n
for the initial linear distribution on c,(x,0) = (c„/l)x
, * aï bl 41« ¡ Dn' Qt ! nUx c,„ ( xt) =— + — + c +—-X expi---— icos-
al ( i„ b
n ¡2 (m - 1) + lj'
for the initial parabolic distribution c,(x,0) = ax2 + bx + c Corresponding specimen bending moments (deflections):
, . ^ i 1 ¿¿(/w + i'Tii'V i 1 (0 = -^exD¡ -¡
/ x / I 1 u(¿n-rlY uí 1
ivi^ () - (aiT --exD,--i-^-,
u (Zrt-1-l) 1 r 1
t(s)-
1200
Fig. 4. Hydrogen concentration distribution c3(x,t) and bending moment M3(t) deflections during hydrogen desorption by
9E-8 anodisation.
I , 4 ^ (-l) (m-I-I)ilX I
c3 (^ü-Cq—> -—— cos----exoi
11 V ¿n t ] // i
\Ln~r I 1 ii
>(n-r l)~
L>\n-rl) n'l
4/2
NOTATIONS IN TABLES 1a, b
c(x,t) ...concentration distributions throughout the thickness of the strip;
M(t) ...bending moments in the given arrangement; the deflection of the free end of
the strip is proportional to the moment M(t); i = 1,2,3 .subscripts for absorption, equilibration, desorption;
x .coordinate perpendicular to the plane of the strip; origin 0 on the left surface of the strip;
t .time;
F0 .constant flux of hydrogen entering the right surface of the strip;
l .thickness of the strip;
D .diffusivity of hydrogen;
lin, par .subscripts referring to the initial linear or parabolic concentration distribution at t=0 for the
equilibration process; a,b,c .coefficients in the expression for parabola;
K1,K2,K3 .factors of proportionality in expressions for bending moments in absorption, equilibration, desorption;
BDT .bending deflection against time curves;
IBC .initial and boundary conditions.
Table 2.
Resummary of selected available data on diffusivity of hydrogen in a-Pd at 50 °C arranged in increasing values
of D(50 °C). Activation energies have been everywhere converted to J/mol.
References D0 m2s-1*107 Activation Energy Q, J/k.mol Range of Temps. 0C D(50 0C) m2s-1*1011
This paper 5.80
(31) 4.5 24060 -50-1000 5.81 (a)
(37) 6.36
(38) 2.908 22270 273-350 7.31
(28) 2.48 21800 5-50 7.42
(19) 2.9 22190 -50-600 7.50 (b)
(32) 3.86 22900 7-60 7.67
(34) 2.6 21840 7-95 7.67
(a) best fit of 11 authors; (b) Best fit of 25 authors. Authors in ref. (37) performed a special measurement at 50 °C.
from the Pd-H sample can not remain constant either theoretically or experimentally, since even at the moment of switching on the reversed current, it would have needed to have been infinitly large if the condition of cright (1.0) = 0 could have been strictly valid.
The experimental desorption curves were therefore accordingly treated in the following ways:
(a) to obtain best fits with theory, all curves were processed with the same program for each of 12 experimental points and an optimum value of diffusivity separately derived from each curve.
(b) the shape of individual experimental curves was checked from beginnings to final stages of the process in order to detect any possible trends of deviations from theoretical.
7. DETAILED ANALYSES OF EXPERIMENTS
Final plots of best fit according to procedure (a) are shown in Figs. 2, 3, 4, for the processes of hydrogen equilibration and desorption respectively, with two scales given along the horizontal axis in each case. Here the first scale in m2 refers to the parameter Dt of the full theoretical curve, while the second scale in s refers to the equidistant experimental points. The experimental curves chosen for these figures were each of average quality and values of measured resistivity were derived from comparisons between both scales.
After assuming a normalised height of 100 mm for all three of the curves (taken either at the plateau or maximum or minimum), the sum of the squares of deviations for all 12 points in our experiments would usually be of the order of 3 - 8 mm2 in the cases of the absorption and equilibration process and of the order of 50 - 70 mm2 for the desorption process. This corresponds to a mean quadratic deviation for one point of 0.5 -0.85 mm for an absorption or equilibration point and of 2.1 - 2.5 mm in the case of the desorption points.
7.1. Comparison of suitability between the three processes
Figures 2-4 represent critical checks of the shape of absorption, equilibration and desorption curves obtained in this way. The values of diffusivity were first determined by referring to the theoretical curves for absorption and equilibration respectively. The same procedure was followed for the desorption curves for the points with relative deviations 0.8 - 0.9 before the minimum and also for points both at the minimum and at series of points 0.9 - 0.1 beyond the minimum.
As visually evident from Figs. 2 and 3, such levels of comparison classify both the absorption and equilibration processes as being very adequately described by this choice of IBC (Table 1a, b). However desorption results would appear rather to differ slightly from this model. In the latter case an overall impression from Fig. 1 and supported by the form of Fig. 4, is of an initial retardation followed by a formal acceleration of the experimental process of desorption in comparison with the theoretical curve.
The information from comparisons of shape and quality of the experimental absorption, equilibration and desorption curves in Figs. 2, 3 and 4 may be summarised as follows:
1. No significant trends exist within the systematic BDT curves for absorption and equilibration cases as functions of time. An apparent slight increase of the values of D vs the relative coordinate time of absorption would seem to lie within experimental scatter.
2. Values of diffusivity determined from the processes of absorption and equilibration of the treated experimental set, of 5.78.10-11 and 5.80.10-11 m2s-1 respectively have been found to coincide with one another beyond what might have been anticipated.
3. Results for equilibration are slightly superior to those of absorption and should br recommened in all cases where this type of measurement is feasible.
4. The trend of deviation of the BDT curves with time (as relative coordinate) for the desorption process seems monotonously high in comparison with the theoretical curve. In terms of an arithmetic mean treatment of three such desorption curves, values of D derived at y/y = 0.8 before the minimum,
■J > -J max '
gave D08 = 7.37.10-11 m2s-1 and for y/ymax at 0.1 beyond the minimum gave a value of D01 = 10.8.10-11 m2s-1, thus differing by a factor of 1.5. The progress of desorption results could thus seem in our conditions to represent evidence of only an apparent gradually accelerating process with reference to the suggested IBC range of accepted values of D cited in Table 2 and would only appear to be speeded up. The overall mean value from our desorption statistics by procedure (a) above is 10.8.10-11 m2s-1, which is nearly twice the diffusivity values derived from the absorption and equilibration processes of the same overall experiment.
7.2. Influences of electrolyte concentration
Bucur [28] has stated in his detailed study that "While contamination of the cathodic (hydrogen entrance) side of the palladium membrane has not appeared seriously to affect the permeation rate, it has, however, been found that significant changes of permeation rate leading to serious errors in calculated diffusion coefficients, can arise from gradual contamination of the anodic (hydrogen detection) side of the membrane". This statement might perhaps be generalised with reference to contamination derived from chemical reactions occurring on the anodic surface, especially if high current densities of the order of 200 Am-2 are being applied during anodisation. This line of evidence seems also effectively to support suggestions of a gradual decrease in efficiency of hydrogen transfer through the electrode over the course of the stages of the anodisation process [26] and so to a gradual appreciable modification of expected hydrogen flux. The regime of anodisation could thus appear to be rather more easily complicated in terms of surface conditions and would thus seem to require further investigations with objectives of finding still further generally improved overall purification conditions. It would therefore suggest replacement of the term for conditions of anodisation, namely c , by a more realistic form
' -1 anodic' -1
such as canodic = f(t), where f(t) would be a decreasing function of time, strongly dependent on conditions of measurement.
7.3. Employments of the diffusion-elastic technique
The experimental curves obtained by the DE technique have so far usually been derived from sequences of the experimental processes as a-e-d (absorption-equilibration-desorption) repeating one after another several times with the same sample specimen. The hydrogen content after each absorption process and each following equilibration process, could then be determined from measurements of the depths of the minimum of each desorption curve analogous to a procedure previously applied to Pt [15]. Taken as examples: two fully completed a-e-d cycles requiring 30 min of hydrogen absorption at a total integrated current of 10 mA should have been expected to correspond with calculated hydrogen contents of (H/Pd = n) transfer values of 0.0055 and 0.0058 respectively. By comparison, a combined a-e-a-e-d cycle, involving a total absorption time of 60 min, under otherwise identical conditions, gave a completed calculated transfer value of H/Pd = n = 0.0091, which is less than, but still comparatively close to, twice the aggregated values of hydrogen content introduced and then completely removed over the shorter time periods. This comparison of efficiency has seemed quite well to support the reliability of the procedure employed for hydrogen content determination, as being satisfactorily reproducible and convenient. It has also served to indicate the accuracy and high quality of measurements of hydrogen diffusion coefficients obtainable by the diffusion elastic technique.
Acknowledgements
Financial support, and flexible arrangement encouragements, from the Czech Republic Academy of Science (involving the Institute of Physics and the Crys-tallographic Association), The Royal Society of London and Johnson Matthey PLC are most gratefully acknowledged.
REFERENCES
[1] Stoney G.G., Proc. Roy. Soc. London, 1909, Vol. A82 P. 172.
[2] Zener C., Elasticity and Anelasticity of Metals. Chicago University Press, 1948.
[3] ©ermäk J., Czech. J. Phys., 1973, Vol. B23, P. 1355.
[4] ©ermäk J., Kufudakis A., Czech. J. Phys., 1973, Vol. B23, P. 1370.
[5] Gorsky W.S., Phys. Z. Sowjetunion, 1935, Vol.8, P. 457.
[6] Alefeld G., Völkl J., Schaumann G., Phys. Stat. Sol., 1970, Vol. 37, P. 337.
[7] Völkl J., Ber. Bunsenges. Phys. Chem., 1972, Vol. 76 P. 797.
[8] Mazzolai F.M., Z. Phys. Chem. Neue Folge. 1985, Vol. 145, P. 199.
[9] ©ermäk J., Gardavskä G., Kufudakis A., Lejek P., Z. Phys. Chem. Neue Folge. 1985, Vol. 145, P. 239.
Lewis F.A., Magennis J.P., McKee S.G. and Ssebu-
wufu P.J.M., Nature, 1983, Vol. 306, P. 673.
Lewis F.A., Kandasamy K., Baranowski B., Platinum
Met. Rev., 1988, Vol. 32, P. 22.
Kandasamy K., Scr. Metal., 1988, Vol. 22, P. 479.
©ermäk J., Kufudakis A., J. Less-Comm. Met., 1986,
Vol. 49, P. 309.
Kufudakis A., Raszynski W., Czech. J. Phys., 1986, Vol. B26, P. 1360.
©ermäk J., Kufudakis A., Gardavskä G., J. Less-Comm. Met., 1979, Vol. 63, P.1.; see also, Das A.K., J. Appl. Phys., 1991, Vol. 70, P. 1355. ©ermak J., Kufudakis A., Gardavskä G., Lewis F.A., Trans. Jap. Inst. Met. Suppl., 1980, Vol. 21, P. 125. ©ermak J., Kufudakis A., L'Hydrogene dans les Metaux, ed. Science et Industrie, 1972, p. 119. Kufudakis A., ©ermäk J., Lewis F.A., Surface Tech-nol., 1982, Vol. 16, P. 57.
Völkl J. and Alefeld G.: Diffusion of Hydrogen in Metals, in Hydrogen in Metals I, Springer, Berlin 1978, P. 321.
Wert Ch.: Trapping of Hydrogen in Metals, in Hydrogen in Metals II, Springer, Berlin, 1988, P. 305. Bucur R.V., J. Mat. Sci. 1988, Vol. 22, P. 3402. Kirchheim R., Progress in Materials Science, 1988, Vol. 32, P. 261.
ermak J., Kufudakis A., Redl Vl., Z. Phys. Chem. Neue Folge, 1939, Vol. 116, P. 9. ©ermak J., Rocz. Chem, Ann. Soc. Chim. Pol., 1986, Vol. 50, P. 1741.
Heuser D.J., King J.S., Summerfield G.C., Brue F. and Epperson J.E., Acta metal, meter., 1991, Vol. 39 P. 2815.
Makhlouf M.M., Sisson R.D., Jr., Met. Trans., 1991, Vol. 22A, P. 1001.
Lewis F.A., Platinum Metals Rev., 1982, Vol. 26, P. 20, 70, 121.
Bucur R.V., Int. J. Hydrogen Energy, 1985, Vol. 10, P. 399. Carslaw H.S., Jaeger J.C.: Conduction of Heat in Solids. Russian translation: 1964. Kufudakis A., ©ermäk J., Lewis F.A., Z. Phys. Chem Neue Folge, 1989, Vol. 164, P. 1013;
ermak J., Kufudakis A. and Lewis F.A., Z. Phys. Chem., 1993, Vol. 181, P. 233.
Birnbaum H.K. and Wert C.A., Ber. Bunsenges. Phys. Chem., 1972, Vol. 76, P. 806.
Ishikawa T. , McLellan R.B., Acta metal., 1986, Vol. 34, P. 1825.
Parrish W., Acta Cryst., 1960, Vol. 13, P. 838. Züchner H., Schöneich H.G., J. Less-Comm. Met., 1984, Vol. 101, P. 363.
Lewis F.A., Kandasamy K. and Tong X.Q., Solid State Phenomena, 2000, Vol. 207, P.73-75. Lewis F.A., In: Progress in Hydrogen Treatment of Materials, ed.
Goltsov V.A., Kassiopeya: Donetsk, Coral Gables 2001, P. 147.
Powell G. L., Kirkpatrick J.R., Phys. Rev. 1991-II, Vol. B-43, P. 6968.
Yoshihara M., McLellan R.B., Acta metall. Mater. 1990, Vol. 38, P. 1007.
