,Journal of the Less-Common Metals, 103 (1984) 81 - 90 81

HYDROGEN IN LIQUID-QUENCHED AND VAPOUR-QUENCHED AMORPHOUS Pd80Si20*

U. STOLZ and R. KIRCHHEIM

Institut fiir Werkstoffwissenschaften, Max-Planck-Institut fiir Metallforschung, D-7000 Stuttgart 1 (F.R.G.)

J. E. SADOC and M. LARIDJANI

Laboratoire de Physique des Solides, Centre d’orsay, Universitk de Paris-Sud, Ba^timent

510, F-91405 Orsay (France)

(Received March 15, 1984)

Summary

Measurements of the chemical potential, diffusion coefficient and partial molar volume of hydrogen in amorphous Pds,Si,, revealed pronounced differences between samples which were prepared by melt spinning and by sputtering. The different concentration dependences of the three quantities can be explained by gaussian energy distributions of different widths, yield- ing a broader distribution for the sputtered sample.

1. Introduction

Computer modelling of amorphous structures [l] has revealed that a variety of interstitial sites (octahedral, tetrahedral etc.) exist where each type of site has edge lengths fluctuating around an average value. The distribution of edge lengths or distances of atoms can be measured and is described by the radial distribution function (RDF). When hydrogen is dissolved in these interstices, a distribution of potential energies of hydrogen atoms is obtained. It has been shown that a gaussian distribution

n(G) dG = & exp\-(VT/ dG (1)

is in agreement with experimental results [2] and theoretical calculations [3]. In eqn. (l), n(G) is the concentration of interstices of free energy G within the interval dG, and o and G” are the width and mean energy respec- tively of the energy distribution. (T is related to the spread Ar of atomic distances by the following relation [ 31:

*Paper presented at the International Symposium on the Properties and Applications of Metal Hydrides IV, Eilat, Israel, April 9 - 13, 1984.

0022-5088/84/$3.00 0 Elsevier Sequoia/Printed in The Netherlands

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3 *= -

N1/2 r0

where N is the number of edges of the interstice, B is the bulk modulus, V, is the partial molar volume and r. is the average atomic distance. Accord- ing to this equation, interstices with a large volume have a low potential energy and vice versa. Thus hydrogen atoms located in low energy sites have a low diffusivity with a high activation energy and their partial molar volume is smaller than that for atoms in the high energy sites. The occupancy of the sites is determined by Fermi-Dirac statistics so that, at low concentrations, only sites of lower energy are occupied. With increasing concentration, sites of higher energy have to be filled and the hydrogen diffusivity and partial molar volume should increase. This behaviour has been measured for amorphous Pd-Si alloys and has been explained quantitatively [ 2, 41.

The purpose of the present study is to compare the solubility, diffusivity and molar volume of hydrogen in two samples of amorphous PdsoSizo. One sample was prepared by melt spinning (liquid quenched) and the other was prepared by sputtering (vapour quenched). We used broader samples than had been used in previous work [ 21 so that electrochemical measurements of the diffusivity could be made, and therefore results for a large number of concentrations and temperatures were obtained. The Fermi-Dirac statistical analysis and the experimental results show an ideal dilute behaviour with a constant activity coefficient, diffusivity and activation energy for very low concentrations or small gaussian distributions, whereas at higher hydrogen contents a step approximation is valid where only hydrogen atoms in sites near the Fermi level (equivalent to the chemical potential) contribute to the hydrogen mobility. For the latter limiting case the decrease in the activation energy for diffusion scales with the increase in the Fermi level caused by an increase in the hydrogen concentration.

2. Experimental details

The solubility of hydrogen was determined from e.m.f. measurements in an electrochemical cell where the sample was used as an electrode and faced two electrolytic compartments. When a current pulse was passed through one of the compartments, protons were discharged on one side of the sample. The increase in hydrogen content is given by Faraday’s law and the e.m.f. response in the other compartment occurs with a time lag after the current pulse. The e.m.f. transient or simply the time lag can be used to evaluate the hydrogen diffusivity. Details of the procedure have been described elsewhere [2, 5,6]. In all equations derived from Fick’s second law it has been assumed that the intrinsic diffusion coefficient is independent of concentration. This is not the case for hydrogen in amorphous metals but we shall show in the following that the measured apparent diffusivities for an initial hydrogen concentration co are close to the intrinsic diffusion coef-

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ficient D for cO, if the change AC in the hydrogen concentration is less than cO. Its actual magnitude depends on the concentration dependence of D.

Computer calculations (for details see ref. 7) to determine the e.m.f. response for a diffusivity which increases with concentration were performed using a numerical method described by Crank [8]. The initial and boundary conditions and all the parameters were varied over a range corresponding to the experimental conditions of this study. It was shown that the measured apparent diffusion coefficient lay between the values of D for co and for cO + AC. This uncertainty was negligible compared with the large changes in D over a wide range of co.

The partial molar volume of hydrogen was obtained from length changes measured during electrolytic doping with hydrogen. The experimental details and the results for the liquid-quenched PdsOSi,, are given in ref. 4.

3. Results and discussion

The total hydrogen concentration c (the ratio of the number of hydrogen atoms to the number of interstices) for a given e.m.f. or chemical potential p is given by Fermi-Dirac statistics as [ 21

1 s m exp[-_((G - G”)P121 dG c=-- ~

a7f”2 .~m 1 + exp{(G - PURTI (3)

The integrand of eqn. (3) is the occupancy of sites of energy G and it depends in a complex way on u, I_L and T. However, there are two limiting cases of physical significance. In the first case of broad energy distributions (u > RT) and of chemical potentials which are not too low, the denominator of the integrand can be approximated by unity for G < p and by infinity for G > /.L. This describes the well-known step behaviour of the Fermi-Dirac function, and integration yields [ 21

1-1 = Go- u erf-‘(1 - 2~) (4)

The physical meaning of this step approximation corresponds to a filling of the low energy sites up to the Fermi level (chemical potential). In the second case of small energy distributions and/or low chemical potentials the numerator of the integrand in eqn. (3) increases faster than the denominator for G > p. Then sites below the Fermi level are still saturated but the main contribution to c comes from hydrogen atoms in sites above /J. For the latter sites we can neglect the first term in the denominator and obtain the ideal dilute approximation

CT2 p=(;“- ~ +RTlnc

4RT (5)

The neglect of unity in the denominator is equivalent to the replacement of Fermi-Dirac statistics by Maxwell-Boltzmann statistics which is reasonable

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because the important sites above the Fermi level are not saturated with hydrogen.

The step approximation has been successfully used [2] to interpret solubility measurements and is used in Fig. 1 to plot the e.m.f. values at 298 K uersus the inverse error function of 1 - 2c. A width of 138 mV = 13.3 kJ mol-’ and 204 mV = 19.7 kJ mall’ is obtained from the slopes of the straight lines for the liquid-quenched and the vapour-quenched samples respectively. Within the temperature range 0 - 70 “C the slope is independent of temperature within the limits of experimental error. As the e.m.f. was measured versus a saturated calomel electrode, the solubility at an e.m.f. of -234 mV corresponds to a hydrogen equilibrium pressure of 1 atm [2]. The hydrogen concentrations under these conditions are 0.017 for the liquid-quenched sample and 0.025 for thevapour-quenched sample. GOvalues can be also obtained from plots such as those shown in Fig. 1, but they will not be presented here because they were not used in the following.

If the integral in eqn. (3) is calculated numerically and the results are used to fit the e.m.f. data in Fig. 1, the exact values for the width u are

concentration, HlPd

erf-’ (1-2~)

Fig. 1. E.m.f. of liquid-quenched (x) and vapour-quenched (0) amorphous Pds&izo as a function of the hydrogen concentration plotted according to the step approximation (eqn. (4)) (SCE, saturated calomel electrode).

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obtained. These are 11 kJ mall’ for liquid-quenched Pds,SizO and 18.5 kJ mol-’ for vapour-quenched Pds0Si2s in good agreement with the step approximation.

The most striking difference between. the two samples investigated in this study can be seen by comparing the hydrogen diffusivities. The experi- mental results obtained at various temperatures are shown in Figs. 2 and 3 as a function of concentration. In all cases the diffusivity increases with the hydrogen concentration. The diffusivity of the vapour-quenched sample is about an order of magnitude smaller than that of the liquid-quenched sample at low concentrations but increases by two orders of magnitude with increasing hydrogen content.

In the following we shall show that the difference in the concentration dependence is due to the different widths of the energy distributions. For this reason we use a model for hydrogen diffusion in disordered metals which was developed in ref. 2. In this model the saddle point energy during the long-range motion of the hydrogen atom was assumed to be the same for all the sites. Then the activation energy for jumps over the saddle point is directly related to the energy of the equilibrium position and the jump rate is very small for the low energy sites. Thus we can qualitatively understand

L. -3 -Lt -3 -L -I -5 -4 -3 -A! -I

log c log c

Fig. 2. various curve.

Fig. 3. various

curve!.

Diffusion coefficient of hydrogen in liquid-quenched amorphous PdseSiza at

temperatures as a function of the hydrogen concentration: -, theoretical

Diffusion coefficient of hydrogen in vapour-quenched amorphous Pds@iZs at temperatures as a function of the hydrogen concentration: -, theoretical

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why the hydrogen mobility increases as the hydrogen atoms are compelled to occupy sites of higher energy at higher hydrogen contents. The quantitative treatment gives a simple relationship between the intrinsic diffusion coeffi- cient D and the chemical potential:

(6)

where D” is the diffusivity in a reference material which contains only sites of energy G” with the same saddle point energy as in the disordered material. The difference between these two energies is the activation energy Q”, and D” can be written as

D”=D”,,exp - f$ ( )

where Do, is a frequency factor which contains the trial frequency and jump distance [2]. All the parameters in eqn. (7) are independent of concentration, and therefore the second factor in eqn. (6) determines concentration dependence.

(7)

the the the

For the ideal dilute approximation, eqns. (5) and (6) give a value for D which is independent of concentration as observed for the liquid-quenched sample at low concentrations. Because of the broader energy distribution of the vapour-quenched sample the validity of the ideal dilute approximation occurs at temperatures above or concentrations below the experimental ranges.

The step approximation is able to explain the concentration dependence shown in Figs. 2 and 3 and yields values for u which are equal to those obtained from the e.m.f. data when the same approximation is applied. How- ever, it will be shown below that the activation energies of diffusion are in some cases much lower than the approximate values. Therefore we have chosen an exact numerical treatment using eqns. (3) and (6) and the calculated value of D as a function of c. When an arbitrary value of D” was chosen curves such as those in Figs. 2 and 3, which depend only on a/RT and are rather sensitive to the choice of this parameter [8], were obtained. It was thus possible to find a unique value of CJ, which describes the con- centration dependence of D at all temperatures with respect to slope and curvature. With this value of (T, the D(c) curves were adjusted parallel to the log D axis to give the best fit to the experimental data. The parallel displacement gave the value of D” for each temperature. This procedure gave widths of 11 kJ mol-’ and 17.5 kJ mol-’ for the gaussian distributions of the liquid-quenched sample and the vapour-quenched sample respectively. These values are in excellent agreement with the e.m.f. data, provided that the different nature of the diffusivity and the e.m.f. are taken into account.

The second fitting parameter D” is presented in Fig. 4 in an Arrhenius plot, from which the prefactor Do0 and the activation energy Q” for the

temperature .“C

L 1

2.8 3 32 3.4 36 3.8

lOOO/T(Kl

Fig. 4. Diffusion coefficient D” of hydrogen in a hypothetical material which contains only sites with the reference energy G”: __ liquid quenched; - - -, vapour quenched. The points were obtained by fitting the curve; in Figs. 2 aild 3 to the experimental data.

reference state were evaluated. The results are as follows: liquid-quenched sample, Do0 = 4.1 X 10P4 cm2 s-l and Q” = 18.9 kJ mol.-‘; vapour-quenched sample, Do,, = 2.4 X 10e4 cm2 s-l and Q” = 9.9 kJ molP’.

The data in Figs. 2 and 3 were also used in Arrhenius presentations, yielding values of the apparent activation energy Q and the prefactor Do. A detailed discussion of these quantities is beyond the scope of the present paper and will be presented elsewhere [ 91. The apparent activation energies at 313 K are plotted uersus the inverse error function in Figs. 5 and 6, because eqns. (4) and (6) predict for the step function that a change in Q with con- cent,ration scales with the change in the Fermi level 12, i.e. straight lines with slope D are obtained in Figs. 5 and 6.

In agreement with prediction the changes in the activation energy are more pronounced for the vapour-quenched sample with the broader distribu- tion. For the liquid-quenched sample, deviations from the step approximation become marked at low concentrations where the ideal dilute approximation is more appropriate.

Measurements of the partial molar volume of hydrogen in the liquid- quenched sample [4] have shown two unusual features which are not observed in crystalline metals. First a negative volume change of about -1 cm3 (mol H)-’ was observed for hydrogen concentrations between zero

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concentration,HIPd concentration,H/Pd

10-2 IF3 lo-’ 10-s 10-6 10-7 /

erf-’ (1-2~1 erf-’ (I-2cl

Fig. 5. Activation energy Q obtained from the data of Fig. 2 at 2’ = 313 K plotted vs. the inverse error function of 1 - 2c in order to obtain a straight line for the step approxima- tion (line 2); line 1, ideal dilute approximation;line 3, numerically calculated from eqns. (3) and (6).

Fig. 6. As for Fig. 5 for the vapour-quenched sample using the data from Fig. 3.

and about 5 X lop5 [H]/[Pd] which was attributed to the presence of large excess volumes or vacancy-like defects. Second the partial molar volume for [H]/[Pd] > 5 X lo-’ was positive and increased steadily with concentration. This concentration dependence can be explained by a distribution of the volume of interstices in an amorphous metal [4] and, because the last quantity also determines the energy distribution [ 31, a relationship between the partial molar volume V, and the chemical potential in the step approxi- mation has been derived:

V, = V,” - Ca erf -I( 1 - 2c) (8)

where C is a constant which can be calculated from measurable quantities. A plot of V, uersus the inverse error function gave a straight line for the liquid-quenched sample with a slope close to the theoretical value [4].

We observed a negative volume change for the vapour-quenched sample for the first 100 at.ppm which is about twice as large as that for the liquid- quenched sample, and therefore the “vacancy-like defects” are present at a higher concentration. The positive partial molar volume given by eqn. (8) is plotted in Fig. 7 and the straight line corresponds to a best fit with V,” = 2.1 cm3 and a slope of -0.32 cm 3. As the theoretical slope is Co we would

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H concentration, HI Pd

lo-*

1 1.5 2 2.5 erf-’ (I -2~1

Fig. 7. Partial molar volume V, of hydrogen in vapour-quenched Pd80Si20 as a function of the hydrogen concentration c plotted us. the inverse error function according to eqn. (8).

expect a slope of about -0.20 for the liquid-quenched sample owing to the smaller width o if a constant value for C is assumed. This agrees with the experimental value of -0.21 given in ref. 4.

We can conclude from these results and their discussion that the differ- ence between the solubilities, diffusivities and partial molar volumes of hydrogen for liquid-quenched and vapour-quenched Pds,Si,, is mainly due to the different energy distribution widths. The sputtered sample has a broader distribution than the sample produced by melt spinning; this is a reasonable result because quenching from the vapour state occurs at a higher cooling rate. The higher cooling rate may also be responsible for the higher concentration in vapour-quenched PdsOSizO of those defects which cause a negative volume change when they are filled with hydrogen.

It is very surprising that the concentration dependence of such different quantities as the chemical potential, the diffusivity and the partial molar volume can all be described by the same parameter, i.e. the width of the energy distribution. The agreement between the experimental results and the theory confirms the usefulness of Fermi-Dirac statistics for interstitials in disordered materials. Thus we can improve our understanding of hydrogen solid solutions and can use hydrogen as a probe for obtaining structural information about the matrix since the diffusion of hydrogen seems to depend very sensitively on structural differences.

Finally we shall show that the width u for the liquid-quenched sample can be calculated from eqn. (2) if we assume isotropic behaviour such that B can be replaced by a Young’s modulus E of 6.7 X lo-” N rn-* [lo] and

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Ar/r, is between 0.10 and 0.14 [ll, 121. Further, assuming octahedral occupancy with N = 12 and V,” = 1.97 cm3 [4] we obtain

a(theor) = 11.5 - 16 kJ mol-’ (9)

which is comparable with the experimental value of 11 kJ mol-’ for the liquid-quenched sample. We measured V,” = 2.1 cm3 for the vapour-quenched sample in this study and the radial distribution function is equal to that of the liquid-quenched samples [ 121, giving Ar/ro = 0.10 - 0.14 again. Therefore the broader energy distribution does not seem to be due to a broader distri- bution of Pd-Pd distances in the vapour-quenched sample. The bulk modulus of the vapour-quenched sample is not known, but it should be about 50% larger than that of the liquid-quenched sample to account for the higher value of u. However, measurements of Young’s modulus E for amorphous Pd-Cu-Si alloys [ 131 indicates that it decreases with increasing cooling rate.

Acknowledgment

U.S. and R.K. are grateful for the financial support provided by the Deutsche Forschungsgemeinschaft.

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