J. CHEM. SOC. FARADAY TRANS., 1994, 90(13), 1979-1982 Effects of Hydrogen and Deuterium Concentration on Measurements of the Solubility and Diffusivity of Hydrogen Isotopes in Yttrium Takeshi Maeda, Shizuo Naito,' Masahiro Yamamoto and Mahito Mabuchi Institute of Atomic Energy, Kyoto University, Uji, Kyoto 61 1, Japan The solubility and diffusivity of hydrogen and deuterium in polycrystalline a-yttrium have been measured in the temperature range 1073-1273 K and at hydrogen and deuterium concentrations, 8 (H :Y and D :Y, atomic ratio), of 0-0.44.The measured solubility obeys the relationship p = k[8/(1 -O)]', where p is the hydrogen pressure and k is a constant independent of 8. The measured diffusion coefficients for hydrogen and deuterium increased with 8.The experimental result has been explained by a model in which a hydrogen atom can occupy a pair of two nearest-neighbour tetrahedral sites of the hcp a-yttrium lattice and move almost freely between the two sites. The yttrium-hydrogen system can retain the hcp a-phase structure in a wide range of temperature and hydrogen con- centration.1-6 The a-phase region extending up to a hydrogen concentration 8 = 0.24 at even 4 K has attracted much research interest because it is closely related to the occurrence of hydrogen pairing?y6-' which was first proposed as a pos- sible cause of the low-temperature resistivity anomaly observed in the yttrium-hydrogen system. At high tem-peratures, however, the hydrogen pairing can no longer be expected to persist and hydrogen atoms are likely to be unpaired.6*'2 In addition, the a-phase region extends up to larger values of 8 as the temperature increases. This makes it possible to study the solubility and diffusivity in a wide range of 6 without paying special attention to additional com-plications resulting from hydrogen ordering and the phase boundary.The purpose of this paper is to find factors that affect the 8 dependence of the solubility and diffusivity in hcp a-yttrium. Several studies have been reported on the solubility of hydrogen and deuterium in a-yttri~m.'-~*'~ Some of the reported values of ~olubility,~~~ however, show considerable deviations from Sieverts law at small 8. In addition, a change in the reported enthalpy of solution of hydrogen with 8 has been shown to exhibit a spread of values over the range 0-0.69 eV.I3 The cause of this observed inconsistency still seems to be unclear.Several researcher^'^-^' have reported mea- surements of diffusion coefficients (D) for hydrogen and deu- terium in a-yttrium. There has been, however, no systematic study of the 8 dependence of D although in some metals other than a-yttrium, effects of 8 on D have been investi- gated.20*21In the fcc yttrium hydride, an observed increase in D with 8 has been related to the sites occupied by hydrogen atoms.22 Decreases in D with 8 in bcc niobium and tantalum have been discussed on the basis of an elastic interaction between the hydrogen atoms.20*2' In the present study we have measured changes in the solu- bility and diffusivity with 8.The 8 dependence of the mea- sured solubility is compared with a model of solution that involves sites available to hydrogen atoms and an interaction between the hydrogen atoms in yttrium. We then discuss a model of diffusion that incorporates the factors of interaction and site occupation, and compare it with the 8 dependence of the measured diffusivity. Experimental Values of the diffusion coefficient, D,were calculated from the rate of gaseous hydrogen absorbed by a yttrium sample and the solubility was calculated from the final amount of hydro- gen absorbed. The apparatus and procedures for measure- ments have been described previo~sly.'~.~~ To obtain D at different values of 8, we repeated the measurement after increasing the hydrogen pressure, p, in a specimen chamber.Values of 8 were defined, for particular values of D obtained from each measurement, as the average of 8 estimated before and after the measurement. The sample used and the heat treatment employed were the same as described previously.' ' Measurements were made in the temperature range 1073-1273 K in steps of 50 K and at hydrogen concentrations of 0.02-0.44. Surface processes were found to affect the measured hydro- gen absorption rate. The values of D were corrected to allow for these effects, as shown previo~sly.'~*~~ Results and Discussion Sohbaity Relationship between Hydrogen Pressure and Hydrogen Con- centrat ion When 8 4 1, Sieverts law has been found to apply to 8 and p.6923*24We are concerned with the solubility at specific values of 8 and consider a relationship that incorporates a consideration of the sites available to hydrogen atoms in the hcp or-yttrium and of the interaction between the hydrogen atoms, both factors which play crucial roles in the relation- ship at finite 8.The relationship has been derived in terms of the chemical potentials, pHZfor hydrogen in the gas phase and pH for hydrogen in yttrium where pH, = 2pH (1) The expression of pH2 is well kn~wn.~*~~ Adopting the lattice- gas model and using the mean-field approximation for the interaction between hydrogen atoms, we can write pH as6i24 pH = pi + w8 + k, T In -e r-8 where pi is the part of the chemical potential pHthat corre- sponds to the energy of hydrogen atoms in yttrium, w is a parameter that characterizes the hydrogen-hydrogen inter-action and r is the number of sites available to a hydrogen atom per yttrium atom in the hcp yttrium lattice.From eqn. (1) and (2) we have p=k -(1 (3) k = k' exp(2w8/kBT) (4) where k' is a constant independent of 8. A little care is needed in choosing the value of r. Hydrogen atoms occupy tetrahedral (T) sites in hcp cr-yttri~m~,~,~-’ (Fig. 1) and the number of the T sites is twice that of yttrium atoms. Now the distance between the nearest-neighbour T sites, which form a T-T pair (Fig. I), is c/4 (where c is the lattice constant along the c axis), i.e.0.14 nm. Variations of this distance corresponding to changes of temperature and 8 are small and can be ignored under the experimental condi- tion in the present study. This distance is rather small as compared to the separation of two hydrogen atoms in metals.6 This suggests that two hydrogen atoms cannot simultaneously occupy the T-T pair and only one of the T sites in it is therefore available to a hydrogen atom. We may thus assume that r = 1. At larger r, the hcp structure is unlikely to be retained and changes to the fcc stru~ture~,~ e.g. at r = 2. The problem of values of r other than unity will be mentioned later. We make a comment here on w. The assumption that one T-T pair is available to only one hydrogen atom means that w is strongly repulsive for nearest-neighbour hydrogen atoms.In eqn. (2) and (3) the values of w should therefore be regard- ed as a long-range part of the interaction, which has been discussed with regard to the origin of spinodal decomposition in some metal-hydrogen system^.^.^' 324 Interpretation of the Measured Solubility To discuss the solubility and to compare critically the result obtained in the present study with those reported earlier, we consider a k-8 plot instead of conventional p8 plots. An In p8 plot in particular shows only insufficiently clearly the extent of deviations of the solubility from eqn. (3). Fig. 2 shows values of k computed from measured values of p and 8 and through eqn. (3), where we have put r = 1.To obtain the values of k precisely at different temperatures we have plotted k on a logarithmic scale. The values of k at small 8, i.e. 8 < 0.1, coincide with those reported previo~sly.’~ At small 8, the difference between the values of k for hydrogen and deu- terium comes from the differences in the partition functions for H, and D, and in the partition functions for hydrogen and deuterium atoms in yttrium.” As can be seen from Fig. 2, values of k obtained in the present study are almost independent of 8. Reported values have been read off graphically from the literat~rel-~ and some of these values have been plotted in Fig. 2. In contrast Fig. 1 Tetrahedral (T) sites and octahedral (0)sites in the hcp a-yttrium.(.)Yttrium atoms, (T) tetrahedral sites and (0)octahedral J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 m5 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 0 0 Fig. 2 8 dependence of k at various temperatures computed from measured values of p and 8 by using eqn. (3) and (4) with r = 1. A, Hydrogen and, B, deuterium. (a) 1073, (b)1123, (c) 1173, (6)1223 and (e) 1273 K. (0,0)This study, (---) Begun et aL3 (*..-) Yanno-poulos et al.’ and (-.-) Lundin et al.’ The reported values plotted in this figure have been read off graphically from plots in the literature. to the values of k obtained in the present study, the reported values show a complicated 8 dependence, particularly at small 8. This inconsistency can also be seen in comparison of the change in the enthalpy of solution of hydrogen with 8.The enthalpy change has been summarized12 for the earlier reported data, for which its 8 dependence has been assumed to be linear. Since this assumption is equivalent to eqn. (4), we can make a comparison in terms of w in eqn. (4).It may be seen that w w 0 for the result of the present study and w = f0.04,’ 0.43,, 0.693 and 0.11 eV12 for the previously reported values.’ The origin of this inconsistency in k and w is presently not clear. All the data except the last one” mentioned above, which has been obtained from calorimetric measurements, have been obtained from measured isotherms. A possible origin may be the influence of impurities such as oxygen and nitrogen that were absorbed by samples in the course of experiments.In fact, it has already been reported that oxygen and nitrogen could cause complicated changes in k in the niobium-and tantalum-hydrogen sys terns.’ 5-2 In the present study we used an ultra-vacuum apparatus in order to prevent impurities from being absorbed by the sample during measurements. We have shown above that the solubility experimentally obtained in this study can be explained for values of r = 1 and w = 0 in eqn. (4). Note, however, that there remains an alternative possibility of explaining the solubility for values of r # 1 and w # 0. In fact, we can show that the experimental result in the present study can also be reproduced if w = -0.05 eV and r = 1.2. (The corresponding computed lines are not shown in Fig.2 because they are horizontal, almost straight lines and would reduce the clarity of the figure.) This value of w = -0.05 eV corresponds to the long- range elastic interaction energy in the yttrium-hydrogen system.6 The negative value, i.e. indicative of the attractive interaction, is required for w to cause spinodal decomposition to occur in the ~ystem.~,~~.~~ It is, however, not clear whether the spinodal decomposition is mainly due to the elastic inter- action in the yttrium-hydrogen system, since yttrium metal has a transition point at 1758 K and its corresponding effect on the phase transition in the system cannot be ruled out. A further discussion on the value of r has been given in the sites. Only a pair of nearest-neighbour T sites (T-T pair) and two of the.0 sites next to these T sites are shown. The thin dotted lines literature.6.28 There is, however, no evidence for r = 1.2 in the indicate reference of a tetrahedron and an octahedron site. The thick solid line is a T-0-T path and the thick dotted line is a T-T-0-T case of hcp cr-yttrium. Thus, it is difficult to conclude from path. only the solubility data that we can determine the particular J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 values of M' and r that reproduce the experimental result. These will be determined after the 8 dependence of the diffu- sion coefficient is discussed. Diffusivity Digusion Coeficients The values of diffusion coefficients have been measured under a hydrogen concentration gradient and the diffusion coeffi- cient values obtained are therefore chemical diffusion coeffi- cient~.~~~~*~~.~~These are related by 29-31 (5) r = rvf (6) where 2 is the jump length, r' is the jump rate, V is the site- availability factor, which is 1 -8, e.g.for r = 1 [eqn. (3)], and f is a mobility correlation factor3' or physical correlation factor,29 indicative of the presence of short-range order due to interactions between hydrogen atoms3' or can represent an additional distortion of the chemical potentialz9 other than that caused by the concentration gradient. The term d(p,/k, T)/aIn 0 in eqn. (5) is called the thermodynamic fa~tor~~-~land r in eqn. (6) represents the effective jump fre- quency. Since values of 0 mainly influence (i) the thermodynamic factor d(pH/k, T)/aIn 8, (ii) the site-availability factor V and (iii) the correlation factor f, we further consider these three factors.In regard to (i) we have from eqn. (2) (7) In the case of (ii), since we have used the mean-field approximation for the hydrogen--hydrogen interaction, we have V = 1 -0/r (8) The presence of a T-T pair (Fig. 1) requires a modification of eqn. (8). The result of quasielastic neutron scattering measurement16 has shown that the rate of hydrogen atom jump between the two T sites in the T-T pair is much larger than that between the 0 and T sites and between the 0 and 0 sites. This is consistent with the theoretical calculation of potential barriers between the various sites available to hydrogen atoms in or-yttrium: that is to say 0.16 eV between the T and T sites in the T-T pair, 0.33-0.46 eV between the 0 and T sites and 0.85 eV between the 0 and 0 sites.32 The large rate of the T-T jump makes it possible for hydrogen to jump through the T-T-O-T path (the thick dotted line in Fig. 1) with a considerable additional rate to jumps through the T-O-T path (the thick solid line in Fig.11, which mainly contribute to the long-range diffusion of hydrogen. We will now calculate for r = 1, i.e. when only one of the T sites in a T-T pair can be occupied by a hydrogen atom, and where the probability is that the hydrogen atom on the T site will find, through either of the two paths, an empty T site in other T-T pairs.For the T-O-T path this probability is 1 -8 and the T-T-O-T path it is, to a crude approximation, a8(1 -O), where O(1 -8) is the probability that the hydrogen atom on the T site finds no empty sites in other T-T pairs but finds an empty site through the other T site in the T-T pairs, and where a is a factor that takes into account the extra T-T jump necessary for the T-T-O-T path. The value of a may be near to but less than unity. We then have V = 1 -0 + a0(l -8) = (1 -0x1 + ae) (9) where the equation has an extra term 1 + a8 in addition to eqn. (8) with r = 1. (iii) The 8 dependence of f is rather complicated and numerical calculations are needed to estimate its precise value.z9-31 We will assume here that correlation is small and put f= 1 for the following reason.In the yttrium-hydrogen system the elastic interaction is one order of magnitude smaller than in other metal-hydrogen systems6 and the mag- nitude of the resulting short-range order, which is responsible for ~orrelation,~' should be small. In addition, the mobility correlation factor should deviate less from unity than the tracer correlation fa~tor.~ Thus, the diffusion coefficient can be written as with the modification of the site-availability factor eqn.(8)] and D = riZ(i + ae) (1 1) with the modification but without the interaction (W = 0). lnterpretation of the Measured Diflusivity Fig. 3 shows values of D measured at various temperatures and values of 8. In order precisely to see the values of D at different temperatures we have plotted D on a logarithmic scale.Measured values of D for hydrogen and deuterium increased with increasing 8. At high temperatures, e.g. at 1273 K, a rather large scatter in D could not be avoided owing to the difficulty in controlling the hydrogen pressure precisely for several seconds at the beginning of the mea~urement.~~ We can now compare measured values of D with eqn. (10) and (11).The solid lines in Fig. 3 show values of D computed from eqn. (ll),where a = 0.8 and rAzhas been adjusted so that D computed at 8 = 0.05 coincides with those reported previo~sly'~and the ratio of D for deuterium and hydrogen is 1/J2. The computed 0 dependence is consistent with the observed increase in D with increase of 8.The adopted value of a = 0.8 seems reasonable on the following considerations. The potential barrier between the T sites in the T-T pair has been calculated as 0.16 eV,32 which would be smaller if relax-ation of metal atoms around the hydrogen atom is con-~idered,~~and is small enough to allow almost free jumps between the T sites at high temperatures. Since the value of a is the factor that takes into account the effect of this T-T jump, and reduces to unity when this jump is completely free, the value of a may be taken as slightly less than unity. A A B 2.0c 0.3 I I I I I I I I I I I 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 0 0 Fig. 3 0 dependence of D measured at various temperatures for A, hydrogen and B, deuterium.The solid lines have been computed from eqn. (11) with a = 0.8.Symbols as Fig. 2. discussion of a possible dependence of a on temperature and 8 is, however, beyond the scope of this paper. In order to reproduce measured values of D by using eqn. (10) we need a value of I larger than unity and a value of w of the order k,T, which is too large as a magnitude of hydrogen-hydrogen interactions in metal-hydrogen systems. Moreover, values of r > 1 and w > 0 cannot reproduce the measured solubility: a value of w c0 is required to repro- duce the measured solubility if r > 1. It is thus impossible to give a reasonable explanation of the measured diffusivity and solubility by using eqn. (3), (4) and (10). A similar result can also be obtained by introducing improved approximations to the interaction such as with the quasi-chemical approx- imati~n.~~.~' A long-range hydrogen-hydrogen interaction factor seems to be less important to high-temperature solu- bility and diffusivity values in the case of the yttrium- hydrogen system, in contrast to the cases of the niobium- and tantalum-hydrogen systems, where 1on.g-range elastic interaction plays a key role in the 8dependence of the diffusi- vity and occurrence of the phase transition.20,21 An observed increase in D with 8 in fcc yttrium has been discussed and its cause ascribed to a partial hydrogen occupation of 0 sites.22 In the hcp yttrium, however, the jump rate is smaller for the 0-0jump than for the 0-T and an increased 0-site occupation is unlikely to make an appreciable contribution to measured values of D.In addition, the fraction of hydrogen atoms on the 0 sites in fcc yttrium has been shown to decrease with increasing tem- perat~re.~~We cannot, therefore, apply the model proposed to explain the increase in D with 8 in fcc yttrium directly to the experimental result obtained in the present study. 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