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A Statistical-Mechanical Method to Evaluate Hydrogen Solubility in Metal Hiroaki Ogawa* Corrosion Resistant Materials DeVelopment Group, DiVision of Fuels and Materials Engineering, Nuclear Science and Engineering Directorate, Japan Atomic Energy Agency, Tokai-mura, Naka-gun, Ibaraki-ken, 319-1195, Japan ReceiVed: April 21, 2009; ReVised Manuscript ReceiVed: NoVember 24, 2009
We propose a site-occupying model (SOM) to evaluate hydrogen solubility in metal because there is a discrepancy between the concept of the site-blocking model and experimental fact, as shown in this work. SOM is based on the experimental fact that hydrogen atoms in metal cannot come closer to each other than about 0.21 nm under hydrogen pressure of 1 MPa or less. We evaluate the hydrogen solubility by developing a statistical-mechanical method which considers partition functions for translational motion, configuration, vibration, rotation, and spin weights. SOM is verified by agreement with experimental data on hydrogen solubility in metal. We also compare SOM with the site-blocking model, and find that the theoretical curves from SOM agree well with the experimental data even at high hydrogen solubility of over 0.5 H/M by atomic ratio (hydrogen/metal). 1. Introduction The present work describes a statistical-mechanical method to evaluate hydrogen solubility in metal. In metal, the total number ns of interstitial sites available for hydrogen atoms per metal atom is 1 for octahedral sites (O-sites) in the face-centered cubic structure (fcc), 2 for tetrahedral sites (T-sites) in the hexagonal closest-packed structure (hcp), and 6 for T-sites in the body-centered cubic structure (bcc). In the ideal case of interstitial solid solutions, hydrogen atoms can be put in all of the sites. However, it is experimental fact that hydrogen atoms in metal cannot come closer to each other than about 0.21 nm under hydrogen pressure of 1 MPa or less.1 Therefore, hydrogen atoms cannot actually be put in all of the sites. It has been shown that the theoretical curve of the hydrogen solution disagrees with experimental values at high hydrogen solubility of over 0.1 H/M (hydrogen/metal) by atomic ratio. Whether or not the theoretical curve can be fitted to the experimental values depends on how the partial configurational entropy of a solute atom (hydrogen) is expressed. One way to correct the expression is to apply the site-blocking model, in which the number of interstitial sites changes depending on the hydrogen concentration in metal: i.e., the blocking effect is stronger at higher hydrogen concentration, and is negligible at lower hydrogen concentration (the ideal case). On the basis of this concept, there are three main corrected expressions for the partial configurational entropy: the expressions of Moon,2 McLellan et al.,3 and Boureau.4 Moon gave an approximate expression applicable to the case of blocking with overlap case. McLellan et al. extended the site-blocking model by considering the reduction in the number of blocked sites at a given concentration due to their overlap. Boureau proposed calculating the partial molar entropy of interstitial solid solution excluding the first and second nearest neighbors of occupied sites. However, even if their expressions are used, the theoretical curves disagree with experimental values at high hydrogen solubility of over 0.5 H/M. Thus, it is insufficient to apply the * To whom correspondence should be addressed. Phone: +81-29-2826829. Fax: +81-29-282-5864. E-mail:
[email protected].
site-blocking model at higher hydrogen solubility. This disagreement arises from the concept of the site-blocking model. At lower hydrogen concentration (the ideal case), the blocking effect is negligible, which means that the hydrogen atoms can come closer to than about 0.21 nm. However, this theoretical result contradicts the experimental fact that the hydrogen atoms cannot come closer than about 0.21 nm. This contradiction has not been known up to now. Veleckis and Edwards estimated ns for T-sites in bcc as 0.779 for the V-H system, 0.904 for the Nb-H system, and 0.702 for the Ta-H system by experiment.5 However, they did not explain the difference between ns ) 6 and their estimated values. Flanagan et al. also estimated ns for Pd alloy-H systems by experiment, and showed that the estimated value differs from the ideal value.6-8 Recently, first-principles methods were used to evaluate the thermodynamic properties and hydride formation in metalhydrogen systems. These methods discussed the energy of metalhydrogen systems, namely the enthalpy,9-13 and the hydrogen solubility was obtained from Sieverts’ law (a conventional method) or from a combination of Sieverts’ law and MonteCarlo calculations.14-16 The expression of the partial configurational entropy assumed the ideal case. It was shown that the theoretical curve obtained by these methods disagrees with the experimental data at high hydrogen solubility of over 0.01 H/M, e.g., at 0.1 H/M, the deviation is about ( 300%, and this deviation tends to increase at higher solubility.16 These methods are valid at low hydrogen solubility of under 0.01 H/M. At high hydrogen solubility of over 0.01 H/M, this means that the configurational entropy is a more significant factor for good agreement than the enthalpy. Thus, the expression of the configurational entropy should be modified from the perspective of statistical-mechanics. There are no models whose theoretical curve agrees with the experimental data at higher hydrogen solubility. The present work proposes a site-occupying model (SOM) whose theoretical curve agrees with experimental values even at high hydrogen concentration of over 0.5 H/M. SOM is based on the experimental fact that hydrogen atoms in metal cannot
10.1021/jp906506z 2010 American Chemical Society Published on Web 01/19/2010
Statistical-Mechanical Solubility in Metal
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Figure 1. A schematic representation of models.
come closer to each other than about 0.21 nm under hydrogen pressure of 1 MPa or less. The hydrogen solubility has been evaluated by developing a statistical-mechanical method which considers partition functions for translational motion, configuration, vibration, rotation, and spin weights. By applying SOM, we can theoretically explain the values given by Velckis and Edwards.
shown in Figure 1: (a) is the ideal case, (b) the site-blocking model, and (c) SOM. In the ideal case, one hydrogen atom occupies an interstitial site. The number of ways of assigning NH distinguishable H atoms from Ns distinguishable sites is given by the following:
2. Theory 2.1. Site-Occupying Model (SOM). A schematic representation of the ideal case, the site-blocking model, and SOM is
Ns ! NH!(Ns - NH)!
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In the ideal case, the partial configurational entropy of a solute atom is given by the following:
(
nH Sc ) -k ln ns - nH
)
(1)
where Ns and NH are replaced by ns and nH, and where ns is the total number of interstitial sites available for hydrogen atoms per metal atom, and nH is the atomic ratio of solute hydrogen atoms to metal atoms (nH ) H/M). For the site-blocking model, the blocking effect acts with hydrogen-concentration dependence, i.e., the blocking effect is stronger at higher hydrogen solubility (concentration), and is negligible at lower hydrogen solubility (concentration). The latter corresponds to the ideal case. Thus, the number of ways that interstitial sites can be available for hydrogen atoms changes depending on hydrogen solubility. At lower hydrogen solubility, the number of ways that interstitial sites can be available for hydrogen atoms approaches that of the ideal case. This means that hydrogen atoms in metal can come closer than 0.21 nm. Figure 1(b) shows an example in which one hydrogen atom blocks a neighboring unoccupied site, and the blocking effect is negligible at lower hydrogen solubility as in the ideal case. Experiments have shown, however, that hydrogen atoms in metal cannot come closer than 0.21 nm. Thus the site-blocking model with hydrogen-concentration dependence is inconsistent with the experimental fact. Figure 1(c) shows an example of SOM. All sites within a radius of 0.21 nm must be unoccupied in order to put one hydrogen atom in interstitial sites. ζ is the total number of interstitial sites within a radius of 0.21 nm in the unit cell. It is shown that two unoccupied sites (i.e., ζ ) 2) are needed to put one hydrogen atom in the interstitial sites. In SOM, it should be noted that the number of ways that interstitial sites can be available for hydrogen atoms is independent of hydrogen solubility. Hence, the total number of ways that interstitial sites can be available to solute atoms decreases from Ns to Ns/ζ. The number of ways of assigning NH distinguishable H atoms from Ns/ζ distinguishable sites can be written as follows:
( )
Ns ! ζ Ns NH! - NH ! ζ
(
)
In SOM, the partial configurational entropy of a solute is given by the following:
( )
Sc ) -k ln
nH
ns - nH ζ
(2)
where Ns and NH are replaced by ns and nH, respectively. Let us consider the total number (ζ) of interstitial sites within a radius of 0.21 nm of one hydrogen atom which was put in an interstitial site of three types of unit cell: fcc, hcp, and bcc. In other words, ζ refers to the total number of unoccupied sites (including the site in which one hydrogen atom is located) within this radius. To estimate the value of ζ, lattice constants are set to a ) b ) c ) 0.3 nm for fcc and bcc, and to a ) b ) 0.3 nm and c ) 0.4 nm for hcp. We actually counted the total number
TABLE 1: Total Number of Interstitial Sitesa,b,c,d crystal structure
fcc
hcp
bcc
site ns ζ ns/ζ
O 1 1 1
T 2 5 0.4
T 6 7.5 0.8
a O: octahedral sites, T: tetrahedral sites. b ns: total number of interstitial sites available for hydrogen atoms per metal atom. c ζ: total number of interstitial sites within a radius of 0.21 nm of one hydrogen atom. d ns/ζ: modified total number of interstitial sites available for hydrogen atoms per metal atom.
(ζ) for the three lattices. Assuming that H atoms occupy mainly O-sites in the fcc lattice and mainly T-sites in the bcc and hcp lattices, the value of ζ is about 1 for fcc, 5 for hcp, and 7.5 for bcc as summarized in Table 1. This means that the unoccupied sites ζ are necessary for putting one hydrogen atom in interstitial sites. Our model should be considered not as a kind of site-blocking model, but as a site-occupying model (SOM). 2.2. Ideal Hydrogen Solution. A statistical-mechanical description of hydrogen solubility was given by Fowler and Smithells.17 They discussed the description using the symmetry number instead of nuclear spin weights, hence the nuclear spin weights have been neglected. We give a statistical-mechanical description with the nuclear spin weights as follows. Let us consider the chemical potential of the ideal hydrogen solution in metal. Assuming Nh3/V(2πmkT)3/2 <
