Hydrogen is ubiquitous and can be found in all commercial metals. Even small amounts of hydrogen can lead to embrittlement, loss of ductility, and premature failure of metal components in service. Thus, techniques to measure low solubilities of hydrogen in metals are desirable.

If unimpeded hydrogen will move within, into, and out of metals. The driving force for hydrogen movement is the negative gradient of the chemical potential. In this study, hydrogen gas will be produced at high pressure (fugacity), between 60 and 280 MPa (70 and 600 MPa), in situ from the confined thermal decomposition of zirconium hydride powder, ZrH2, distributed on the surface of a probe made of zirconium metal. The hydrogen in the gas will move into the metal and the probe until the zirconium hydride powder is depleted or equilibrium is reached. In both instances, the chemical potential gradients are zero and there is no further net movement of hydrogen:

  • The chemical potential gradients are zero when the hydride powder is depleted because the means of transporting hydrogen between the metal and the probe is lost when the production of hydrogen gas ceases; the chemical potentials could be different, but there are no gradients if there is no means of transport.

  • Chemical potential gradients at equilibrium are zero for hydrogen as atoms in solid solutions, in the gas phase as H2, and as metal hydrides if these forms of hydrogen can interact in a mixture.

The probe introduced in this study exploits these features of chemical potentials to determine the solubility of atomic hydrogen in metal in equilibrium with hydrogen gas: the Sieverts solubility. At equilibrium, the chemical potentials of hydrogen gas and atomic hydrogen in solid solution are equal. Adding a probe introduces more chemical potentials to consider, but the probe does not affect the metal-hydrogen equilibrium. The chemical potentials of hydrogen gas and atomic hydrogen in the metal will simultaneously have to equal the chemical potentials of all interacting forms of hydrogen in the probe and in any residual zirconium hydride powder. Whether hydrogen movement stops because of depletion of the zirconium hydride or because of equilibrium, the hydrogen that partitions to the probe provides a proxy measurement of the solubility of hydrogen in the metal in equilibrium with hydrogen gas by mass balance.

In this study the solubility of hydrogen in copper is examined. Copper has been proposed for use as the outer shell of containers for spent nuclear fuel. The effects of hydrogen on copper are being studied to identify diffusion-controlled degradation mechanisms that might take place during storage over thousands of years.

The Sieverts solubility, S, of hydrogen in copper at equilibrium with hydrogen gas has been described by an Arrhenius expression determined from experimentally measured values that span 2 orders of magnitude between 1000 to 400 °C[1,2,3,4,5,6,7] and values calculated from permeability data and optimized thermodynamic parameters from a CALPHAD study[7]:

$$ S = 2600\exp \left( { - \frac{48550}{RT}} \right){\text{ [mol H}}_{2} /{\text{m}}^{3} \sqrt {\text{MPa}} ] $$

where R is 8.314 J/mol K. The solubility of hydrogen in copper is low and there are few direct measurements below 500 °C and no data below 400 °C,[7] so mass-transport calculations that require solubilities at temperatures relevant to storage of spent fuel (< 200 °C) require extrapolation of Eq 1. The effectiveness of this extrapolation is challenged by the apparent incompatibility of Eq 1 with a similar equation determined from diffusivity and hydrogen-gas permeation studies.

Diffusivities for hydrogen in copper spanning 6 orders of magnitude have been measured between 1000 °C and room temperature using electrochemical permeation and outgassing techniques.[1,8,9,10,11,12] The diffusivities were fitted to an Arrhenius expression[7]:

$$ D = 1.74 \times 10^{ - 6} \exp \left( { - \frac{42000}{RT}} \right){\text{ [m}}^{2} /{\text{s}}]. $$

Hydrogen-gas permeation studies in copper have been reported spanning 10 orders of magnitude between 700 °C and room temperature.[9,11,12,13,14,15] In these studies, high-pressure hydrogen gas is on one surface of a thin foil of copper, and low pressure is on the other surface. Hydrogen gas dissociates at the metal surface on the high-pressure side into atomic hydrogen that moves into solid solution in the metal:

$$ \frac{1}{2}H_{2} \rightleftharpoons H_{\text{solid\,solution}}. $$

The activity coefficient for hydrogen in the metal is assumed to be unity because the concentrations are low (< 3 ppm by mass in copper, wppm). The concentration of hydrogen at the metal surface is related to the fugacity of hydrogen gas, \( f_{{{\text{H}}_{ 2} }} , \) by a modified Sieverts equation[16]:

$$ c_{\text{H\,in\,Cu}} = S\left( {\frac{{2M_{\text{H}} }}{{\rho_{\text{Cu}} }}} \right)\sqrt {f_{{{\text{H}}_{2} }} } {\text{ [wppm]}} $$

where MH is the molar mass of hydrogen, \( \rho_{\text{Cu}}\) is the density of copper. The flux of atoms through the metal is given by an approximation to Fick’s Law for thin foils with thickness \( \delta x \):

$$ {\text{J}} = - D\frac{\Delta c}{\delta x} $$

where Δc is the difference in the concentrations of hydrogen in solution at the two surfaces of the foil.

Substituting Eq 4 into Eq 5 yields the equation for the permeation flux through a thin foil:

$$ {\text{J}} = - DS\left( {\frac{{2M_{\text{H}}}}{{\rho_{\text{Cu}}}}} \right)\frac{{\sqrt {f_{{{\text{H}}_{2} }} } }}{\delta x} $$

where the assumption has been made that the fugacity of hydrogen on the low-pressure side of the foil is zero and the hydrogen that moves through the metal is replenished from the gas at steady state. The permeability constant, P, determined from experiments is defined:

$$ P \equiv DS. $$

Experimental permeabilities were fitted to an Arrhenius expression[7]:

$$ P = 1.4 \times 10^{ - 3} \exp \left( { - \frac{84320}{RT}} \right){\text{ [mol H}}_{2} /{\text{m s }}\sqrt {\text{MPa}} ]. $$

The solubility can be determined from Eq 7 and the permeability and diffusivity relations given by Eqs 2 and 8:

$$ S_{\text{H\,in\,Cu}} \equiv P/D = 812 \cdot \exp \left( { - \frac{42300}{RT}} \right){\text{ [mol H}}_{2} /{\text{m}}^{3} \sqrt {\text{MPa}} ]. $$

Comparing the values from Eq 9 with those from Eq 1 demonstrates the inconsistency of these Arrhenius equations for S, D and P: they do not satisfy the defining relationship P = DS (Eq 7). It is commonly believed[7] that this discrepancy can be attributed to the solubility because of trapping of hydrogen at crystalline defects such as vacancies, dislocations, grain boundaries, porosity, but these defects would affect values of diffusivity and permeation, not just solubility. The equation for solubility is the most likely of S, D and P to be in error because it was partially calculated from ab initio models that do not include crystalline defects, whereas the equations for diffusivity and permeation were determined solely from experimental data. Experimentally determined solubilities will be higher in metals containing large numbers of defects because more hydrogen will partition to the defects. Ab initio models that do not include partitioning to defects will always underpredict experimental solubilities.

In this study we introduce an indirect method to determine hydrogen solubilities in copper between 350 and 450 °C that will support empirical extrapolations to temperatures more relevant to nuclear spent fuel storage (< 200 °C). The method based on chemical potential uses a zirconium metal probe, in situ generation of molecular hydrogen, and high-pressure partitioning between molecular hydrogen and atomic hydrogen in copper and zirconium at various temperatures. A Sieverts expression written in Arrhenius form will be determined from the solubilities measured in this study and compared with Eqs 1 and 9.


Test assemblies were composed of a zirconium (Zircaloy-2) disk covered with zirconium hydride powder (ZrH2, Sigma-Aldrich) between two copper plates. Two grades of copper were used: 110 (electrolytic tough pitch, 99.90 Cu min/0.05 Oxygen Max); and 101 (oxygen-free high-conductivity (OFHC), 99.99% Cu min). A set of test assemblies are represented schematically in Fig. 1.

Fig. 1
figure 1

An exploded view of the test assembly

Test assemblies were placed between cylinders of M42 steel, compressed using an MTS 810 load frame, and heated within an electric furnace to a setpoint between 350 and 450 °C for 20–140 h while under compression, which was sufficient to dissociate the ZrH2 powder used in this study into zirconium metal and presumably hydrogen gas. Following heat treatment, the load was removed, and the assemblies were water quenched and the zirconium disks recovered and analysed for hydrogen concentration with a Netzsch Pegusus 404C Differential Scanning Calorimeter (DSC). Three scans of the endothermic heat flow relative to a disk with 14 wppm of hydrogen were recorded between 100 and 550 °C at 10 °C/min. The average of the temperatures where the maximum slope occurred for the endothermic heat flow for the second and third scans was used to calculate the corresponding hydrogen concentration in the disk using the calibration correlation[17]:

$$ c_{\text{H in Zr}} = 106446.7\exp \left({-\frac{4328.67}{{T_{\text{max-slope}} }}} \right)[{\text{wppm}}]. $$

The hydrogen concentrations determined with DSC and Eq 10 are expected to be reliable to 5% when calibrated with Hot Vacuum Extraction Mass Spectroscopy (HVEMS).[18] The sensitivity of the current method relies on these established techniques developed by the nuclear power industry used to measure concentrations of hydrogen in zirconium alloys. Masses were measured with an Acculab ALC-210.4 (± 0.0001 g).


The chemical potential probe is a disk of zirconium that is placed alongside the metal; copper is the metal in this study. A test assembly was composed of a zirconium disk, and zirconium hydride (ZrH2) powder spread on the disk, both situated between two copper sheets that were pressed together with platens attached to a load frame to form a leak-tight seal. When the temperature was raised by the oven enclosing the assembly, the zirconium hydride decomposed forming hydrogen gas. The hydrogen gas remained within the assembly because of the seal and oxides that formed on the exterior metal surfaces. Under the reducing conditions within the sealed region, the oxides on the internal metal surfaces of the test assembly apparently failed, most likely first at the bottoms of cracks and pores, and were not replenished when they dissolved into the metals or maybe reacted with hydrogen forming water that later reacted with the metals. Without barrier oxides, hydrogen gas dissociates into atomic hydrogen on the metallic surfaces. Ingress of hydrogen into the metals occurs via Eq 3 and hydrogen partitions between the zirconium and copper.

The test assemblies were heated at the test temperature to equilibrium where the chemical potentials are the same for hydrogen in all forms, i.e., molecular hydrogen within the sealed region and atomic hydrogen in the metals. The assembly was then quenched to temperatures where hydrogen diffusion is negligible during the time to remove the probe. During removal, an oxide rapidly formed on the probe that captured the hydrogen partitioned to it during the test. The concentration of hydrogen in the probe provides a record of the high-temperature state of depletion or equilibrium that can be measured using standard techniques to quantify hydrogen in metals—in this study DSC was used.

Sieverts solubility constants for hydrogen in copper were determined from regressions to hydrogen masses inferred from mass-balance equations that account for partitioning of hydrogen from the thermal decomposition of ZrH2 into the components of the test assemblies: copper and zirconium. Before the load and the temperature were applied, hydrogen was mainly in the zirconium hydride powder, but also small amounts were in the zirconium metal and the copper. The zirconium metal used for these test assemblies was Zircaloy-2 with (14 ± 1) wppm of initial hydrogen determined with HVEMS (Experimental Section). The initial concentration of hydrogen in the as-received copper at room temperature and pressure for these tests was neglected based on calculated values of 10−6 wppm from determined Sieverts relations. The concentration of hydrogen in the zirconium can be determined from conservation of mass:

$$c_{\text{H\,in\,Zr}}^{\text{final}} = \frac{{m_{\text{H\,in\,Zr}}^{\text{initial}} + F \cdot m_{{{\text{ZrH}}_{2}}}^{{}} - m_{{{\text{H\,in\,ZrH}}_{\text{X}}}}^{\text{final}} - m_{\text{H\,in\,Cu}}^{\text{final}} }}{{m_{\text{Zr}}}},$$

where F is 0.0216, the mass fraction of hydrogen in ZrH2. In Eq 11, m is the mass of the material named in the subscript: initial and final refers to before and after the experiment. ZrHx is the residual zirconium hydride powder remaining after the process, where \( 0 \le {\text{x}} < 2 \). In this study, the hydride powder completely dissociates into hydrogen gas and zirconium metal. The final mass of hydrogen in the hydride powder is zero, i.e., \( m_{{{\text{ZrH}}_{\text{X}} }}^{\text{final}} = 0, \) because \( {\text{x}} = 0, \) and the final mass of zirconium, \( m_{\text{Zr}} , \) is the mass of the zirconium disk plus the mass of zirconium metal powder created from the complete dissociation of the hydride powder: \( m_{\text{Zr}} = [m_{\text{Zr}}^{\text{initial}} + (1 - F) \cdot m_{{{\text{ZrH}}_{2}}}^{{}} ] \).

The final mass of hydrogen in copper is related to Sieverts’ equation:

$$ m_{\text{H\,in\,Cu}}^{\text{final}} = m_{\text{Cu}} \left[ {S_{\text{H\,in\,Cu}} \exp \left( { - \frac{{Q_{\text{s}} }}{RT}} \right)} \right]\left( {\frac{{2M_{\text{H}} }}{{\rho_{\text{Cu}} }}} \right)\sqrt {f_{{{\text{H}}_{ 2} }} }.$$

The Sieverts parameters, SH in Cu, and QS, were determined from χ2 regression of the final concentrations of hydrogen in zirconium experimentally determined with DSC, as described in the experimental section, to values predicted by Eq 11 with Eq 12. The fugacity was calculated from pressure using an empiricalFootnote 1 relation that agrees with the tabulated results in.[19] Pressures were equal to the force from the load frame divided by the surface area of the zirconium disks; values ranged from 60 to 280 MPa, Table 1. In the regressions, the squared deviation of a measurement from its calculated value was weighted by the inverse of the sum of the squares of the error in the determined concentration of hydrogen in zirconium and the corresponding contributions from the errors in the mass measurements in Eq 11.

Table 1 Experimental data

Results and Discussion

Twenty-one test assemblies were used to determine the solubility of hydrogen in copper using the regression method described in the previous section:

$$ S_{\text{H\,in\,Cu}} = (1000 \pm 200)\exp \left( { - \frac{43000 \pm 1300}{RT}} \right){\text{ [mol H}}_{2} /{\text{m}}^{3} \sqrt {\text{MPa}} ]. $$

The χ2 per degree of freedom for the regression was 1.7. Following ± in Eq 13 is one standard error, which was determined for each parameter by changing the χ2 per degree of freedom by 1. The data are presented in Table 1. The result of this study shown by Eq 13 compares well with Eq 9 determined from experimentally derived equations for P and D: the determined parameters are within one standard error. Equation 13 does not compare as well with Eq 1.[7]

Figure 2 shows observed concentrations of hydrogen in the zirconium metal and values predicted from the regression to Eqs 11 and 12. Figure 2(b) shows the results of another regression where the mass-balance equation was rearranged so that the dependent variable was the concentration of hydrogen in copper. In this regression, Eq 12 was fitted to solubilities inferred from the mass balance. Both regressions produced the same Sieverts parameters. Figure 2(b) shows the inferred concentrations of hydrogen in copper at these temperatures and pressures are small, a few wppm, compared with the measured concentrations of hydrogen in zirconium shown in Fig. 2(a).

Fig. 2
figure 2

Observed concentrations of hydrogen in zirconium were determined from DSC endothermic maximum-slope temperatures and Eq 10. Predicted concentrations of hydrogen in zirconium were determined from regression to the mass balance Eq 11 to determine the Sieverts parameters in Eq 12. Concentrations of hydrogen in copper were predicted with the determined Sieverts equation and compared with concentrations inferred from mass balance

Figure 3 shows extrapolations of several reported solubility equations for copper derived from measurements made at higher temperatures. The discrepancy between Eq 1 suggested by Magnusson and Frisk and Eq 13 determined in this study is difficult to see given the scales in Fig. 3. Figure 4 shows the relative deviation of the solubility determined in this study from the defining relation, Eq 7:

$$ \Delta = \frac{S - P/D}{P/D} \times 100\%. $$
Fig. 3
figure 3

Predicted solubilities of hydrogen in copper determined from extrapolations of equations determined from high-temperature measurements. The regression line of this study (Eq 13) is shown in red; it is within error of the solubility determined by permeability (Eq 8) divided by diffusivity (Eq 2) as described by Eq 9

Fig. 4
figure 4

The deviation given by Eq 13 for hydrogen solubility predictions and observations relative to the defining relation S = P/D (Eq 7). The yellow band bounded by the dashed lines shows one standard error from the Sieverts relation determined in this study (Eq 13) and is consistent with the ratio of permeability and diffusivity indicated by 0%

The agreement of the current Sieverts solubility equation with the equation inferred from permeation and diffusivity studies over 10 and 6 orders of magnitude is remarkable. The current Sieverts solubility equation was determined from experiments performed between 350 and 450 °C. Further experiments at lower temperatures would help to show if this agreement persists to temperatures relevant to copper-clad containers of spent nuclear fuel (< 200 °C).

The current results suggest that the reason S, D and P did not satisfy the defining relation given by Eq 7, P = SD, is because the formula for solubility given by Eq 1 was incorrect. The apparent failure to satisfy the defining relationship prompted speculation about diffusible hydrogen and hydrogen trapped in crystalline imperfections such as dislocations, grain boundaries and cavities interfering with hydrogen movement. These speculations might no longer be required now that the experimentally determined relation for solubility has been revised with Eq 13 and the defining relation satisfied (Eq 7). These results also suggest that revisiting the assumptions of the CALPHAD model might prove beneficial.[7]


A chemical potential probe has been developed to measure the solubility of hydrogen in metals. The method is based on mass balance and high-pressure hydrogen produced in situ partitioning between a zirconium probe and the metal, which was copper in this study. The sensitivity of the method was demonstrated between 350 and 450 °C where solubilities of hydrogen in copper are too low to measure with conventional techniques. The current results can reduce the uncertainty of extrapolations used to estimate the service life of copper-clad spent fuel containers. Solubilities were fitted to an Arrhenius relation:

$$ S_{\text{H\,in\,Cu}} = (1000 \pm 200)\exp \left( { - \frac{43000 \pm 1300}{RT}} \right){\text{ [mol H}}_{2} /{\text{m}}^{3} \sqrt {\text{MPa}} ]. $$

This relation was determined from solubilities that varied by less than a factor of 10, yet it agrees with solubilities determined from the ratio of permeability and diffusivity relations determined from data spanning 10, and 6, orders of magnitude between room temperature and 700, and 1000 °C, respectively. Previously, these relationships for solubility, permeability and diffusivity were not mutually consistent. This agreement provides confidence that these relationships can be applied in diffusion-controlled models of degradation mechanisms following hydrogen ingress in copper containers intended for long-term storage of spent nuclear fuel.