Enhanced gaseous hydrogen solubility in ferritic and martensitic steels at low temperatures

(cid:1) Improvement of Sievert ' s law to consider trapping sites and stress states. (cid:1) Calculation and validation of gaseous hydrogen solubility under extreme conditions. (cid:1) Hydrogen uptake measured for L450 steel at 200 and 1000 bar. (cid:1) Calculation of the hydrogen solubility in pipes and pressure vessels. (cid:1) Comprehensive literature survey of gaseous hydrogen solubility. high pressure hydrogen detrimental failure by embrittlement. Understanding the mechanisms and driving forces of hydrogen absorption on the surface of metals for avoiding hydrogen embrittlement. In this study, the effect of stress-enhanced gaseous hydrogen uptake in bulk metals is investigated in detail. For that purpose, a generalized of Sievert ' s is derived from thermodynamic potentials considering the effect of microstructural sites and multiaxial This is parametrized veriﬁed data for which were charged under gaseous hydrogen atmosphere at pressures up to 1000 The trapping sites on the parameter identiﬁcation is critically discussed.


a b s t r a c t
Metals that are exposed to high pressure hydrogen gas may undergo detrimental failure by embrittlement. Understanding the mechanisms and driving forces of hydrogen absorption on the surface of metals is crucial for avoiding hydrogen embrittlement. In this study, the effect of stress-enhanced gaseous hydrogen uptake in bulk metals is investigated in detail.
For that purpose, a generalized form of Sievert's law is derived from thermodynamic potentials considering the effect of microstructural trapping sites and multiaxial stresses.
This new equation is parametrized and verified using experimental data for carbon steels, which were charged under gaseous hydrogen atmosphere at pressures up to 1000 bar. The role of microstructural trapping sites on the parameter identification is critically discussed.

Introduction
Green gaseous hydrogen is one of the most promising energy carriers in near future [1]. Thermal combustion of pure hydrogen gas is clean, as it produces only water. Hydrogen gas is also used in fuel cell systems converting the stored chemical energy into electricity. However, at atmospheric pressure of 1 bar and temperature of 25 C, hydrogen gas has relatively low volumetric energy density (12.7 MJ/m 3 ), especially compared to methane (40 MJ/m 3 ), which is the main constituent of natural gas [2]. In order to distribute large volumes of gaseous hydrogen in a cost-efficient way [3], compression or liquification of the gas is required [2], which increase the volumetric energy density to a practical level [4]. Hydrogen partial pressures up to 200 bar are common in industry (e.g. for storage in gas bottles), while pressures up to 1000 bar will be considered in future to increase the energy density (e.g. for compressed hydrogen tube trailers [5] or gas stations). One of the challenges for using gaseous hydrogen is establishing reliable and safe structures including pipelines and pressure vessels for transmission and storage. However, in particular high-strength steels exposed to hydrogen gas at high pressures may suffer from loss in ductility and strength, which is known as hydrogen embrittlement (HE) [6]. Different mechanisms of macroscopic brittle and time-delayed HE have been explained based on microplasticity or atomistic theories [7e9]. However, even after more than 100 years of research in this field [10] no conclusive theory is available in literature. Some authors [11] proposed a simultaneous contribution of different "classical" mechanisms, such as hydrogen-enhanced localized plasticity (HELP) and hydrogen-enhanced decohesion (HEDE). For example, HEDE possibly predominates for intergranular fracture, while HELP contributes to dimpled fracture. Other authors including Lynch [12], Nagumo [13] or Kirchheim [14] proposed new aspects of hydrogen-related failures to improve the understanding of the rate dependency [15,16] and of different occurrence of fracture surfaces, such as cleavagelike morphologies, depending on hydrogen content and stress intensity [17]. Anyway, each of the proposed HE mechanisms suppose atomic hydrogen absorption on the surface to reach the critical hydrogen content of the material [18,19]. In case of hydrogen gas, high driving forces are necessary to dissociate hydrogen molecules [20,21] and to diffuse hydrogen atoms into the bulk. Under equilibrium conditions, the analytical calculation of the hydrogen solubility c H within a given material requires the environmental hydrogen partial pressure p H2 and the temperature T of the investigated system. Sievert's law [22,23] allows calculating the equilibrium hydrogen content at high temperatures or low partial pressures [24,25]: K is the temperature-dependent equilibrium coefficient. Nevertheless, at high pressures and low temperatures "real" hydrogen gas behavior becomes crucial and the hydrogen partial pressure must be replaced by the fugacity f [25]. The entry of hydrogen into steels causes a small expansion of the lattice volume. Thus, the hydrogen absorption is affected by elastic stress fields which can be related to the increase of the hydrogen solubility [18,26,27]: c H and c H;0 are the hydrogen solubilities in the stressed and unstressed material at identical partial pressure, s 1 is the uniaxial tensile stress, V H is the partial molar volume of interstitial lattice sites and R is the universal gas constant. However, the roles of multiaxial stresses and microstructure on the hydrogen solubility of steels at low temperatures and high gas pressures need further investigations. Microstructures of high-strength steels contain high densities of trapping sites which increase the hydrogen solubility.
For that reason, the present work aims to derive a new thermodynamic relationship for gaseous hydrogen solubility from generalized chemical potentials. This new relationship considers the influence of trap densities, binding energies, temperature, fugacity and hydrostatic stresses. Parametrization was performed using experimental data. Applying a simple single-trap approach enabled the reasonable calculation of hydrogen solubility for a wide range of carbon steels at low temperatures and gas pressures up to 1000 bar. The new findings improve the basic understanding of material tests performed at low temperatures and high gas pressures on lab scale. Based on these tests the influence of hydrogencontaining gas mixtures on structural steel components and the hydrogen solubility in steel for pipeline and pressure vessels during service were studied.

Gaseous hydrogen absorption
Absorption of diatomic hydrogen gas molecules H 2 at the metal surface can be described by the reversible reaction where V L represents a vacant interstitial lattice site next to the surface and H L is a freely diffusive lattice hydrogen atom in the metal phase. V L can also be neglected in Eq. (3), if the number of interstitial lattice sites is much higher than the number of absorbed lattice hydrogen atoms. Only the interstitial lattice sites are assumed to contribute to the dynamic i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( x x x x ) x x x exchange with the environmental gas phase. A vacant interstitial lattice site is necessary, that an adsorbed hydrogen atom can jump from the surface into bulk. Once hydrogen is absorbed by the interstitial lattice, it diffuses into the metal and can be trapped at a microstructural defect site [28], which can be expressed as where V T is a vacant trapping site and H T is a trapped hydrogen atom. For simplification, it is assumed that V T is directly accessible from the crystal lattice [29], which is fulfilled for, e.g., vacancies, dislocations and carbide interfaces. Especially trapping plays a crucial role in steels [30], as it increases the local hydrogen solubility and decreases the chemical diffusion coefficient [31,32]. This will be discussed in more detail in section 2.2. As illustrated schematically in Fig. 1, a hydrogen atom must pass different energy states for being absorbed in the metal phase. In thermodynamic equilibrium, the hydrogen content absorbed in the metal phase is constant for a given temperature and hydrogen partial pressure. This hydrogen content only depends on the solution enthalpy DH s , which represents the heat release per bulk of absorbed hydrogen atom. In the present paper, the solution enthalpy is negative for an exothermic reaction (DH s < 0) and positive for an endothermic absorption (DH s > 0). According to the theory of chemical kinetics, the energy states at the surface only affect the absorption rates, but do not change the hydrogen content in thermodynamic equilibrium. This equilibrium content is named hydrogen solubility throughout the entire work. Different to the work of San Marchi et al. [25], hydrogen solubility includes both lattice and trapped hydrogen atoms and, thus, represents the hydrogen content as determined by extractive hydrogen measuring methods. For gas mixtures (e.g., hydrogen blended methane), the solubility of hydrogen can be assumed as unaffected, as long as the gas mixture does not dissociate and interact with the hydrogen molecules. If this is case, the gas mixture behaves like an ideal gas.

Hydrogen in metals
Assuming that hydrogen can diffuse freely in the crystal lattice where it is trapped at microstructural defects [33] (e.g. grain boundaries, dislocations and vacancies). Thus, the total molar hydrogen concentration c includes the lattice concentration c L and the trap concentration c T : The total hydrogen content c H in wppm can be calculated from the total molar concentration in mol/mm 3 using c H ¼ 10 6 ,Mc=r. M is the molar mass of hydrogen (1.008 g/mol) and r is the density of steel (7.910 À3 g/mm 3 ). The density of lattice sites, N L , and the density of trap sites, N T , are constant in the present work, which allows defining the lattice site fraction y L ¼ c L =N L and the trap site fraction y T ¼ c T =N T . Both site fractions, y L and y T , can take values from zero to unity depending on the solution enthalpy DH s [25] and on the binding energy E b [34]. In ferritic iron crystals, interstitial lattice hydrogen can either occupy tetrahedral or octahedral sites. According to Kholtobina et al. [29], the calculated solution enthalpy DH s of tetrahedral sites (0.22 eV or 21.2 kJ/mol) is smaller than of octahedral sites (0.37 eV or 35.7 kJ/mol). Hence, the occupation of tetrahedral sites in ferritic iron is preferred at low temperatures [9]. Nevertheless, octahedral sites are thermally activated if the temperature increases. E b of different crystal defects has been well studied in the last decades [28,31,35,36], and typical values are in the range from 0 to about 100 kJ/mol in ferritic or martensitic steels. E b depends on the type of defect, on the change of interatomic forces between hydrogen and iron at the defect sites with respect to interstitial lattice sites, and on the dilation of the crystal lattice in the immediate vicinity of crystal defects. Hydrogen in steels can be treated as ideal solution interacting with the hydrostatic stress field on the macroscale [18,37] as well as on the microscale [18,38]. Micro-residual stresses [39] are in the range of the Young's modulus E and, thus, these stresses are one magnitude larger than macrostresses. While Fig. 1 e Sketch of the hydrogen energy states in a metal-gas equilibrium. i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( x x x x ) x x x hydrostatic tension dilates the crystal lattice and increases the lattice hydrogen concentration, hydrostatic compression of the lattice decreases the lattice hydrogen concentration [40]. For instance, Hussein et al. [41] demonstrated how heterogeneous fields of microstresses contribute to hydrogen segregation at grain boundaries. In thermodynamic equilibrium, Oriani's theory applies [33,42] and the trapped hydrogen concentration can be calculated from the lattice concentration at given temperature T as =RTÞ is the inverse equilibrium constant resulting from Eq. (4) by applying the law of mass action [28]. Inserting Eq. (6) into Eq. (5) yields the total molar hydrogen concentration c as function of the lattice concentration c L [42]: According to Kirchheim [43], Eq. (7) can be simplified in practice as is to the total number of possible hydrogen positions in the microstructure per volume unit. Fig. 2 compares calculations of c by using Eq. (7) and Eq. (8). Similar results were obtained from both equations, if N T ≪N L , y L ≪ K T and K T ≪ 1 [42], which is fulfilled for binding energies less than 40 kJ/mol and for y L ¼ 1:2,10 À7 .

Hydrogen solubility
This section presents a newly established relationship between hydrogen solubility c H , microstructure, multiaxial stress state, temperature and gas pressure, which is derived based on thermodynamic expressions. In thermodynamic equilibrium, the generalized chemical potentials of environmental diatomic hydrogen molecules, m H2 , and of hydrogen atoms dissolved in ferritic iron, m L , must be equal: Considering the activity of real hydrogen gas as f ¼ p H 2 g, which is apparent at high pressures and low temperatures, the generalized chemical potential of environmental hydrogen molecules is g is the fugacity coefficient. Based on the Abel-Noble relationship, San Marchi et al. [25] derived a simple analytical expression for the fugacity coefficient, where b represents the constant finite volume of hydrogen gas molecules (van der Waals constant). Basically, all gases behave ideal at very low pressures (p≪ 1 bar) and the fugacity coefficient g, as defined in Eq. (11), needs to converge to unity. Since atomic hydrogen dissolved in ferritic iron can be treated as ideal solution, the generalized chemical potential m L [38,42,44,45] is s H is the hydrostatic stress which is 1/3 of the trace of the stress tensor s ij . The first terms m H 2 ;0 and m L;0 in Eq. (10) and Eq.
(12), respectively, are the temperature-dependent chemical potentials of pure hydrogen molecules and atoms, respectively, and the second term corresponds to the entropy of mixture. The third term results from the energy gain by hydrogen diffusing to distorted interstitial lattice sites [46]. Inserting Eqs. (10) and (12) into Eq. (9) yields which can be reformulated to The right-hand side of Eq. (14) describes the change of the solution enthalpy DH s and of the entropy DS as m L;0 À 1 Solving Eq. (15) for the lattice hydrogen concentration c L yields the following analytical expression where the pre-factor K 0 0 is constant. Combining Eq. (16) with Eq. (8) allows expressing the solubility c H of a metal in thermodynamic equilibrium as Fig. 2 e Comparing calculations of the total hydrogen concentration according to Eqs. (7) and (8).
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( x x x x ) x x x Throughout this work K 0 yK 0 0 ,gðN L ; N T ; K T Þ is the solubility coefficient and the term DH s À s H V H is the effective solution enthalpy. Eq. (17) is an analytical expression of the hydrogen solubility relating the hydrogen content of a metal to the partial pressure of hydrogen, to the temperature, to the microstructural trapping capacity of the metal and to the multiaxial stress state. Introducing the equilibrium coefficient , one can write Eq. (17) directly in a more simplified form as Obviously, if the hydrostatic stress is zero, at high temperatures and at low partial pressures, Eq. (18) becomes Sievert's law [22,23].

Mechanical problem
Multiaxial stresses s i are induced below the surface of structural components (e.g., pipes) exposed to environmental gas pressure p G . The total gas pressure is the sum of the partial pressures of different gas species. Neglecting interaction between the different molecules, the partial pressure is the pressure a single gas species would have, if it would occupy the entire volume at given temperature without the presence of any other species. In the present work, the hydrogen partial pressure p H2 is either 10% or 100% of p G . While a hydrogen partial pressure of 10% of p G may already exist in state-of-theart natural gas grids, grids for distributing pure hydrogen gas of 100% of p G will be built in near future. For relating the gas pressure to the mechanical stress state in a certain specimen or component, respectively, analytical equations were collected in Table 1 [47]. All equations assume linear elastic material behavior and thus, the effect of plastic yielding on the gaseous hydrogen solubility [15,48] was not considered in the calculations. The role of stress-enhanced hydrogen solubility is studied for three different cases: 1. Externally pressurized bulky specimens, 2. Internally pressurized thin-walled pipes (up to 200 bar) and 3. Internally pressurized thick-walled pressure vessels (up to 1000 bar).
Gas charging of bulky specimens on laboratory scale causes compressive stresses which are equal to the outer gas pressure, s i ¼ À p G , where s i is the principal stress in each of the three directions i. The compressive stresses are homogenously distributed within the specimens and they are independent from the shape and size of the specimens. The compressive hydrostatic stress s H decreases linearly with increasing gas pressure p G , as illustrated in Fig. 3. According to the basics of plasticity theory, a homogenous outer pressure cannot cause any plastic yielding and thus the equivalent stress according to von Mises s eq in the specimens is zero, as shown in Fig. 3. Anyway, the effect of the hydrostatic compressive stress is neglected in the experimental evaluation of gaseous hydrogen uptake of bulky specimens using autoclave systems. Therefore, for comparing theoretical and experimental results it is necessary to define the stress-free state of s H ¼ 0 as reference throughout this work.
Due to the cylindrical symmetry of pipes, stress components s i act in different directions, namely in radial, r, tangential, t, and axial, a, directions. In case of linear elastic material behavior, analytical expressions exist for each of the stress components s r , s t and s a [47]: r is the radius, r i is the inner radius and r a is the outer radius of the pipe. Whenever t 2r 0 ≪1, Barlow's equations [47] can be applied and Eq. (19) are simplified to where t ¼ r a À r i is the thickness of the pipe wall and r 0 ¼ ðr a Àr i Þ=2 is the average pipe radius. Barlow's equations are considered in various standards such as DIN EN 13480 and DIN EN 13445 to determine dimensions of pipelines or pressure vessels. In Fig. 4a the stress components s i acting in different directions are plotted as function of the radius r at the internal gas pressure of p G ¼ 100 bar. Realistic pipeline dimensions with the inner radius of r i ¼ 400 mm and the outer radius of r a ¼ 410 mm are applied. Hence, the wall thickness t ¼ 10 mm and the ratio t=2r 0 ¼ 0.012 are considered. According to Barlow's theory, the thin-wall approximation applies and the stress components given in Eq. (19) agree with Eq. (20). Tangential tensile stress of 400 MPa and hydrostatic stress of 200 MPa occur. The positive hydrostatic stress s H ¼ p G r=2t, which influences the stress-enhanced hydrogen solubility, is almost constant throughout the pipe wall. Fig. 4b shows the dependency of the multiaxial stress occurring in the pipe wall from internal gas pressure. According to Barlow's equations, each stress component increases linearly with increasing p G . Basic design criteria require that the equivalent stress s eq is smaller than the yield strength R p0:2 to avoid any plastic deformation. In Fig. 4b s eq according to von Mises increases linearly as function of p G and reaches 400 MPa and 700 MPa at gas pressures of 100 bar and 200 bar, respectively. s eq ¼ 700 MPa exceeds R p0:2 of commercial carbon steels. s H reaches 200 MPa at 100 bar and 400 MPa at 200 bar. Note that s H ¼ s t =2. According to Eq. (19), increasing the wall thickness t decreases the stresses at the given gas pressure of p G ¼ 100 bar and for the given inner radius of r i ¼ 200 mm, as shown in Fig. 5a. Tangential stress and axial stress are tensile throughout the wall and they decrease steadily with increasing wall thickness. The radial stress of À10 MPa is negative (compression stress) and does not change with increasing wall thickness. Thus, the hydrostatic stress decreases significantly from 99 MPa to 5 MPa if the wall thickness increases from 10 mm to 200 mm.
For thick-walled pressure vessels with the ratio of r a =r i ¼ 2, the tangential stress at the inner surface radius is s t ¼ 5p G =3, as indicated in Fig. 5a. At higher pressures the radial stress cannot be neglected, and it is equal to the negative gas pressure at the inner radius, s r ¼ À p G . The radial stress decreases linearly through the wall and becomes zero at the outer radius r a . As shown in Fig. 5b, the stresses in thick-walled pressure vessels increase linearly with increasing inner gas pressure.

Experimental procedure
Cylindrical specimens with diameter of 3 mm and height of 15 mm were machined out of a L450 pipe steel. Before the specimens were placed inside of an autoclave, their surfaces were grinded, polished and rinsed with acetone. Highpressure charging was applied at 200 bar and 80 C as well as at 1000 bar and 50 C. The duration of hydrogen gas charging was 3 weeks. When the specimens were removed from the autoclave, they were stored directly inside of a liquid nitrogen bath to suppress any hydrogen diffusion. For the quantification of the hydrogen content, a Bruker G8 Galileo coupled with an infrared furnace and with a mass spectrometer was used, as described in Ref. [49]. The specimens were Table 1 e Multiaxial stress states in bulk specimens exposed to an environmental gas pressure p G and cylindrical stress components [47].

No.
Short description Stress state Schematic of the problem Ref.
Bulky specimen (Reference state) Externally pressurized bulky specimen Internally pressurized thin-walled pipe (Barlow's formula for pipes with t≪r 0 ) Internally pressurized thick-walled vessel with r a r i ¼ 2 i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( x x x x ) x x x thawed in acetone for 20 s before there were dried using inert nitrogen gas. Afterwards the samples were directly put into the infrared furnace and they were immediately heated up to 400 C.

Parameter identification
For calculating the hydrogen solubility c H of a given material as function of total gas pressure p G , temperature T and applied hydrostatic stress s H , a set of four constant parameters needs to be identified, namely.
1. The solubility coefficient K 0 , 2. The solution enthalpy DH s , 3. The partial molar volume V H , 4. And the van der Waals constant b.
As shown in Fig. 6a, K 0 was measured for different materials [50e53]. Smaller values ranging from 1.7 to 33 wppm/ bar 0.5 were determined by different authors [50e52] using a permeation setup at temperatures above 100 C, which allows recording the transient increase of the pressure as function of time. Disc-shaped specimens were used to separate a pressurized charging cell from an evacuated measurement cell equipped with a mass spectrometer. The applied pressure was very low, but it was even increased to 30 bar. At temperatures above 100 C hydrogen trapping in bulky ferrite is de facto thermally suppressed, and lattice-controlled diffusion can be assumed [55]. Once all trapping sites are thermally activated, the equilibrium coefficient K ¼ F=D eff , where F is the permeability coefficient and D eff is the effective diffusivity. However, at lower temperature hydrogen diffusion is bulk-controlled [55,56] rather than lattice-controlled, and trapping sites in the microstructure retard long-range chemical diffusion [16,31,55,57]. Since the chemical diffusivity D chem is not independent from the pressure [16,58], the equilibrium coefficient will be significantly affected by the specific method of determination. More recent approaches enable measuring the solubility coefficient directly by reliable autoclave charging techniques at low temperatures combined with subsequent thermal desorption analysis (TDA) [53,54,59e61] without any restriction to lattice-controlled diffusion. However, this technique has some crucial limitations, such as increase of diffusion coefficients at elevated temperatures and relatively long time needed for opening the pressurized autoclaves. Especially at higher temperatures the hydrogen content in bulky specimens can easily follow the pressure drop during evacuation, which may lead to an underestimation of the hydrogen solubility. A solubility coefficient of about 300 wppmbar À0.5 was evaluated from the measured data at 80 C provided by Trautmann et al. [53,54]. They used autoclaves to charge cuboid specimens with pure hydrogen gas before measuring their hydrogen contents using TDA at 950 C. The difference in the solubility coefficient K 0 by using the two different methods may be due to the difference of the applied temperatures. In Fig. 6b calculated solubility coefficients according to Eq. (17) were plotted as function of temperature, binding energy and trap density. Trap densities between 10 À9 and 10 À7 mol/mm 3 Fig. 3 e Multiaxial stress state in bulky specimens exposed to an external gas pressure on laboratory scale. Fig. 4 e Multiaxial stresses in a thin-walled pipe a) as function of the radius at given gas pressure of 100 bar and b) as function of internal gas pressure for the inner surface radius r i ¼ 400 mm and the wall thickness t ¼ 10 mm. i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( x x x x ) x x x represent hydrogen trapping in cold rolled ferritic iron [31] or in dual phase steel [16]. The higher the binding energy and the higher the trap density, the higher is the hydrogen solubility coefficient which is in the range of 10 2 -10 4 wppm/bar 0.5 at ambient temperature. With increasing temperatures trapping sites get thermally activated and the solubility coefficient decreases steadily. Once all trapping sites are fully activated, diffusion in the specimens is lattice-controlled and the solubility coefficient reaches a minimum. In the present work the minimum was 10 wppm/bar 0.5 , which is in good agreement with the results of permeation measurements [51,52]. Nevertheless, further experimental studies are necessary to clarify the role of trapping sites on the equilibrium coefficient. A combination of both techniques would be beneficial for compensating the limitations caused by high temperatures and high gas pressures.
As shown in Fig. 7a, the solution enthalpy DH s of ferritic iron was well studied in literature [50e52, 62,64] and a value of 27 kJ/mol was chosen in the present work. This value agrees with data published by Nelson and Stein [52], who measured the solution enthalpy of different heat-treated 4130 steels. Only Gadgeel and Johnson [51] found even lower values of 19e25 kJ/mol for different carbon steels. Although Trautmann et al. [53] published hydrogen solubilities of carbon steels only for 25 C and 80 C, a rough estimation using Eq. (1) gives solution enthalpies less than 15 kJ/mol (red points in Fig. 7b). Depending on the temperature range and thus on the measurement technique the determined solution enthalpies can differ significantly. Therefore, the temperature dependency of the hydrogen solubility according to Eq. (17) was plotted in Fig. 7b. While the solution enthalpy is typically measured in the lattice diffusion-controlled temperature regime (» 100 C) to be around 27 kJ/mol, at low temperatures the thermal activation of trapping sites at low temperatures can affect the evaluation significantly. DH s decreases with decreasing temperature and can also change from negative (endothermic absorption) to positive at temperatures below zero. In other words, the gaseous hydrogen solubility can increase with decreasing temperature below a critical temperature of around À30 C. Since this behavior is rather attributed to the increase of trapped hydrogen concentration c T than to the change of DH s , it is named "quasi-exothermic" absorption in the present work. Further measurements for validating the increase of the hydrogen solubility with decreasing temperature are necessary in the low temperature regime.
Dissolved hydrogen causes a lattice expansion in ferritic steels [46], and the partial molar volume V H can directly be obtained from dilatation measurements. Fig. 8a shows  i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( x x x x ) x x x measured values of V H of iron and steels in the range of 1220e3000 mm 3 /mol published in literature [26,63,65e67]. This corresponds approximately to the dilation of Dlz 0.1 nm for each hydrogen atom. According to Wipf [30] or Nagumo [9] the dilation per atom is similar for many metals and seems to be insensitive to temperature and microstructure [9,26,30]. Wagenblast and Wriedt determined V H ¼ 2000 mm 3 /mol [65], and they measured the dilation of iron wires at hydrogen partial pressures of about 0.02e0.97 bar and at high temperatures of 450e800 C. For low-temperature and high-pressure applications the partial pressure p H2 of the ideal gas needs to be replaced by the fugacity f , as given in Eq. (10). Numerous models and empirical relationships accounting for the real behavior of hydrogen gas by considering intermolecular van der Waals forces and the finite volume of hydrogen molecules exist [25,69e72]. The fugacity calculated using some of these models are plotted in Fig. 8b. Since f zp H2 for pressures up to 150 bar and at ambient temperatures, the ideal gas behavior can be assumed with reasonable accuracy. Extrapolating the fugacity up to 1000 bar shows large difference from the ideal gas behavior. For example, San Marchi et al. [25] fitted the van der Waals constant b to hydrogen data provided by Michels et al. [73] from ambient temperature to 150 C and for pressures up to 3000 bar, and they confirmed that bz 15.8410 3 mm 3 /mol. Thus, f increases exponentially to 1984 bar, whereas p H2 increases linearly to only 1000 bar (f z2p H2 ). Krom and Bakker [69] suggested another relationship for describing the real behavior of hydrogen gas by calculating f . As shown in Fig. 8b, the obtained curves were quite similar. The constant, pressure-and temperature-independent parameters used for calculating the stress-enhanced hydrogen solubility in pressurized specimens, pipes and pressure vessels are summarized in Table 2. Since these parameters were not fitted to any specific material or hydrogen solubility measurement, they can be considered as rough estimation for ferritic and martensitic steels. The temperature dependency of the solubility coefficient is approximated by a single trap approach according to Eq. (9) with the binding energy of 30 kJ/ mol and the trap density of 10 À7 mol/mm 3 . This corresponds to the trapping capacity of highly deformed ferritic iron [31] or of ferritic-martensitic steels [15,16]. The solubility coefficient K 0 ¼ K 0 0 in the lattice diffusion-controlled regime at high temperatures was assumed as constant and corresponded to measurements of Quick and Johnson [50] for pure iron.

Externally pressurized bulky specimens
For characterizing the hydrogen solubility of materials, often cylindrical or cuboidal specimens are charged with gaseous hydrogen under compressive conditions. Lack of knowledge exists about the hydrogen solubility of ferritic or martensitic steels at low temperatures and high hydrogen gas pressures. Therefore, the present work compares analytical predictions with measured hydrogen solubilities of ferritic-pearlitic L450 steel and other commercial carbon steels published by Trautmann et al. [53]. As shown in Fig. 9, the analytical predictions are in reasonable agreement with the measurements even at total gas pressures of 1000 bar. The difference of calculated and measured hydrogen solubilities is about ±0.1 wppm at 25 C and 50 C, as shown in Fig. 9a and b, respectively, and thus comparable with the accuracy of state-of-theart hydrogen extraction techniques. As shown in Fig. 9c, the calculations underestimate the solubility at 80 C. This is most likely related to the temperature dependency of the solubility coefficient rather than to the measured solution enthalpy. Hydrogen trapping in commercial steels is very complex, especially in the presence of carbides [28,35], and requires a multiple trap approach rather than an effective single trap approach ( Table 2) to capture the temperature dependency of the solubility coefficient more accurately. Temperature-  Fig. 9 e Hydrogen solubility of externally pressurized bulky specimens as function of total gas pressure and temperature: a) 25 C, b) 50 C and c) 80 C [53]. Kissinger's theory [55,56] and/or (ii) recorded thermal desorption spectra measured after electrochemical charging [28,35,55]. As shown in Fig. 9a, the hydrogen solubility increases with the square root of the external total gas pressure at 25 C and strongly depends on the hydrogen partial pressure. While the hydrogen content of 10% in a hydrogen blended gas causes a hydrogen solubility of 0.13 wppm at 1000 bar, pure hydrogen gas increases the hydrogen solubility to 0.55 wppm. Nevertheless, if the total gas pressure is limited to 200 bar the hydrogen solubility of bulky specimens stays below 0.2 wppm, as summarized in Table 3. The present calculations also consider the effect of compressive stresses acting on the specimens by the outer total gas pressure. Compressive stresses reduce the hydrogen solubility by increasing the effective solution enthalpy DH s À s H V H . However, as shown in Fig. 9 this effect is negligible even at gas pressures of 1000 bar, as the difference in the hydrogen solubility (compared with the reference without considering the stress effect) is less than 0.05 wppm. This is less than the detection limit of most hydrogen measuring devices.

Hydrogen solubility of internally pressurized pipes and vessels
The hydrogen solubility c H of internally pressurized pipelines or pressure vessels can differ from the laboratory test results by the effect of stress-enhanced hydrogen uptake. The internally applied total gas pressure causes tensile stresses at the inner surface radius r i of the pipes which decrease the effective solution enthalpy DH s À s H V H . The magnitude of the hydrostatic stress s H depends only on the dimensions of the pipe and of the internal total gas pressure, as described in section 2.4. Note that macroscopic residual stresses resulting from, e.g., thermomechanical treatments [76], are not considered in the present work. In Fig. 10a the hydrogen solubility in thinwalled pipes is plotted as function of the total gas pressure. In general, the hydrogen solubility in pipes increases with the total gas pressure and it also increases with the hydrogen content in the gas mixture. For both conditions, namely hydrogen blended gas and pure hydrogen gas, the hydrogen solubility in the pipes shows a slight underestimation compared to the experimentally hydrogen solubility, which is determined under hydrostatic compression. Therefore, the experimental hydrogen solubility is indicated as "autoclave" in Fig. 10. The difference increases with the total gas pressure. The maximum difference was observed for pure hydrogen gas. As summarized in Table 3, the hydrogen solubility increased by 0.01 wppm at the gas pressure of 70 bar, and by 0.08 wppm at 200 bar compared to the externally pressurized bulky specimens. In Fig. 10b, the hydrogen solubility of thickwalled pressure vessels is illustrated as function of the total gas pressure, and characteristic values are summarized in Table 3. They range from 0.12 wppm at 70 bar to 0.61 wppm at 1000 bar. Due to the higher radius ratio of r a =r i ¼ 2, the hydrostatic stress caused by the internal total gas pressure was much smaller than in thin-walled pipes. Hence, the effect of stress-enhanced hydrogen uptake in thick-walled pressure vessels was only minor with less than 0.06 wppm even at highest pure gas pressures of 1000 bar. Pipelines or pressure vessels enable distribution of hydrogen gas over long distances. Pipelines usually operate at low pressures with a maximum operating pressure (MOP) of 70 bar. According to the present calculations, this corresponds to a hydrogen solubility of around 0.1 wppm. Increasing the gas pressure in future gas grids could be necessary for compensating the lower volumetric energy density of hydrogen gas compared to natural gas. Therefore, the calculated hydrogen solubility in pipeline materials was plotted in Fig. 10a up to the gas  pressure of 200 bar. Even higher pressures can be applied in pressure vessels to increase the amount of stored hydrogen gas, which also increases the hydrogen solubility, as shown in Fig. 10b. Based on the present calculations, the hydrogen contents absorbed during service of pipelines and pressure vessels do not seem to be negligible. However, the contents are acceptable with respect to HE, if the mechanical strength of the pipe materials is low. This would limit stress-enhanced hydrogen uptake and increase the HE resistivity. Although results of the laboratory tests underestimate slightly the hydrogen solubility in pipelines and pressure vessels, the effect of stress-enhanced hydrogen solubility is negligible if the surfaces are smooth. This is also in agreement with the predictions of Wipf [30], who argued that the gain in solution energy is negligible in most calculations. However, hydrostatics stresses are strongly intensified at micro-notches or at crack tips. There the hydrostatic stresses can reach several Gigapascal, which needs further detailed investigations [38].

Conclusions
This work calculated and measured the hydrogen solubility in ferritic steel components at high gas pressures up to 1000 bar and at temperatures from À100 to 900 C. The following conclusions can be drawn from the present work: a new thermodynamic relationship was proposed, which extends Sievert's law to low temperatures and high gas pressures considering microstructural trapping sites, fugacity and multiaxial stress states as following This equation was parametrized using a set of four constant parameters. Two of them, namely the solution enthalpy and the partial molar volume of hydrogen in steels, are similar for iron and many steels and independent of temperature and microstructure. Challenges in the parametrization of Sievert's law at low temperatures and high gas pressures were identified using the new equation. In particular, the solution enthalpy should be determined at high temperatures, while the solubility coefficient needs to be measured over a wide range of temperature using a combination of ex-situ and in-situ techniques. The parametrized equation was validated using measured hydrogen contents at 200 and 1000 bar. The accuracy of the calculated hydrogen contents was in the range of ±0.1 wppm. Measured and calculated hydrogen contents in pipes and pressure vessels up to 0.3 wppm at 200 bar and 0.6 wppm at 1000 bar are not negligible and need careful examination with respect to hydrogen embrittlement.
Future work should focus on the stress-enhanced hydrogen solubility at micro-notches and on the role of compressive residual stresses as mitigation strategy against hydrogen embrittlement.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. r e f e r e n c e s