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Junichiro Moriyama, Osamu Takakuwa, Masatake Yamaguchi, [Yuhei Ogawa](https://orcid.org/0000-0003-2713-9822), [Kaneaki Tsuzaki](https://orcid.org/0000-0003-2400-7605)

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[The contribution of Cr and Ni to hydrogen absorption energy in Fe-Cr-Ni austenitic systems: A first-principles study](https://mdr.nims.go.jp/datasets/761a3e84-7a12-49d9-a43a-c71991372821)

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1   1  2 The contribution of Cr and Ni to hydrogen absorption energy in Fe-Cr-Ni 3 austenitic systems: A first-principles study 4  5  6 Junichiro Moriyamaa, Osamu Takakuwab,c, Masatake Yamaguchid,e,f,  7 Yuhei Ogawag, Kaneaki Tsuzakic,f,g,h 8  9 a Graduate School of Mechanical Engineering, Kyushu University, 10 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan 11  12 b Department of Mechanical Engineering, Kyushu University, 13 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan 14  15 c Research Center for Hydrogen Industrial Use and Storage (HYDROGENIUS), Kyushu 16 University, 17 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan 18  19 d Center for Computational Science and e-Systems, Japan Atomic Energy Agency 20 (JAEA), 2-4 Shirakata, Tokai-mura, Naka-gun, Ibaraki 319-1195, Japan 21  22 eDepartment of Materials Science and Engineering, The University of Tokyo, 7-3-1 23 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 24  25 f Elements Strategy Initiative for Structural Materials, Kyoto University, Yoshida-26 honmachi, Sakyo-ku, Kyoto 606-8501, Japan 27  28 g National Institute for Materials Science (NIMS), 1-2-1 Sengen, Tsukuba, Ibaraki 305-29 0047, Japan 30  31 h Professor emeritus, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, 32 2  Japan 33  34 Corresponding author: Junichiro Moriyama 35 E-mail: moriyama.junitiro.953@s.kyushu-u.ac.jp 36  37 Abstract 38 The present study focuses on a novel hydrogen-improved strength-ductility balance 39 in several practical Fe-Cr-Ni-based austenitic alloys that directly depends on the 40 dissolved hydrogen content. The hydrogen absorption energy of Fe-Cr-Ni model alloys 41 that have a face-centered cubic structure was examined using first-principles 42 calculations to verify the contribution of Cr and Ni substitutions from Fe to hydrogen 43 solubility in these alloys. Chromium substitution reduced the hydrogen absorption 44 energy to a substantially greater degree than does Ni substitution, with increased Cr : Ni 45 ratios resulting in higher hydrogen solubility. This pattern seen in the calculations 46 supports the previously obtained experimental results in practical alloys with various 47 Cr : Ni ratios. The pronounced reduction in hydrogen absorption energy that results 48 from Cr substitution was mostly attributed to a decrement in the chemical effect derived 49 from the electronic binding state rather than a mechanical effect caused by changed 50 interatomic spacing due to these substitutions. We propose here a Cr-equivalent index to 51 predict hydrogen solubility in Fe-Cr-Ni-based alloys that have varying Cr and Ni 52 content. 53  54 Keywords: Austenitic alloy; First-principles calculation; Hydrogen; Absorption energy; 55 Occupancy 56  57 1. Introduction 58 Hydrogen (H) causes a deterioration in various mechanical properties of metallic 59 materials: of ductility [1], strength [2], and fracture toughness [3]. This is the well-60 known phenomenon of hydrogen embrittlement (HE). Since HE is a barrier to the 61 progress of using H as an energy carrier, a substantial number of studies have been 62 conducted to understand the underlying mechanisms of HE. It is hoped that metals and 63 alloys with higher resistance to HE can be developed by suppressing these multiple HE-64 3  triggers. 65 Fe-Cr-Ni-based austenitic alloys with a face-centered cubic (FCC) structure have a 66 higher resistance to HE than body-centered cubic (BCC) structure alloys such as carbon 67 steels [4] and martensitic steels [5]. However, if the stability of the austenitic phase is 68 low, the structural changes in crystals transforming from FCC to BCC make them 69 susceptible to HE [6,7]. The susceptibility to HE of austenitic alloys has been 70 represented by the Ni-equivalent, which is given by the sum of the relative contributions 71 of alloying elements to the stability of the austenitic phase [8]. In 18 mass%Cr-8 72 mass%Ni-based austenitic stainless steel (18Cr-8Ni), which has a metastable FCC 73 phase, HE appears as a reduction in ductility and crack propagation resistance [9], while 74 18Cr-12Ni and 24Cr-19Ni have superior resistance to HE owing to their high Ni-75 equivalent [9,10]. 76 Dissolved H hardens some FCC metals and alloys, as seen in pure Ni [11,12], Cu-Ni 77 alloy [13], and Fe-Cr-Ni-based austenitic alloys [14]. Dissolved H has also been shown 78 to improve the ductility of Fe-Mn alloy [15] and high-entropy Fe-Mn-Ni-Cr-Co alloy 79 [16]. These facts suggest that dissolved H can be used as a key alloying element to 80 develop novel structural materials with better mechanical performance. Ogawa et al. 81 revealed that, in 24Cr-19Ni, the higher the dissolved H content, the greater the 82 improvement in the strength-ductility balance [17]. Mechanical improvements such as 83 this in Fe-Cr-Ni-based austenitic alloys depend on Cr and Ni content: alloys with a 84 higher Cr content and Cr : Ni ratio achieve higher dissolved H content, amplifying the 85 positive impacts of H [18]. Notably, dissolved H has potential similar to that of carbon 86 and nitrogen for solid-solution hardening in the austenitic phase [18]. Consequently, the 87 interrelation between dissolved H content and Cr and Ni in Fe-Cr-Ni-based austenitic 88 alloys needs to be rationalized on an atomic scale to make dissolved H more effective as 89 a beneficial alloying element. 90 First-principles calculation based on density functional theory (DFT) is one of the 91 most suitable ways to characterize the atomic-scale interaction between metals and 92 solutes. Dissolved H content can be evaluated by the energy required to solidify H in a 93 Fe-Cr-Ni atomic system, i.e., H-absorption energy. We can therefore derive a 94 quantitative understanding of dissolved H content in Fe-Cr-Ni-based austenitic alloys by 95 unveiling the contributions of the alloying elements (Cr, Ni) to the H-absorption energy. 96 4  It has been reported that the alloying elements and their magnetic states alter the H-97 absorption energy across Fe-Cr-Ni-based austenitic stainless steels (AISI304 and 316L), 98 Co-Cr-Fe-Ni, and Co-Cr-Fe-Mn-Ni high entropy alloys [19]. In Fe-Mn [20,21] and Fe-99 Al [22] systems, replacing a Fe atom surrounding O-sites with Mn or Al decreases H-100 absorption energy. Although the contribution of certain alloying elements to the H-101 absorption energy remains a concern in HE-relevant research, no systematic studies 102 have yet directly compared dissolved H content from experiments with those from the 103 H-absorption energy in various Fe-Cr-Ni-based austenitic alloys. 104 In the present study, we aim to examine the contributions of Cr and Ni to the 105 dissolved H content of Fe-Cr-Ni-based austenitic alloys based on a systematic 106 calculation of the H-absorption energy at interstitial positions in the vicinity of Cr or Ni 107 atoms using DFT calculations. The H-absorption energy was divided into elastic energy, 108 caused by the movement of the solvent atoms invoked by dissolved H, and chemical 109 energy, stemming from the change in electronic binding states, which makes it possible 110 to identify the role of each alloying element in the H-absorption energy. The dissolved 111 H content, i.e., H-occupancy, was calculated using the H-absorption energy while 112 varying Cr and Ni content, and then compared to the experimental results. 113  114 2. Computational methodology 115 2.1 Calculation method 116 The DFT calculations were performed using the Vienna Ab Initio Simulation 117 Package [23–25] within the generalized gradient approximation of the Perdew-Burke-118 Ernzerhof form for electron exchange and correlation [26]. The projector-augmented 119 wave (PAW) method was used for plane wave expansion [27]. Fe-Cr-Ni atomic systems 120 with FCC structure were simulated using a 2×2×2 supercell with 32 atoms (Fe31X1, 121 Fe30X2(X: Fe, Cr, Ni)). For example, the system with two atoms replaced by Cr denotes 122 Fe30Cr2. We calculated the H-absorption energy when Cr or Ni replaces one or two of 123 the 32 Fe atoms using these supercells. It was also calculated for Cr32 and Ni32, in which 124 Cr or Ni replaces all the Fe atoms. The bulk properties were calculated using a 125 sufficiently high plane-wave cutoff energy of 360 eV with Fermi surface smearing [28] 126 to obtain accurate interatomic forces, employing a smearing width of 0.2 eV. A 6×6×6 127 k-point Monkhorst-Pack grid [29] was used. We employed VESTA (Visualization for 128 5  Electric and Structural Analysis) [30] to visualize the atomic structure. 129  130 2.2 DFT calculations of H-absorption energies in Fe31X1, Fe30X2 (X: Fe, Cr, Ni) 131 Both cell geometry and atomic positions were fully relaxed in the Fe31X1, and 132 Fe30X2 (X: Fe, Cr, Ni) unit cells when calculating the H-absorption energy. The 133 magnetic state should be considered, as it impacts the H-absorption energy, as reported 134 by Zhou et al [19]. The atomic structure in the Fe-Cr-Ni-based austenitic alloy is stable 135 over a finite temperature range when in a paramagnetic (PM) state; it is unstable at T = 136 0K. When assuming the PM state in the DFT calculation, the H-absorption energy 137 should include excess energy to stabilize the magnetic state. Hence, the 138 antiferromagnetic double layer (AFMD) state (Fig. 1 (a)), i.e., a stable state at T = 0K, is 139 appropriate for investigating the H-absorption energy using the DFT calculation [31]. 140 Even though a recent study revealed that a spin-spiral state is the most stable in Fe with 141 an FCC structure at T = 0K [32], the total energy of the AFMD state is similar to the 142 spin-spiral state [33]. In the present study we therefore employed the AFMD state for 143 analyzing the H-absorption energy. 144 The initial magnetic moment of Fe, Cr, and Ni atoms is ± 3.00μB, ± 5.00μB, and ± 145 1.00μB, where positive and negative values respectively mean up- and down-spin. After 146 the calculations, the value of the magnetic moment within the PAW sphere of Fe, Cr, 147 and Ni atoms changed to ± 2.03μB, ± 0.230μB, and ± 0.950μB, closely consistent with 148 the results of previous studies [19,34]. The H-absorption energy in the non-magnetic 149 (NM) state (Fig. 1 (b)) was also calculated. The H-absorption energy 𝐸ab is given by: 150 𝐸ab=𝐸tot[Fe32-nXnH1] − 𝐸tot[Fe32-nXn] −12𝐸tot[H2] + 𝐸ZP (1) 151 𝐸tot[Fe32-nXnH1], 𝐸tot[Fe32-nXn], and 𝐸tot[H2] were total energy of Fe32-nXnH1, 152 Fe32-nXn, and H2 molecule. The part of zero-point energy 𝐸ZP, of H atom is given by: 153 𝐸ZP =12ℎ𝜈 − 𝑍𝑃𝐸[1 2⁄ H2] (2) 154 ℎ, 𝜈 denote the Plank constant, and the vibrational frequency of the H atom in 155 Fe32-nXnH1, respectively. The 𝑍𝑃𝐸[1 2⁄ H2] is half the zero-point energy of the H2 156 molecules. The vibrational frequency was calculated throuth harmonic approximation. 157 The hessian of the vibration of the H atom was obtained with the location of the other 158 6  fixed atom to decouple the vibration of the H atom from that of the other atom. 159  160 2.3 Deviation of elastic and chemical energy in the H-absorption energy 161 The H-absorption energy can be divided into elastic and chemical parts. The elastic 162 part stems from the movement of the solvent atoms invoked by dissolved H, and the 163 chemical part stems from changes in electronic binding state. Each was separately 164 defined as “elastic energy”, 𝐸abela, and “chemical energy”, 𝐸abchem, in the H-absorption 165 energy, 𝐸ab. The 𝐸abela and 𝐸abchem are given by  166 𝐸abela=𝐸totpulloutH[Fe32-nXn] − 𝐸tot[Fe32-nXn] (3) 167 𝐸abchem=𝐸ab − 𝐸abela − 𝐸ZP (4) 168 𝐸totpulloutH[Fe32-nX2] is the total energy of Fe32-nXn unit cell comes from removing the 169 H atom from fully relaxed Fe32-nXnH1. 170  171 2.4 DFT calculations of the H-absorption energy in Cr32 and Ni32 systems 172 More precisely, the H-absorption energy in pure Fe, Cr, and Ni systems should be 173 resolved to estimate the intrinsic effect of Cr and Ni on the H-absorption energy. The H-174 absorption energy in Fe and Ni with FCC structure at absolute zero was estimated by 175 extrapolation from the experimental data obtained at high temperatures [35]. Since the 176 Cr element only shows a BCC structure in a solid state over the conventional 177 temperature range, a Cr32 system with an FCC structure was hypothetically constructed. 178 The ferromagnetic (FM) state (Fig. 1 (c)) for the Ni32 system with the initial magnetic 179 moment was 1.00𝜇B. Both cell geometry and atomic positions were fully relaxed in the 180 unit cell. As a result, the magnetic moment changed to 0.642𝜇B (within the PAW 181 sphere) a value that is consistent with a past study that gave a value of 0.641𝜇B[36]. The 182 magnetic state (AFMD), antiferromagnetic single layer (AFMS) state (Fig. 1 (d)), and 183 nonmagnetic (NM) state were employed for the Cr32 systems with an initial magnetic 184 moment of ± 5.00𝜇B. As a result, in all magnetic states of Cr32 systems, the magnetic 185 moment changed to 0.00𝜇B (within the PAW sphere) a value that is consistent with a 186 past study that gave 0.00𝜇B [36].  187 7   188 Fig. 1. The magnetic state of the unit cell with 32 atoms. (a) Antiferromagnetic double layer (AFMD), (b) 189 Non-magnetic (NM), (c) Ferromagnetic (FM), and (d) Antiferromagnetic single layer (AFMS) state. The 190 up and down arrows in AFMD, FM, and AFMS represent up- and down-spin. 191  192 2.5 Calculation of H occupancy 193 H-occupancy, 𝜃L, describes the dissolved H content, which represents the ratio of 194 the internal sites occupied by the H atom, given by the following equation: 195 𝜃L=11+exp (𝐸ab − 𝜇*𝑘B𝑇)(5) 196 𝜇*= −74𝑘B𝑇ln𝑇𝑇*+12𝑘𝐵𝑇ln𝑓𝑓0(6) 197 𝑓=𝑝exp (𝑝𝑉𝐻𝑅𝑇) (7) 198 The chemical potential of the H atom, 𝜇*, can be calculated using Eq. (6). 𝑘B is the 199 Boltzmann constant and 𝑇 is the temperature. 𝑇* is the reference temperature of 7.55 200 K [35], and f denotes the fugacity of H2 gas given by Eq. (7) with the gas pressure, 𝑝, 201 the gas constant 𝑅 (= 8.31 𝐽/(𝐾 ∙ 𝑚𝑜𝑙)), the molar volume of H2, 𝑉𝐻  (=202 1.584 × 10−5 m3/mol) [37], and the reference fugacity, 𝑓0 (= 0.1MPa). We calculated 203 the H-occupancy with 𝑝 = 100 MPa at 𝑇 = 543 K for comparison with the 204 experimental data obtained by the authors [18,38]. The method of calculating the H-205 occupancy using H-absorption energy will be described in Section 4.2. 206  207 3. H-absorption energy of Fe31X1, Fe30X2 in the Fe-Cr-Ni atomic system 208 3.1 Validity of computational settings 209 Convergence tests were performed to validate the calculation conditions in the 210 present study using a unit cell size of 32 atoms, K-point of 6×6×6, and cutoff energy of 211 8  360 eV. When changing the unit cell size to four, 32, and 108 atoms in an NM state, the 212 H-absorption energy was 0.145 eV for four atoms, 0.0662 eV for 32 atoms, and 0.0820 eV 213 for 108 atoms. A unit cell with 32 atoms was therefore employed to acquire accurate H-214 absorption energy, taking into account calculation costs. Figure 1 shows the H-absorption 215 energy obtained by the convergence test by varying K-point and cutoff energy in the 216 unit cell consisting of 32 Fe atoms without a zero-point energy correction. The H-217 absorption energy is equivalent in the order of 10–1 eV regardless of the K-point and 218 cutoff energy, similar to past studies giving 0.09 eV (Nazarov et al.) [34], and 0.13 eV 219 (Ismer et al.) [20]. The unit cell size of 32 atoms with a K-point of 6×6×6, and the 220 cutoff energy of 360 eV therefore ensures the accuracy of this study in investigating the 221 effects of Cr and Ni solute atoms with various configurations. 222  223 Table 1. H-absorption energy of the unit cell consisting of 32 atoms with various K-224 point and cutoff energy without the zero-point energy correction 225 Condition K-point cutoff energy (eV) H-absorption energy (eV) Employed conditions  6×6×6 360 0.132 K-point (1) 7×7×7 360 0.138 K-point (2) 8×8×8 360 0.139 K-point (3) 9×9×9 360 0.138 cutoff energy (1) 6×6×6 400 0.132 cutoff energy (2) 6×6×6 500 0.123 cutoff energy (3) 6×6×6 600 0.122  226 3.2 H-absorption energy at O-sites and T-sites 227 Figures 2 and 3 show the H-absorption energy 𝐸ab in the O-sites and T-sites, 228 respectively, when replacing one Fe atom, i.e., Fe31X1, as represented in (a) or two Fe 229 atoms, i.e., Fe30X2, as shown in (b). The part of the zero-point energy correction, 𝐸ZP, is 230 shown as bars filled with a hatched pattern. The up and down arrows in the figures 231 denote the magnetic moment as up- and down-spin. The 𝐸ab fell to a minimum at 0.118 232 eV for Fe31Cr1 and to 0.125 eV for Fe31Ni1, and 0.0490 eV for Fe30Cr2 and 0.0961 eV 233 for Fe30Ni2. 𝐸ab represents the stability of the H atom, since the lower the energy, the 234 9  more stable the site. The H atom can remain more stable in the Fe-Cr system than in the 235 Fe-Ni system. The O-sites have a noticeably lower 𝐸ab than the T-sites, regardless of 236 the substituted element (Cr or Ni), for both Fe31X1 and Fe30X2, except when two Ni 237 atoms are placed diagonally in Fe30Ni2 (type 2-2 and 3-2 in Fig. 3 (b)). In 2-2 and 3-2 in 238 Fig. 3 (b), the structure relaxation invoked the extensive movement of atoms, i.e., 239 change in the lattice constant, substantially affecting the H-absorption energy. As 240 reported in past studies, the O-sites work as a solution site for H atoms [39] and the T-241 sites act as a diffusion pathway [20], i.e., the saddle point of activation energy. In the 242 present study, therefore, we focus on 𝐸ab at the O-sites in our calculations. 243 The following: 1) magnetic state, 2) location of substitutional atoms, and 3) number 244 of substitutional atoms are dominant factors that control the H-absorption energy, 𝐸ab. 245 In 1) magnetic state, AFMD decreased the 𝐸ab by replacing Fe with Ni and Cr, and Cr 246 decreased it in the NM state. In 2) location of substitutional atoms, when replacing Fe 247 with Ni, 𝐸ab changed as a function of the location of Ni. However, the location of Cr 248 did not influence 𝐸ab. Cr and Ni thus appear to contribute in different ways to the 249 reduction of 𝐸ab. The insights from 1) and 2) indicate that 𝐸ab is strongly affected by 250 magnetic moments but not by their direction (up or down). In 3) number of 251 substitutional atoms, one-atom substitution invoked the reduction in 𝐸ab of 0.0614 eV 252 by Ni and 0.068 eV by Cr, which became more pronounced in two-atom substitution, to 253 0.0901 eV by Ni and 0.130 eV by Cr. Chromium therefore plays a more influential role 254 than Ni in reducing 𝐸ab. This conclusion corresponds to the experimental results, which 255 show that the higher the Cr content, or the higher the Cr : Ni ratio, the higher the 256 dissolved H content in Fe-Cr-Ni-based austenitic alloys [18].  257 10  Fig. 2. H-absorption energy, 𝐸ab, at the O-sites of (a) Fe31X1 and (b) Fe30X2 (X: Fe, Ni, Cr) in the AFMD 258 and NM states. Up and down arrows in the AFMD state represents the up- and down-spin. The part of 259 zero-point energy correction, 𝐸ZP, is shown as bars filled with a hatched pattern. 260  261 11  Fig. 3. H-absorption energy, 𝐸ab, at the T-sites of (a) Fe31X1 and (b) Fe30X2 (X: Fe, Ni, Cr) in 262 the AFMD and NM states. Up and down arrows in the AFMD state represent the up- and down-263 12  spin. The part of zero-point energy correction, 𝐸ZP, is shown as bars filled with a hatched 264 pattern. The values of 2-2 and 3-2 of Fe2Ni2 in (b) did not represent correct H-absorption energy 265 at the T-sites because of the significant movement of the H atoms and the change in the lattice 266 constant of the unit cell during the structure relaxation calculation. 267  268 3.3 Contribution of elastic and chemical energy to H-absorption energy 269 Figures 4 and 5 respectively show elastic energy, 𝐸abela, and chemical energy, 𝐸abchem, 270 in (a) Fe31X1 and (b) Fe30X2 systems. Ni-substitution increased or decreased 𝐸abela 271 depending on their positions, e.g., 0.0690 eV and 0.0793 eV in Fe31Ni1 as represented in 272 type 1-1 and 1-3 in Fig. 4 (a). Cr-substitutions increased 𝐸abela in all cases (Fig. 4). In 273 contrast, both Ni- and Cr-substitution decreased 𝐸abchem in all cases (Fig. 5). 274 Interestingly, the effect on 𝐸abchem became more pronounced as the number of 275 substituted atoms increased, irrespective of magnetic moment, especially for Cr-276 substitutions, where 𝐸abchem became negative, e.g., –0.0164 eV in Fe31Cr1. The reduction 277 in total H-absorption energy therefore stems from a marked decrease in the chemical 278 energy part owing to the change in the electronic state, which is more pronounced in the 279 Cr-substitutions. 280 To more precisely demonstrate the contribution of Cr and Ni to the reduction in the 281 H-absorption energy, Fig. 6 shows the H-absorption energy, 𝐸ab, in (a), the elastic 282 energy, 𝐸abela, in (b), and the chemical energy, 𝐸abchem, in (c) when Cr and Ni replace all 283 Fe atoms in the unit cell. The part of zero-point energy correction, 𝐸ZP, is represented 284 as bars filled with a hatched pattern in (a). Due to Cr-substitutions, 𝐸abela showed a slight 285 increase during the remarkable reductions in 𝐸abchem, i.e., –0.321 eV from Fe5Cr1 to Cr6. 286 The reduction in 𝐸abchem was independent of magnetic state because it transformed from 287 AFMD and AFMS to NM during the structure relaxation. In contrast, 𝐸abchem did not 288 change with Ni-substitution, while a slight reduction emerged in 𝐸abela (–0.0192 eV, 289 attributable to the change in the magnetic state, AFMD, and FM): Cr reduces 𝐸ab by 290 potentially lowering 𝐸abchem, and Ni by 𝐸abela. Cr-substitution is more effective in reducing 291 𝐸ab. The experimental values of the H-absorption energy for Fe with an FCC structure 292 and Ni are positive, at 26.2 and 14.6 kJ/mol [35]. Chromium, therefore, as demonstrated 293 by the present study, contributes more to reducing the H-absorption energy. 294 13  Fig. 4. The elastic energy, 𝐸abela, at O-sites of (a) Fe31X1 (X: Fe, Ni, Cr) (b) Fe30X2 (X: Fe, Ni, 295 Cr) in the AFMD and NM states. Up and down arrows in the AFMD state represent up- and 296 down-spin. 297 14   298 Fig. 5. The chemical energy, 𝐸abchem, at O-sites of (a) Fe31X1 (X: Fe, Ni, Cr) (b) Fe30X2 (X: Fe, 299 Ni, Cr) in the AFMD and NM states. Up and down arrows in the AFMD state represent up- and 300 down-spin. 301  302 15  Fig. 6. The difference between the contribution of Cr and Ni to the H-absorption energy at the 303 O-site for Fe5Cr1 in Fe31Cr1 cell, Cr6 in Cr32 cell, Fe5Ni1 in Fe31Ni1 cell, and Ni6 in Ni32 cell. (a) 304 16  H-absorption energy, 𝐸ab. The part of zero-point energy correction, 𝐸ZP, is represented as 305 shown as bars filled with a hatched pattern. (b) Elastic energy, 𝐸abela, (c) chemical energy, 𝐸abchem. 306 Up and down arrows represent up- and down-spin. 307  308 4. H-occupancy calculated by H-absorption energy 309 4.1 Mean H-absorption energy of the O-sites in Fe-Cr-Ni-based austenitic alloys with 310 any chemical composition 311 Figure 7 shows the correlation between the H-absorption energy, 𝐸ab, described in 312 Figs. 2 and 6 and the number of Cr and Ni atoms in the O-sites. Please Note that the 313 𝐸ab in Fig. 7 is the average value of the various O-site patterns generated by the 314 different locations of substitution atoms and magnetic moment shown in Figs. 2 and 6. 315 Assuming that 𝐸ab changes linearly with the number of Cr and Ni atoms, 𝐸ab at the 316 O-sites in Fe-Cr and Fe-Ni systems is given by the following equations. 317 𝐸ab = 0.186 − 0.0646𝑛Cr (8) 318 𝐸ab = 0.186 − 0.0178𝑛Ni (9) 319  320 𝑛Cr and 𝑛Ni are the numbers of Cr and Ni atoms in the O-sites, and 0.186 eV is the 321 𝐸ab at the O-sites consisting of Fe atoms only. 322 We predicted the H-occupancy, 𝜃L, of the various Fe-Cr-Ni-based austenitic alloys 323 by the H-absorption energy at the O-sites. Neglecting the interaction between Cr and Ni 324 atoms constituting the same O-site, 𝐸ab is roughly described by the following equation. 325 𝐸ab = 0.186 − 0.0646𝑛Cr − 0.0178𝑛Ni (10) 326 17  Fig. 7. Variation in the mean H-absorption energy, 𝐸ab, at O-sites under AFMD as a function of 327 the number of Cr atoms, 𝑛Cr, and Ni atoms, 𝑛Ni. When 𝑛Cr and 𝑛Ni = 1, each value of 𝐸ab 328 in Fig. 7 is the average value of 𝐸ab of the O-site patterns under AFMD shown in Fig. 2 (a), 329 i.e., 𝑛Cr and 𝑛Ni = 2 in Fig. 2 (b), and 𝑛Cr and 𝑛Ni = 6 in Fig. 6 (a). 330  331 4.2 H-occupancy in alloys with various atomic ratios of alloying elements 332 To calculate the H-occupancy 𝜃L of Fe-Cr-Ni-based austenitic alloys with various 333 atomic ratio, we generated 100,000 random combinations of six atoms consisting of the 334 O-sites along with the atomic ratio of each alloy, and calculated each 𝐸ab using Eq. 335 (10) in Section 4.1. The 𝜃L of each alloy is the average of H-occupancies from each 336 𝐸ab according to Eq. (5) in Section 2.5. 337 Figure 8 shows the H-occupancy 𝜃L with an arbitrary atomic ratio, derived using 338 the above method, together with the experimental results in Table 2. The 𝜃L values 339 were sorted in ascending order by the sum of the atomic Cr and Ni ratio in Table 2. Over 340 the wide range of totals of Cr and Ni ratios (0 – 36.0 at %), e.g., 16Cr-10Ni to 22Cr-341 12Ni in Fig. 8, the 𝜃L values in the present study are closely consistent with the 342 experimental results for the absolute value. In contrast, when the sum of Cr and Ni ratio 343 exceeds 36.0 at %, as represented by 23Cr-13Ni to 18Cr-35Ni, the calculated result is 344 higher than the experimental one, although the patterns are the same. 345 18  To examine the above discrepancies between the experimental and calculated values 346 for H-occupancy 𝜃L, Fig. 9 shows the relationship between the discrepancy and (a) the 347 totals of the Cr and Ni atomic ratios, (b) the Cr atomic ratio, and (c) the Ni atomic ratio. 348 The discrepancy became more marked as the totals of the Cr and Ni atomic ratios 349 increased. Specifically, the H-absorption energy 𝐸ab derived from Eq. (11) was 350 increasingly underestimated with rising totals of Cr and Ni content. Figures 9 (b) and (c) 351 show that the discrepancy did not correlate with Cr content: it appears to be a function 352 of Ni, except for 0Cr-36Ni. It was more pronounced in 18Cr-35Ni than in 0Cr-36Ni, 353 even with a similar Ni content. Thus, interaction between Cr and Ni atoms is significant 354 if the Fe-Cr-Ni-based austenitic alloy contains a high amount of Cr and Ni, e.g., the sum 355 value exceeds 36.0 at%. Although the interaction between Cr and Ni has been well 356 reported, the interaction between certain alloying elemenets has been studied from the 357 perspective of the phase stability of Nb-Si alloy [40]. The Zr and Hf atoms reduced the 358 stability of the Nb phase in the alloy. Only when another atom (Al, Ni) was contained in 359 the Nb phase, Zr and Hf atoms stabilized the Nb phase dur to the synergetic effect with 360 Al and Ni atoms. Therefore, there is a possibility that the Cr and Ni atoms have such a 361 synergetic effect on the H-absorption energy. 362 Fig. 8. H-occupancy 𝜃L of Fe-Cr-Ni-based austenitic alloys with various Cr and Ni contents 363 obtained by DFT calculations in the present study and experiments in past studies [18,38]. 364  365 19  Table 2. Overall data of the H-occupancy, the Cr-equivalent in the various Fe-Cr-Ni-based austenitic alloys, generated by the experiments and DFT calculations, together with the discrepancy between the DFT and experiments[18,38].  Materials mass Cr % mass Ni % atomic Cr atomic Ni H-occupancy (×10–3) in Fig. 8 Discrepancy  % in Fig. 9 Cr-equivalent in Fig. 10 Experiments DFT calculation 16Cr-10Ni (AISI 316) 16.2 10.1 0.172 0.0958 5.32 5.35 0.612 0.199 18Cr-8Ni (AISI 304) 18.2 8.18 0.193 0.0770 5.32 6.46 21.4 0.214 17Cr-12Ni (AISI 316L) 16.8 12.1 0.179 0.114 5.37 5.92 10.2 0.210 21Cr-10Ni (HPI 160) 20.7 9.71 0.219 0.0914 8.37 8.74 6.58 0.245 0Cr-36Ni (UNS K93600) 0.00 36.1 0.00 0.349 1.52 0.975 -35.8 0.0963 22Cr-12Ni (XM 19) 22.4 12.5 0.238 0.117 9.77 10.8 11.1 0.270 23Cr-13Ni (AISI 309S) 22.5 13.3 0.239 0.125 7.06 11.2 58.6 0.274 20  24Cr-19Ni (AISI 310S) 24.2 19.1 0.257 0.179 7.22 14.4 99.9 0.307 18Cr-35Ni (UNS K08330) 18.4 35.1 0.198 0.335 3.20 10.4 227 0.290 21  Fig. 9. The discrepancy between the H-occupancy 𝜃L of Fe-Cr-Ni-based austenitic alloys with various Cr and Ni content obtained by DFT calculations and experiments in past studies [18,38] in Table 2 versus (a) the totals of the Cr and Ni atomic ratios, (b) the Cr atomic ratio, and (c) the Ni atomic ratio.  As shown in Fig. 8, the H-occupancy of Fe-Cr-Ni-based austenitic alloys can be estimated by the H-absorption energy when the total Cr and Ni content is below 36.0 at%, without taking into account the interaction between Cr and Ni. As an index of H-occupancy of Fe-Cr-Ni-based austenitic alloys, we propose the Cr-equivalent, Creq-H, as follows. Creq-H=1[Cr]+0.276[Ni] (11) [Cr] and [Ni] denote the Cr and Ni atomic ratios. The coefficient of 0.276 represents the contribution of Ni to the H-absorption energy based on that of Cr, i.e., 0.0178 / 0.0646. Figure 10 plots the experimentally-derived H-occupancies [18,38] and the calculated H-occupancies in the present study as a function of Creq-H. The calculated H-occupancies were closely consistent with the results of experiments in the range Creq-H < 0.250. The Creq-H influences the H content derived from the change in the H-absorption energy in the Fe-Cr-Ni sytem. Consequently, the higher the H content, i.e., the lower the H-absorption energy, the greater the improvement in the strength and ductility balance in the Fe-Cr-Ni alloy. Although H-occupancy is overestimated in the range of Creq-H greater than 0.250, Creq-H can be used as an index of the H-occupancy of Fe-Cr-Ni-based austenitic alloys when Creq-H < 0.250. The interaction between Cr and Ni atoms should be incorporated such that Creq-H can estimate the H-occupancy accurately without any limitations on Cr and Ni content. 22   Fig. 10. The H-occupancy 𝜃L of Fe-Cr-Ni-based austenitic alloys with various Cr and Ni contents obtained by DFT calculations and experiments in past studies [18,38] versus the Cr-equivalent, Creq-H in Table 2.  5. Conclusions To understand the details of various dissolved H content in Fe-Cr-Ni-based austenitic alloys as a function of the alloying elements, we investigated the effects of Cr and Ni on H-absorption energy at O-sites by employing DFT calculations in AFMD and NM states. The H-occupancy calculated based on H-absorption energy was directly compared with the experimental results for various Fe-Cr-Ni-based austenitic alloys. The primary insights are summarized as follows. (1) Replacing Fe with Cr and Ni reduces the H-absorption energy, i.e., increases the H-occupancy. This reduction becomes more pronounced with substitutions by Cr than by Ni.  (2) The reduction in the H-absorption energy stems primarily from a decrease in chemical energy rather than a decrease in elastic energy. Chromium reduces the chemical energy in H-absorption to a substantially greater degree than does Ni. (3) The H-absorption energy 𝐸ab at the O-sites in the Fe-Cr-Ni austenitic alloy system is given as follows. 23  𝐸ab = 0.186 − 0.0646𝑛Cr − 0.0178𝑛Ni  The calculated H-occupancy by 𝐸ab strongly supports the experimental results as absolute values over a wide range of totaled Cr and Ni content (< 36.0 at%). When it exceeds 36.0 at%, the calculated H-occupancy is overestimated compared to the experimental results, likely due to the calculation not including the effects on H-absorption energy of interactions between Cr and Ni atoms. (4)  The Cr-equivalent Creq-H is thus a potentially useful index of H-occupancy of Fe-Cr-Ni-based austenitic alloys when Creq-H < 0.250. When Creq-H > 0.250, a divergence is generated between the calculated and experimental results due to interactions between the Cr and Ni atoms.  Acknowledgments This study was partly supported by JSPS KAKENHI (JP20K04161; JP21K04702; JP21K14045) and the Yoshida Gakujutsu Foundation. 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