# Modeling of Phase Equilibria in Ni-H: Bridging the Atomistic with the Continuum Scale

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## Abstract

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## 1. Introduction

## 2. Methods

## 3. Results

#### 3.1. Monte Carlo Modeling

#### 3.2. Free Energy Formulation

#### 3.3. Solvation Energy and H–H Interaction

#### 3.4. Configurational Entropy

#### 3.5. Maxwell Construction

#### 3.6. Elastic Effects

#### 3.6.1. Linear Elastic Effects

#### 3.6.2. Nonlinear Elastic Effects

#### 3.7. Interfacial Effects

#### 3.8. High Concentration Limit

#### 3.9. Conversion to Partial Pressures

#### 3.10. Volume Relaxation

#### 3.11. Phase Diagram

## 4. Discussion and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

MC | Monte Carlo |

## Appendix A. Thermodynamic Framework

## Appendix B. Reference State

## Appendix C. Elastic Constants

#### Appendix C.1. Bulk Modulus

**Figure A1.**Energy density as function of the external hydrostatic strain $\u03f5$. With the relation in Equation (A7), we get the bulk modulus K in quadratic approximation. The atomistic data are taken from simulations at $T=0\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$. For high strains deviations from the linear elastic response can be seen.

#### Appendix C.2. Monoclinic Distortion

**Figure A2.**The results for the monoclinic distortion are shown. The atomistic data is fitted via a quadratic ansatz.

#### Appendix C.3. Orthorhombic Distortion

**Figure A3.**The results for the orthorhombic distortion are shown. The atomistic data are fitted via a quadratic ansatz.

#### Appendix C.4. Nonlinear Elastic Coefficients

**Figure A4.**The free energy density as a function of the external strain $\u03f5$ for the pure Nickel. The quadratic ansatz is only true for small strains, whereas for higher strain we have to use a nonlinear function to fit the Monte Carlo data.

**Figure A5.**Monoclinic deformation of pure nickel for different bulk lattice constants. The fits (curves) are parabolic, and the data (points) are well described without further nonlinear corrections, in agreement with the statement that corrections depend only on $\mathrm{tr}\u03f5$. The latter is contained in the lattice constant dependence of the curvatures, leading to strain-dependent elastic constants ${C}_{44}\left(\mathrm{tr}\u03f5\right)$. The energy of the bulk compression is subtracted in the plots.

**Figure A6.**The elastic constant ${C}_{44}$ as function of the trace of the strain $\mathrm{tr}\u03f5$ for Nickel and the hydride is shown. For $\mathrm{tr}\u03f5=0$, we obtain the values: ${C}_{44,Ni}=134.3$$\mathrm{GPa}$ and ${C}_{44,H}=33.5$$\mathrm{GPa}$.

**Figure A7.**The difference ${C}_{11}-{C}_{12}$ depending on the trace of strain $\mathrm{tr}\phantom{\rule{0.166667em}{0ex}}\u03f5$ for Nickel and the hydride is shown.

## Appendix D. The Isotropic Model

## Appendix E. Interfacial Effects

**Figure A8.**Surface energy depending on the concentration for different geometric cases. The thick line segments indicate the lowest energy configurations.

## References

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**Figure 1.**Selected atomistic simulation results for different hydrogen concentrations with fixed volume at $T=300\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$. The Ni matrix is not shown, and the hydrogen atoms are visualized by the dots. The equilibrium shape of the precipitates in a periodic system depends on the H concentrations. First, hydride spheres appear at low concentrations, here for $c=0.22$ (

**a**); followed by a hydride tube (

**b**) with $c=0.23$. For higher saturations (here $c=0.78$), slabs are observed (

**c**). Beyond $c=0.5$, the situation is inverted and Ni precipitates form inside the hydride matrix. Snapshot (

**d**) shows a Ni tube for $c=0.85$. The last possible shape—a Ni sphere inside a hydride matrix—is not shown.

**Figure 2.**Chemical potential of hydrogen as function of H concentration at $T=300\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$. The S-shaped curve is the single phase prediction, the horizontal dashed line the Maxwell construction. For them, elastic effects are not taken into account, in contrast to the Monte Carlo data, which is for fixed volume (lattice constant of pure Ni). For high concentrations, compressive elastic effects become dominant. Apart from the dilute regions $c\ll 1$ and $1-c\ll 1$, the Monte Carlo data correspond to two-phase configurations.

**Figure 3.**Energy as function of the H concentration for a homogeneous and pressure free system, to determine the solvation energy ${\mu}_{0}$ and the H–H interaction. Notice that neither entropic effects are present at $T=0$, nor elastic or interfacial effects. The Monte Carlo data is averaged over several configurations.

**Figure 4.**The chemical potential contribution ${\mu}_{\mathrm{H-H}}$ as function of the hydrogen concentration in a homogeneous system.

**Figure 5.**Dilute limit of the chemical potential, which is dominated by the configurational entropy. The data are for $T=300\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$ and for fixed volume with the equilibrium lattice constant of pure Ni. The chemical potential is (apart from the offset ${\mu}_{0}$) dominated by the configurational contribution ${\mu}_{\mathrm{c}}$, whereas elastic and H–H interaction terms, ${\mu}_{\mathrm{el}}$ and ${\mu}_{\mathrm{H-H}}$ are negligible there. The accuracy of the continuum description in comparison to the Monte Carlo data are finally about 20 meV per atom in the entire concentration range $0<c<1$.

**Figure 6.**Chemical potential in the single phase state for different temperatures. The curves are the analytical descriptions based on the expression ${\mu}_{0}+{\mu}_{\mathrm{H-H}}+{\mu}_{c}$ using the parameters of the H–H interaction and the solvation energy. The points are the data from the grand canonical Monte Carlo simulations of homogeneous, single-phase states. At low temperatures, the solubility is very low, and therefore the concentration range of the single phase equilibrium states limited to the dilute regions.

**Figure 7.**Lattice constant as function of the hydrogen concentration. The Monte Carlo results are obtained from homogeneous free volume simulations, averaged over several configurations. The fit describes the nonlinear concentration dependence, in contrast to a linear interpolation (Vegard’s law).

**Figure 8.**Elastic contribution to the chemical potential in various approximations: The solid curve expresses the isotropic prediction by Equation (6), using equal elastic constants for both phases (we use the values of pure Ni). The long-dashed curve is the same isotropic model, but this time using the different elastic constants of both phases. This prediction is almost identical to a numerically calculated expression based on cubic elasticity, which assumes a spherical inclusion in a cubic box with periodic boundary conditions (short-dashed curve). In contrast to all these linear elastic approximations, the dotted line also considers elastic nonlinearities. Its prediction is very close to the Monte Carlo data.

**Figure 9.**Energy per Ni atom as function of the hydrogen concentration. The atomistic data are calculated at $T=0\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$ without Monte Carlo steps but atomic relaxation for fixed volume. A spherical precipitate is placed in the center to the simulation cell (with periodic boundary conditions). The irrelevant integration constant ${E}_{0}$ for pure Ni is subtracted, as well as the trivial term ${\mu}_{0}$, thus only the H–H interaction, elasticity and surface contributions remain. The shape of the precipitate is not important for the elastic energy; we used both Ni-H and Ni inclusions in the atomistic simulation, visualized by open and filled spheres.

**Figure 10.**The chemical potential of hydrogen at $T=300\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$. The dots are the data from the Monte Carlo simulations. The chemical potential including nonlinear elastic effects is very close to the Monte Carlo data, and, together with interfacial effects, the agreement is excellent. The contribution from the interfacial terms is largest for low concentrations, as there the precipitates are small and therefore have the largest surface-to-volume ratio. The dotted curves near $c=0$ and $c=1$ are the analytical predictions for the single phase dilute limits. The H${}_{2}$ partial pressure is based on an ideal gas model, which is not realistic for higher pressures and only serves for illustrational purposes.

**Figure 11.**Comparison of the chemical potential for fixed volume and in a stress-free situation, $P=0$, $T=300\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$. The volume constraint leads to a positive slope of the chemical potential in the two-phase region, thus stabilizing phase separation of given chemical potential. In contrast, two phase states are unstable for given ${\mu}_{H}$, as shown here using the continuum prediction.

**Figure 12.**Evolution of the hydrogen concentration as function of the number of Monte Carlo steps for $T=300\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$. Grand-canconical simulations with given chemical potential ${\mu}_{H}$ are used. (

**Top**) Fixed volume using the equilibrium lattice constant of Ni. Due to the positive slope of the chemical potential in the two-phase region (solid curve and data points in Figure 11) stable two phase states with arbitrary average concentrations can be obtained in thermodynamic equilibrium. (

**Bottom**) For free volume, the miscibility gap is inaccessible for the given chemical potential in the grand canonical simulations, and instead the equilibrium state is almost free of hydrogen or almost fully saturated, as the solubilities are low at low temperatures (see phase diagram Figure 13). The reason for this behavior (which is accompanied by a hysteresis) is the negative slope of the chemical potential in the two-phase region (see Figure 11), which implies that these configurations are unstable. All simulations are started as a hydrogen free system, and formation of the hydride therefore demands to overcome a nucleation barrier.

**Figure 13.**Phase diagram of the Ni-H systems, based on parameters extracted from Monte Carlo simulations. The solid line is the binodal without consideration of elastic effects, whereas the dash-dotted line is the same with elastic effects for fixed volume.

**Figure 14.**Sketch of the hydride phase (light grey) forming as a slab in the nickel matrix with fixed volume. Due to translational invariance, we can assume the hydride phase to be located in the left part of the cube, starting at $x=0$.

**Figure 15.**The phase diagram depending on the eigenstrain is shown. The binodal and elastic binodal curves are the same as in Figure 13. The other two curves show the tremendous influence of the concentration dependence of the eigenstrain or lattice constant. If, instead of the concave dependence ${\epsilon}_{0}^{\u2033}\left(c\right)<0$, a linear relationship is assumed (Vegard’s law), the solubility of hydrogen is drastically enhanced. In contrast, the concentration dependence of the elastic constants plays only a minor role: The case of different elastic constants anticipates a linear interpolation of the elastic constants between the values for Ni and Ni-H, whereas, for equal constants, both phases are assumed to have the same elastic constants (using values for Ni).

**Table 1.**Elastic constants of pure nickel and the fully saturated hydride, as obtained from the EAM potential (see Appendix C for details). We note that the relaxed configuration of Ni is used as reference state.

${\mathit{C}}_{\mathit{ij}}$ | Ni | Ni-H |
---|---|---|

${C}_{11}$ | 250.7 GPa | 295.3 GPa |

${C}_{12}$ | 145.5 GPa | 197.3 GPa |

${C}_{44}$ | 134.3 GPa | 33.5 GPa |

**Table 2.**Coefficients for the nonlinear elastic corrections. If higher order coefficients are missing, the expansion is truncated already at lower order.

Elastic Constant | Material | ${\mathit{d}}_{1}$ | ${\mathit{d}}_{2}$ | ${\mathit{d}}_{3}$ | ${\mathit{d}}_{4}$ | ${\mathit{d}}_{5}$ |
---|---|---|---|---|---|---|

${C}_{44}$ | Ni | −3.25 | −7.04 | – | – | – |

${C}_{11}-{C}_{12}$ | Ni | −2.01 | −12.51 | – | – | – |

K | Ni | −1.00 | 0.53 | −4.49 | 2.57 | 50.03 |

${C}_{44}$ | Ni-H | −22.80 | −47.20 | – | – | – |

${C}_{11}-{C}_{12}$ | Ni-H | −9.77 | −14.69 | – | – | – |

K | Ni | −2.25 | 3.37 | 11.51 | 3.55 | −51.00 |

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**MDPI and ACS Style**

Korbmacher, D.; Von Pezold, J.; Brinckmann, S.; Neugebauer, J.; Hüter, C.; Spatschek, R. Modeling of Phase Equilibria in Ni-H: Bridging the Atomistic with the Continuum Scale. *Metals* **2018**, *8*, 280.
https://doi.org/10.3390/met8040280

**AMA Style**

Korbmacher D, Von Pezold J, Brinckmann S, Neugebauer J, Hüter C, Spatschek R. Modeling of Phase Equilibria in Ni-H: Bridging the Atomistic with the Continuum Scale. *Metals*. 2018; 8(4):280.
https://doi.org/10.3390/met8040280

**Chicago/Turabian Style**

Korbmacher, Dominique, Johann Von Pezold, Steffen Brinckmann, Jörg Neugebauer, Claas Hüter, and Robert Spatschek. 2018. "Modeling of Phase Equilibria in Ni-H: Bridging the Atomistic with the Continuum Scale" *Metals* 8, no. 4: 280.
https://doi.org/10.3390/met8040280