Oriani equilibrium for interstitial hydrogen

1. Introduction

The classic paper: Oriani RA. The diffusion and trapping of hydrogen in steel. Acta metallurgica, 18 (1970) 147-57.

In the context of hydrogen embrittlement, the Oriani equation (or Oriani's thermodynamic equilibrium model) describes the equilibrium distribution of hydrogen between different types of sites within a metal lattice. Specifically between perfect crystal interstitial lattice sites and hydrogen trapping sites such as dislocations, grain boundaries, vacancies, or interfaces.

Oriani postulated that the transfer of hydrogen between the bulk lattice and a specific population of traps is rapid enough compared to macroscopic diffusion that a local thermodynamic equilibrium is maintained. The relationship is treated like a chemical reaction where nascent hydrogen (H) moves from a lattice site ($\mathrm{L}$) to a trap site ($\mathrm{T}$):

The standard form of the Oriani equation relates the fractional occupancy of the traps ($\theta_\mathrm{T}$) to the fractional occupancy of the lattice ($\theta_\mathrm{L}$):

where:

• $\theta_\mathrm{T}$ is the fractional occupancy of the trap sites ($0 \le \theta_\mathrm{T} \le 1$).
• $\theta_\mathrm{L}$ is the fractional occupancy of the lattice sites ($0 \le \theta_\mathrm{L} \le 1$).
• $\Delta E_\mathrm{B}$ is the trap binding energy (defined as a positive value for attractive traps, typically in $\text{J/mol}$ or $\text{eV}$).
• $R$ is the universal gas constant.
• $T$ is the absolute temperature in Kelvin.

Dilute solution approximation

In practical scenarios involving structural steels, the concentration of hydrogen in the perfect crystal is incredibly small, meaning $\theta_\mathrm{L} \ll 1$. Under this dilute solution assumption, $1 - \theta_\mathrm{L} \approx 1$. The equation simplifies to a Langmuir-type isotherm (Developed by Irving Langmuir in 1916, for which he won the Nobel Prize in Chemistry, it predicts the equilibrium fractional occupancy or amount of material adsorbed on a surface as a function of the pressure of the gas or concentration of the fluid above it. There are no interactions (attraction or repulsion) permitted between neighbouring adsorbed atoms. The energy change associated with a atom binding to an open site remains completely identical regardless of whether adjacent sites are empty or completely occupied.):

If we express the occupancies in terms of actual concentrations (where $c_\mathrm{L}, c_\mathrm{T}$ are concentrations and $N_\mathrm{L}, N_\mathrm{T}$ represent the respective site densities), we can substitute $\theta_\mathrm{L} = c_\mathrm{L} / N_\mathrm{L}$ and $\theta_\mathrm{T} = c_\mathrm{T} / N_\mathrm{T}$ to yield:

2. Derivation via chemical potential

To derive Oriani's equation from fundamental principles, we equate the chemical potentials of hydrogen in the two states. For local equilibrium to exist, the chemical potential of hydrogen in the lattice ($\mu_\mathrm{L}$) must equal the chemical potential of hydrogen in the traps ($\mu_\mathrm{T}$):

Both sub-lattices can be modelled as statistical interstitial systems where configurational entropy dominates mixing behavior. The chemical potential of an interstitial species in an ideal solution framework is given by:

Applying this to both populations:

Equating $\mu_\mathrm{L} = \mu_\mathrm{T}$ and isolating the standard chemical potentials:

Defining the trap binding energy as the net energy difference $\Delta E_\mathrm{B} = \mu_\mathrm{L}^0 - \mu_\mathrm{T}^0$, and taking the exponential of both sides leads directly to Oriani's classic formulation:

Caveat

We have assumed that equilibrium in Oriani's model is determined by equating the chemical potentials of hydrogen. However, equilibrium also requires the chemical potentials of iron to be uniform.

The Oriani model is a local equilibrium model, not a global one. It relies on the massive disparity between the diffusion rate of interstitial hydrogen and the self-diffusion rate of substitutional iron atoms. Because hydrogen moves orders of magnitude faster than iron can rearrange itself, the hydrogen population reaches a state of relaxed thermodynamic equilibrium with respect to the local, frozen-in configuration of the iron lattice.

Even though the iron lattice cannot diffuse to homogenise its chemical potential, $\mu_\mathrm{Fe}$ is absolutely varying locally across the microstructure. Trapping sites are defined by structural disruptions that alter the local energy state of iron. This might be captured through the spatial variation of $\mu_\mathrm{Fe}$ via the trap binding energy $\Delta E_\text{B}$.

In summary, the reason one can safely isolate and equate only the hydrogen chemical potentials comes down to the separation of time scales. There are many similar examples of constrained equilibria.

3. Non-ideal extensions

The standard derivation relies entirely on an ideal interstitial solution model (or a regular solution with an ideal entropy of mixing). This implies the absence of hydrogen-hydrogen ($\text{H-H}$) interactions:

1. Enthalpic independence: $\mu^0$ is constant; the energy change when a hydrogen atom enters a site depends solely on the host metal environment ($\Delta E_\mathrm{B}$) and is independent of neighboring solute atoms.
2. Entropic independence: The configurational entropy term assumes that the only restriction on site entry is simple, single-site blocking ($1-\theta$).

Incorporating solute interactions ($\omega$ parameter)

When local concentrations become high, $\text{H-H}$ strain or electronic interactions can no longer be ignored. Under a Bragg-Williams or regular solution approximation, a lateral interaction parameter $\omega_\mathrm{TT}$ can be introduced into the trap chemical potential:

Equating this to an ideal, dilute lattice yields a modified Oriani equation where the effective binding energy varies dynamically with occupancy:

4. Modelling physical coverage limits ($\theta_{\max} = 0.25$)

To limit coverage within a macro-scale continuum model, two main thermodynamic strategies exist:

Approach 1: coverage-dependent interaction enthalpy $\omega\{\theta\}$

If we maintain the ideal entropy formulation, the interaction term $\omega\{\theta\}$ must scale dynamically to represent the sudden onset of strong repulsive forces after a stable superstructure is filled:

• Step function formulation:
  $$\omega\{\theta\} = \begin{cases} 0 & \text{for } \theta \le 0.25 \\ \Delta E_{\text{repulsion}} & \text{for } \theta > 0.25 \end{cases}$$

  If $\Delta E_{\text{repulsion}} \gg RT$, the exponential term drops rapidly to zero when $\theta$ attempts to exceed $0.25$.

• Asymptotic expansion: For numerical continuity in finite element codes, an asymptotic penalty parameter can be constructed:
  $$\omega\{\theta\} = \omega_0 + \frac{\gamma}{(0.25 - \theta)^n}.$$

  As $\theta \rightarrow 0.25$, the enthalpic penalty $\omega\{\theta\}\cdot\theta \rightarrow \infty$, creating an uncrossable thermodynamic barrier.

Approach 2: Multi-site blocking (configurational entropy modification)

Statistical mechanics indicates that immediate, high-energy nearest-neighbor repulsion is often more elegantly captured by changing the site-blocking factor in the configuration entropy term rather than using a shifting enthalpy parameter.

If one hydrogen atom effectively excludes its 4 nearest neighbors from being occupied, the configurational phrase alters from $(1-\theta)$ to $(1-4\theta)$. The chemical potential transforms to:

Equating this to the ideal bulk lattice potential leads to a modified multi-site Oriani relation:

As $\theta \rightarrow 0.25$, the denominator $(1-4\theta) \rightarrow 0$, causing the left side to diverge asymptotically to infinity. This strictly limits maximum coverage to $0.25$ without needing an infinitely spiking interaction parameter.

H-H interaction in iron

The interaction between hydrogen atoms in $\alpha$-iron has been estimated using first principles calculations:

• Counts, Wolverton, Gibala, Binding of multiple H atoms to solute atoms in bcc Fe using first principles. Acta materialia 59 (2011) 5812.
• Song, Bhadeshia, Suh. Effect of hydrogen on the surface energy of ferrite and austenite. Corrosion Science 77 (2013) 379.

The studies confirm that when hydrogen atoms residing in tetrahedral interstices, approach in close proximity, they repel. This is when their separation in less than 0.2 nm, which corresponds to to the 4th nearest neighbour separation. If two hydrogen atoms atoms approach in adjacent tetrahedral sites (first near neighbours), the pair is unstable and the atoms will spontaneously relax into a third nearest neighbour configuration. A H-H pair in iron is not energetically favourable until the atoms are separated by a distance greater than 0.2 nm. Even then, any interaction is weak.

There are other studies (experimental and theoretical calculations) that indicate a repulsive interaction between neighbouring hydrogen atoms on iron surfaces as the hydrogen coverage increases. Wang et al., The Journal of Physical Chemistry C 118 (2014) 4181.