Rate Equation for the Transfer of Interstitials across Interfaces between Equilibrated Crystals

Technical Summary: Rate Equation for the Transfer of Interstitials across Interfaces between Equilibrated Crystals

Problem Statement
The paper addresses a discrepancy between experimental observations and existing kinetic models regarding the charging and discharging of interstitial solid solutions, specifically metal hydrides (e.g., Pd, PdPt). Experiments indicate that the characteristic time ($T$) for composition changes increases drastically (by orders of magnitude) when the composition range includes a phase transformation or the vicinity of a miscibility gap critical point. The standard Butler-Volmer (BV) equation, widely used to analyze electrochemically controlled hydriding kinetics, fails to explain this observation. The BV framework assumes transfer between sites with fixed a priori occupancy and ground-state energy, treating the driving force solely as the difference in chemical potentials ($\Delta \mu_S$). However, in nanoscale metal hydrides, internal diffusion is orders of magnitude faster than interfacial transfer, meaning the bulk phases reach internal equilibrium quasi-instantaneously. Consequently, the interfacial transfer occurs between extended, ergodic ensembles of solute particles where occupancies and ground-state energies fluctuate, a scenario not captured by standard jump-probability models.

Methodology
The author derives a new kinetic rate law based on statistical mechanics and the principle of detailed balance, rather than phenomenological jump probabilities. The system is modeled as a composite of two interstitial solid solution phases (A and B) separated by an energy barrier with a transition state ($T$).

1. Statistical Framework: The phases are treated as canonical systems at constant temperature ($T$) with conserved numbers of solute particles ($N_S$) and matrix sites ($N_M$). The bulk diffusion is assumed fast enough to maintain internal ergodicity within each phase.
2. Transition State Analysis: The probability of a "transfer configuration" ($C_p$)—where a particle is in an activated state at the source site and the target site is vacant—is calculated by summing over all microstates of the constrained system.
3. Free Energy Decomposition: The free energy change ($\delta F$) required to reach the transfer configuration is computed via a four-step virtual process:
   • Compressing the solute ensemble in the source phase to free a site (entropic cost).
   • Compressing the solute ensemble in the target phase to free a site (entropic cost).
   • Removing a particle from the source phase to a reference state (chemical potential work).
   • Inserting the particle into the specific transition state (energy cost).
4. Flux Derivation: The net flux density ($J_B$) is derived as the difference between forward and backward fluxes, utilizing the calculated probabilities of the transfer configurations.

Key Contributions and Results
The primary contribution is the derivation of a new rate equation (Eq. 15) for the interfacial flux of charge-neutral particles:
$$J_B = \nu_0 \rho_T (1 - x_A)(1 - x_B) \exp\left(-\frac{\epsilon_T}{k_B T}\right) \left( \exp\left(\frac{\mu_S^A}{k_B T}\right) - \exp\left(\frac{\mu_S^B}{k_B T}\right) \right)$$

Key features of this result include:

• Explicit Dependence on Chemical Potentials: Unlike the BV equation, which depends on the difference in chemical potentials ($\Delta \mu_S$), this rate law depends explicitly on the individual chemical potentials ($\mu_S^A$ and $\mu_S^B$) of the two phases.
• Vacancy Fraction Dependency: The flux is explicitly modulated by the vacancy fractions $(1-x_A)$ and $(1-x_B)$, reflecting the statistical probability of finding an empty target site in an ergodic ensemble.
• Detailed Balance: The equation naturally satisfies the principle of detailed balance, vanishing when $\mu_S^A = \mu_S^B$.
• Explanation of Kinetic Slow-down: The paper demonstrates that near a miscibility gap critical point or spinodal, the solute susceptibility ($\chi_S = dx/d\mu_S$) diverges. Since the characteristic time $T$ is inversely proportional to the product of the exchange flux density and the susceptibility ($T^{-1} \propto J_{eq}^\mu / \chi_S$), the divergence of $\chi_S$ leads to a divergence in $T$. This provides a theoretical explanation for the experimentally observed drastic slow-down in charging times near critical points.
• Asymmetry: The derived law predicts an asymmetry in the characteristic time $T$ on either side of the critical point, a feature observed experimentally but not reproduced by the symmetric BV equation.

Significance
The paper claims that its derived rate law offers a more physically consistent description of interfacial transfer in systems where bulk phases equilibrate rapidly compared to the interface. By treating the transfer sites as coupled to ergodic canonical systems rather than fixed sites, the model accounts for the fluctuating ground-state energies and occupancies inherent to interacting solid solutions. The work suggests that the failure of the Butler-Volmer equation to explain kinetic anomalies in metal hydrides stems from its neglect of the explicit dependence on equilibrium compositions and the statistical nature of vacancy availability. The results provide a theoretical basis for understanding charging/discharging kinetics in energy storage materials, particularly near phase transformations, without requiring ad-hoc adjustments to the transfer coefficient or barrier energy.