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

Abstract

This work inspects the thermally activated transfer of solute particles across the interface between two interstitial solid solution phases that equilibrate internally by fast diffusion on conserved arrays of sites. When each phase is considered as an ergodic ensemble of particles, statistical mechanics predicts the occupancy of the transition states at equilibrium to depend on the barrier energy and on the chemical potentials and vacancy fractions in each of the phases. A rate law for the non-equilibrium interfacial transfer, based on a constant transition probability between activated states, naturally satisfies the principle of detailed balance. Contrary to Butler-Volmer-type laws, values of the particle chemical potentials enter explicitly rather than through their difference. This, along with the dependency on the vacancy fractions, implies here an exchange flux density that depends explicitly on the compositions at equilibrium. The results can explain experimental observations of a drastic slow-down in the charging of metal hydrides near phase transformations or miscibility-gap critical points.

Figures (5)

• Figure 1: Scheme of energy landscape, variation of single-particle potential energy with position. Phases $\sf A$ and $\sf B$ exhibit periodic energy variation, with minima defining sites available to particles. Phases are separated by an energy barrier, and transition states $\sf T$ represent local saddle points between neighboring sites on opposite sides of the barrier.
• Figure 2: Virtual steps in computing the transfer configuration free energy. Sites in phases $\sf A$ (circles, orange shade ) and $\sf B$ (squares, blue shade) are occupied by solute particles (gray discs). Top: freeing the transfer sites by compressing the respective sets of particles to the smaller sets of remaining sites. Bottom: removing a particle (red disk) from the bulk ensemble in $\sf A$ and inserting it into the source site of the transfer configuration ("=" and white rectangle). Red arrows correspond to the four $\delta \mathcal{F}$ in the main text.
• Figure 3: Scheme of per-particle energy $\epsilon$ versus reaction coordinate for transitions between phases $\sf A$ and $\sf B$ across a barrier at transition state $\sf T$.
• Figure 4: Schematic diagrams showing the solute fraction ($x$) dependence of free energy, $\mathcal{F}$, solute chemical potential, $\mu_{\rm S}$, inverse solute susceptibility, $\chi^{-1}_{\rm S}$, and characteristic charging time, $\mathcal{T}$, in response to an infinitesimal increment in $\mu_{\rm S}$. Schematic represents a solid solution with a miscibility gap (as apparent from the double-well free energy function) at temperatures $T$ below a critical temperature $T_{\rm C}$. Graphs refer to $T$ below, at and above $T_{\rm C}$, as indicated in legend. Below $T_{\rm C}$, extrema in $\mu_{\rm S}$ locate the spinodals, where $\chi^{-1}_{\rm S} \rightarrow 0$ and, in the consequence, $\mathcal{T}$ diverges. Solid solution is instable between the spinodals.
• Figure 5: Implications of Butler-Volmer (BV) law and present approach, for an interacting regular solution in contact with a nearly saturated solution. (a), current-potential response, $J_{\mu}$, versus solute fraction, $x$. All graphs are normalized to their respective prefactor ($J_0$ or $\tilde{J}_0$) and to their value ($J^{\rm C}_{\mu}/J^{\rm C}_0$ or $J^{\rm C}_{\mu}/\tilde{J}^{\rm C}_0$) at the regular solution critical point. Solid lines, present approach; dashed lines, BV law. Color code and legend: graphs for three different temperatures $T$, specified relative to the critical temperature $T_{\rm C}$. (b), experimental BapariWeissmüller2025 halftime, $t_{1/2}$, for H equilibration in nanoporous Pd-Pt-H after incremental jumps in the electrode potential, $E$, at $T$ just above $T_{\rm C}$. Blue dots: data; gray line: trendline symmetric about the maximum. (c), predictions of present (orange line) and BV (green line; symmetric around maximum) approach for halftime ${\mathcal{T}}$, here at $T = 1.1 \, T_{\rm C}$. Axes are referred to values at maximum; note identical axes range span as in (b). Red arrows in (b) and (c) mark asymmetry, larger halftime on the low-$E$ (high-$x$) side of the maximum.