Boron Diffusion Activation Energies in FCC Alloys

• The paper details the quantitative determination of boron migration activation energies using an Arrhenius-type relation and spectral sampling methodology.
• It reveals how substitutional Cr and Mo alter energy distributions in both bulk and grain boundary environments, directly impacting defect mobility.
• The study demonstrates that Cr enables rapid in-boundary redistribution while Mo acts as a kinetic trap, informing dopant strategies in alloy design.

Boron diffusion activation energies quantify the kinetic barriers encountered by light interstitial boron atoms as they migrate within metallic alloys, particularly face-centered cubic (FCC) Ni and its substitutional alloys. These energy distributions determine both the rates and pathways for boron transport in the bulk lattice and along crystallographic defects such as grain boundaries (GBs). Accurate characterization of these activation energies, especially in the presence of substitutional solutes like Cr and Mo, is critical for predicting defect mobility, mediating long-term stability, and guiding microstructure evolution in complex alloys (Doležal et al., 31 Dec 2025).

1. Theoretical Framework and Key Formulas

Boron transport is described by an Arrhenius-type relation for diffusivity: $D = D_0 \exp\biggl(-\frac{E_a}{k_B T}\biggr),$ where $D_0 = \nu a^2$ is the diffusion prefactor ($\nu \approx 10^{13}$ s$^{-1}$, $a = 3.52$ Å), $E_a$ is the activation energy, $k_B$ is Boltzmann’s constant, and $T$ is the absolute temperature.

To resolve the distribution of kinetic barriers instead of reporting a single $E_a$, a spectral sampling methodology is employed. The probability density $P(E; n, \mathrm{chem})$ for a migration barrier $E$ within a local chemical coordination $n$ is constructed from transition-state searches: $$P\bigl(E;\,n,\mathrm{chem}\bigr) = \frac{1}{N_{\rm samp}} \sum_{i=1}^{N_{\rm samp}} \delta\left(E - E_i^{(n)}\right),$$ where $E_i^{(n)}$ are barriers computed for cages with $n$ solute atoms (Cr or Mo).

Grain boundary segregation energy is defined as: $$E_{\rm seg}(n) = [E_{\rm GB}(n)-E_{\rm GB}^{0}] - [\langle E_{\rm bulk}^{(n)}\rangle - E_{\rm bulk}^{0}],$$ quantifying the thermodynamic preference for B to occupy interfacial versus bulk sites at fixed solute coordination.

2. Spectral Sampling Methodology

The spectral sampling approach systematically enumerates and computes migration barriers across all unique solute-decorated environments near a B interstitial. In FCC Ni, up to six nearest-neighbor Ni atoms can be replaced by either Cr or Mo for each migration event. Each permutation (coordination $n$) is relaxed, then characterized via a 6-image climbing-image nudged elastic band (NEB) calculation (spring constant 0.5 eV/Å, force tolerance 0.01 eV/Å).

For grain boundary studies—specifically at Σ5 ⟨100⟩{210} symmetric tilt GBs—candidate interstitial sites are identified using Voronoi analysis and filtered by geometric constraints. Every feasible B hop (≤2.5 Å) connecting voids is sampled for each unique chemical and topological environment.

Computational parameters include a 1,372-atom bulk supercell (24 Å cubic), 4,800-atom GB cell (tilt direction ∼17.6 Å), and the universal neural-network interatomic potential (PFP v5.0 + D3Dispersion). This framework resolves the rugged, multi-modal activation energy spectra characteristic of chemically complex alloys.

3. Bulk Boron Activation Energy Distributions

Boron diffusion in pure Ni is dominated by a single, symmetric activation energy value: $E_a = 1.484$ eV for both forward and reverse hops. However, alloying with Cr or Mo alters the profile markedly:

System	Mean $\langle E_a\rangle$ (eV)	$\sigma(E_a)$ (eV)	Modalities
Ni (pure), $n=0$	1.484	0.00	1 (single)
Ni–Cr (all $n$)	1.62	0.20	2–3 peaks (bi/tri)
Ni–Mo (all $n$)	2.03	0.50	2–4 peaks (multi)

Cr substitution raises the mean bulk migration barrier to $\sim$1.62 eV while inducing bi- or tri-modality associated with different solute configurations (steric, off-axis, symmetric-shared). Mo substitution has an even stronger effect, increasing the mean barrier to $\sim$2.03 eV and broadening the distribution further (range: 1.35–3.04 eV). This underscores that boron encounters a rugged, solute-dependent migration landscape in disordered alloys (Doležal et al., 31 Dec 2025).

4. Grain Boundary Activation Energies

At Σ5⟨100⟩{210} GBs, the activation energy spectrum becomes highly anisotropic, with distinct mechanisms for in-plane versus out-of-plane hops. Barriers are summarized (for $n=5,6$ decorated cages, at 800 °C):

Path/Region	$\langle E_{\rm in\text{-}plane}\rangle$ Cr	$\langle E_{\rm out\text{-}of\text{-}plane}\rangle$ Cr	$\langle E_{\rm in\text{-}plane}\rangle$ Mo	$\langle E_{\rm out\text{-}of\text{-}plane}\rangle$ Mo
Low-energy midplane	0.25	1.85 (n=6)	0.75	2.61 (n=6)
Central region	0.55	2.02	1.10	2.80
Escape (x-hops)	1.80	2.51	2.56	3.28

Cr preserves low in-plane barriers ($\sim$0.19–0.30 eV) even at maximal decoration ($n=6$), while out-of-plane (escape) barriers reach ∼2.5 eV. In contrast, Mo elevates all activation energies: in-plane ($\sim$0.6–0.9 eV) and out-of-plane (∼3.0 eV), with out-of-plane barriers exceeding those for Cr by ∼0.76 eV. Diffusivity in the GB midplane at 800 °C is $D_{\rm GB}\sim10^{-15}$ m$^2$/s (Cr) and $D_{\rm GB}\sim10^{-20}$ m$^2$/s (Mo), with Mo imposing a $10^5$-fold greater restriction for out-of-plane motion.

5. Segregation Energies

Grain-boundary segregation energetics further differentiate Cr and Mo effects. For each solute, incremental addition lowers $E_{\rm seg}(n)$, increasing the thermodynamic driving force for B affinity at GB sites:

System	$E_{\rm seg}(n=0)$ (eV)	$E_{\rm seg}(n=6)$ (eV)	Distribution shape
Pure Ni	–0.10	—	Weak, broad
Ni–Cr	—	–0.45	Deeper, less sharp
Ni–Mo	—	–0.75	Deep, nearly delta-like

Cr generates a moderate, relatively broad well for segregation, while Mo yields a sharp, nearly delta-like minimum of $-0.75$ eV for $n=6$. Both promote B segregation, but their characteristic site-dependence and well-depth differ significantly.

6. Functional Implications and Dopant Design

Cr and Mo shape boron transport via distinct kinetic and thermodynamic mechanisms. Cr’s combination of low in-plane barriers and large out-of-plane barriers at GBs enables rapid in-boundary redistribution coupled with strong confinement, conditions favoring Cr-rich boride (e.g., Cr$_5$B$_3$) nucleation. Mo, by contrast, uniformly suppresses boron mobility across all directions, acting as a kinetic trap while also anchoring B via deeper, site-insensitive segregation wells, a factor that accounts for observed Mo-rich boride and carbide formation at interfaces.

Dopant strategies can thus be informed by these findings: moderate Cr additions are optimal for applications requiring accelerated boundary healing and redistribution, whereas Mo is more suitable when the goal is to immobilize B at critical interfaces and suppress boundary embrittlement. These effects consistently emerge from the multi-modal, rugged activation energy landscapes resolved by spectral sampling.

7. Broader Significance and Framework Extension

The spectral sampling methodology demonstrates that boron migration in complex alloys cannot be represented by a single activation energy; instead, the full spectrum, shaped by local chemistry and topology, dictates mobility and segregation. This approach is transferable to other interstitial-solute and defect transport phenomena in chemically complex alloys and provides a systematic protocol for correlating composition and microstructure with transport properties, central to designing advanced superalloys and interface-engineered materials (Doležal et al., 31 Dec 2025).