Chemical-Dependent Diffusion Activation Energies

• Chemical environment-dependent diffusion activation energies are defined by the sum of variable vacancy formation and migration barriers that change with local atomic configurations.
• Simulations reveal that increasing vacancy concentrations lower activation energies by up to 72-78%, thereby enhancing atomic mobility and reducing thermal stability.
• Understanding these energy distributions aids in engineering material properties, as they provide predictive insights for diffusion kinetics and phase stability in complex systems.

Chemical environment-dependent diffusion activation energies describe how the energy barriers for atomic or molecular migration processes—in solids, surfaces, or complex environments—are modulated by variations in local atomic, electronic, or structural surroundings. Instead of a single, well-defined activation energy, systems characterized by chemical or structural heterogeneity present a distribution or envelope of activation energies that strongly influence macroscopic diffusivity and related kinetic phenomena. This dependence is critical for modeling real materials (such as alloys, functional thin films, or disordered networks), where vacancy concentration, local strain, and the presence of different atomic species introduce site-to-site variations that cannot be captured by a mean-field or single-point approach.

1. Microscopic Basis for Chemical Environment Dependence

The diffusion activation energy for a given atomic jump is the sum of the vacancy formation energy ($E_f$) and the migration barrier ($E_b$). Both components can vary substantially with the local chemical environment:

• Vacancy formation energy $E_f$: Defined as $E_f = E_i - E_0 + \mu_i$, where $E_i$ is the energy of a supercell containing a vacancy, $E_0$ is the ground-state supercell energy, and $\mu_i$ is the chemical potential of the removed species. The value of $E_f$ depends on local bonding and the chemical identity of neighboring atoms.
• Migration barrier $E_b$: Determined via methods such as nudged elastic band (NEB), $E_b$ is sensitive to lattice relaxation—especially in systems with pre-existing vacancies or anisotropic strain—and the local arrangement of adjacent species.

Because both $E_f$ and $E_b$ are functions of the immediate atomic configuration, their sum $E_a = E_f + E_b$ defines a spatially heterogeneous energy landscape for diffusion. In compositionally complex nitrides such as fcc-(Ti,Al)N$_x$, sampling different local environments across a statistically significant ensemble produces an "activation energy envelope" that captures this intrinsic variability (Nayak et al., 18 Oct 2025).

2. Vacancy Concentration and Activation Energy Envelopes

In stoichiometric compounds, all lattice sites are nominally occupied, and vacancies must be generated thermally, resulting in diffusion activation energies that are the sum of the averaged $E_f$ and $E_b$ over all possible chemical environments. As the vacancy concentration increases (either due to non-stoichiometry or intentional defect introduction), the activation barrier for atomic migration changes in the following ways:

• Pre-existing metal vacancies: For metal-rich diffusion pathways, the $E_f$ term is removed from the activation energy calculation, because such vacancies already exist in the lattice. This process reduces $E_a$ nearly to $E_b$ alone. The simulations show that the average migration barrier for Al is reduced by up to 72%, and for Ti by approximately 78%, in the presence of metal vacancies due to lattice strain relaxation.
• Nitrogen vacancies: While their formation energy may be higher, once present, nitrogen vacancies also enhance diffusion primarily by relieving local lattice strain, thereby decreasing $E_b$ and further accelerating mass transport.

For a range of vacancy stoichiometries ($x = 0.47, 0.5, 0.53$ in fcc-(Ti$_{0.5}$Al$_{0.5}$)$_{1-x}$N$_x$), the predicted ensemble of activation energies shifts to lower values as vacancy content increases, with the spread (standard deviation, min/max) in activation energy rising due to the larger heterogeneity in local atomic configurations (Nayak et al., 18 Oct 2025). This effect is summarized conceptually as: $$\begin{align*} E_a &\approx \langle E_f \rangle + \langle E_b \rangle \quad \text{(stoichiometric)} \ E_a &\approx \langle E_b \rangle \quad \text{(pre-existing vacancies)} \end{align*}$$

3. Impact on Thermal Stability and Mass Transport

The thermal stability of metastable phases such as fcc-(Ti,Al)N$_x$ hinges on the energetics of atomic migration. Lowered $E_a$ due to increased vacancy concentration or lattice strain relaxation directly translates to higher atomic mobility and thus faster decomposition kinetics or phase transformation rates. The experimental observation of maximum thermal stability in stoichiometric fcc-(Ti,Al)N is explained by the larger average $E_a$ in the absence of extrinsic vacancies. In non-stoichiometric compounds ($x \neq 0.5$), both reduced $E_a$ and a broader activation energy distribution account for the more facile decomposition observed in real systems (Nayak et al., 18 Oct 2025).

This relationship is summarized in the table below:

Compound/Defect State	Average $E_a$	Mobility / Stability
Stoichiometric (no vacancies)	High	Low, stable
Metal vacancies present	Reduced	High, less stable
Nitrogen vacancies present	Reduced	High, less stable

4. Local Lattice Relaxation and Mechanistic Insights

The presence of metal or nitrogen vacancies not only eliminates the energetic cost for forming a defect on the relevant sublattice, but also facilitates strain relaxation throughout the lattice. This relaxation effect systematically lowers the migration barrier $E_b$, as the diffusing atom faces less resistance from neighboring atoms and the potential energy landscape flattens along the migration coordinate. Lattice relaxation is especially significant in chemically complex systems, where local environments vary in terms of both atomic species and mechanical constraints, further broadening the activation energy envelope.

5. Limitations of Single-Point Activation Energies

The strong dependence of both $E_f$ and $E_b$ on local chemistry challenges the utility of single-point activation energy calculations. A single value cannot capture:

• Site-to-site variations across different chemical environments
• Pathway-dependent barriers due to local strain fields or clustering
• The effects of varying vacancy concentrations, which may shift system behavior from the vacancy formation + migration regime into the migration-alone regime

Statistical sampling using supercell approaches, combined with methods such as nudged elastic band calculations for migration and supercell energetics for formation, is therefore required to construct a physically meaningful activation energy envelope. This approach provides a more accurate and predictive description of mass transport and decomposition kinetics in chemically heterogeneous materials.

6. Broader Implications and Materials Design

Environment-dependent diffusion activation energies have general implications for materials performance:

• Design for high stability: Fine control of stoichiometry and minimization of both metal and nonmetal vacancies maximize $E_a$ and thus thermal stability.
• Tailoring mobility: Vacancies introduced deliberately may enhance the kinetics where phase transformations or fast diffusion are needed, as in catalytic supports or non-equilibrium processing.
• Vacancy engineering: Understanding the mechanisms by which vacancies relax strain and effect changes in $E_b$ enables rational design of diffusion profiles.
• Predictive modeling: For systems where local configurations are highly variable (such as high-entropy or complex concentrated alloys), the environment-dependent approach is essential for reliable computational predictions.

The environment-sensitive diffusion activation energy paradigm thus underpins advances in the rational design and stability optimization of coatings, ceramics, alloys, and functional materials, especially where mass transport and phase stability are tightly coupled to device performance (Nayak et al., 18 Oct 2025).