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Stability Scoring

Stability scoring quantifies the thermodynamic resistance of a catalyst surface to dissolution and reconstruction.

Physical Basis

Surface energy (\(\gamma\)) is the energy required to create a unit area of surface:

\[\gamma = \frac{E_{slab} - n \cdot E_{bulk}}{2A}\]

Where:

  • \(E_{slab}\) = Total energy of the slab
  • \(E_{bulk}\) = Energy per atom in bulk
  • \(n\) = Number of atoms
  • \(A\) = Surface area
  • Factor of 2 accounts for two surfaces

Stability Correlation

Lower surface energy indicates:

  • Stronger metal-metal bonds at the surface
  • Reduced tendency to reconstruct
  • Better resistance to dissolution
  • Longer operational lifetime

Mathematical Formulation

ASCICat uses inverse linear normalization:

\[S_s(\gamma) = \frac{\gamma_{max} - \gamma}{\gamma_{max} - \gamma_{min}}\]

Where:

Parameter Description Source
\(\gamma\) Surface energy (J/m²) DFT calculation
\(\gamma_{min}\) Minimum in dataset Data-driven
\(\gamma_{max}\) Maximum in dataset Data-driven

Properties:

  • Lowest \(\gamma\) → \(S_s = 1\) (most stable)
  • Highest \(\gamma\) → \(S_s = 0\) (least stable)
  • Linear interpolation between extremes

Python Usage

Basic Usage

from ascicat.scoring import score_stability
import numpy as np

# Single value
gamma = 1.5  # J/m²
# Need reference range for normalization
score = score_stability(gamma, gamma_min=0.5, gamma_max=4.0)
print(f"Stability score: {score:.3f}")  # 0.714

# Array of surface energies
gammas = np.array([0.52, 1.0, 2.0, 3.0, 4.5])

# Automatic min/max from data
scores = score_stability(gammas)
print("Surface Energy | Stability Score")
print("-" * 35)
for g, s in zip(gammas, scores):
    print(f"    {g:.2f} J/m²   |      {s:.3f}")

Output:

Surface Energy | Stability Score
-----------------------------------
    0.52 J/m²   |      1.000
    1.00 J/m²   |      0.880
    2.00 J/m²   |      0.628
    3.00 J/m²   |      0.377
    4.50 J/m²   |      0.000

With Fixed Ranges

# Using configuration-defined ranges
from ascicat.config import HER_CONFIG

gamma_min, gamma_max = HER_CONFIG.stability_range
scores = score_stability(gammas, gamma_min, gamma_max)

Typical Surface Energy Values

Pure Metals (Low-Index Surfaces)

Metal (111) γ (J/m²) (100) γ (J/m²) (110) γ (J/m²)
Pt 0.52 0.58 0.67
Pd 0.49 0.54 0.62
Au 0.32 0.37 0.42
Ag 0.35 0.40 0.46
Cu 0.58 0.65 0.75
Ni 0.62 0.70 0.81
Co 0.76 0.85 0.98
Fe 0.78 0.88 1.02

Surface Orientation

Close-packed surfaces (111 for FCC, 110 for BCC) have lower surface energies and higher stability.

General Ranges

Category γ Range (J/m²) Examples
Very stable 0.3 - 0.6 Au, Ag, Pt
Stable 0.6 - 1.2 Cu, Ni, Pd
Moderate 1.2 - 2.0 Co, Fe, Mo
Less stable 2.0 - 3.5 W, Ta, open surfaces
Unstable > 3.5 Highly stepped surfaces

Data Validation

ASCICat validates surface energy inputs:

# Negative values trigger warning
gammas_invalid = np.array([0.5, -0.2, 1.5])  # -0.2 is invalid
scores = score_stability(gammas_invalid)
# Warning: Found 1 negative surface energy values...
# Negative values clipped to zero

Physical Validity

Surface energies must be non-negative (thermodynamic requirement). Negative values in your data may indicate calculation errors.

Score Interpretation

Score Range Stability Practical Implications
0.9 - 1.0 Excellent Long operational lifetime
0.7 - 0.9 Good Suitable for most applications
0.5 - 0.7 Moderate May require protective measures
0.3 - 0.5 Poor Limited durability
0.0 - 0.3 Very poor Rapid degradation expected

Edge Cases

All Identical Values

# When all surface energies are the same
identical = np.array([1.5, 1.5, 1.5])
scores = score_stability(identical)
# Returns: [1.0, 1.0, 1.0]  # All equally "most stable"

Single Value with Data-Driven

# Single value without reference range
single = np.array([1.5])
scores = score_stability(single)
# Returns: [1.0]  # Single value is its own best

Integration with ASCICalculator

When using ASCICalculator, stability scoring is handled automatically:

from ascicat import ASCICalculator

calc = ASCICalculator(reaction='HER')
calc.load_data('data/HER_clean.csv')

# Stability scores computed automatically
results = calc.calculate_asci()

# Access individual scores
print(results[['symbol', 'surface_energy', 'stability_score']].head())

Limitations

What Stability Scoring Doesn't Capture

  • Electrochemical dissolution - pH and potential effects
  • Adsorbate-induced reconstruction - Surface rearrangement under operation
  • Oxidation/corrosion - Chemical degradation
  • Poisoning - Irreversible adsorption of contaminants
  • Agglomeration - Nanoparticle sintering

Surface energy is a thermodynamic descriptor that correlates with stability but doesn't capture all degradation mechanisms.

References

  • Hansen, H. A. et al. Surface energies of late transition metals. Phys. Chem. Chem. Phys. 10, 3722 (2008)
  • Vitos, L. et al. The surface energy of metals. Surf. Sci. 411, 186 (1998)