Chapter 5: Applications and Future Perspectives - Introduction to High-Entropy Materials

Learning Objectives

Identify key structural applications in aerospace and nuclear industries
Understand energy-related applications including batteries and fuel cells
Explain catalytic applications for electrochemical reactions
Describe protective coating applications including thermal and environmental barriers
Evaluate biomedical applications and biocompatibility considerations
Analyze future directions including machine learning and sustainability

5.1 Structural Applications

flowchart TB Struct[Structural Applications] --> Aero[Aerospace] Struct --> Nuclear[Nuclear] Struct --> Auto[Automotive] Struct --> Tool[Tooling] Aero --> Turb[Turbine components] Aero --> Cryo[Cryogenic tanks] Aero --> Heat[Heat exchangers] Nuclear --> Fuel[Fuel cladding] Nuclear --> Struct2[Structural materials] Nuclear --> Waste[Waste forms] Auto --> Eng[Engine components] Auto --> Body[Body panels] Tool --> Cut[Cutting tools] Tool --> Dies[Dies and molds] style Struct fill:#e7f3ff style Aero fill:#d4edda style Nuclear fill:#fff3cd style Auto fill:#f8d7da style Tool fill:#e2d5f1

5.1.1 Aerospace Applications

The aerospace industry represents one of the most promising sectors for HEA adoption due to the demanding requirements for high-temperature performance, specific strength, and environmental resistance.

High-Temperature Structural Materials

Refractory HEAs offer the potential to replace or extend the operating range of nickel superalloys in turbine engines:

Operating temperature: Target >1100°C (vs. ~1050°C for current Ni superalloys)
Creep resistance: Sluggish diffusion enhances high-T stability
Oxidation resistance: Requires Al/Cr additions or coatings

Application	HEA Type	Key Properties Required	Status
Turbine blades	Refractory HEAs	High-T strength, creep, oxidation	Research
Combustor liners	AlCoCrFeNi variants	Oxidation, thermal cycling	Prototype
Cryogenic tanks	CoCrFeMnNi	Low-T toughness, H₂ resistance	Evaluation
Fasteners	Al-containing HEAs	Specific strength, corrosion	Commercial

Cryogenic Applications

The exceptional cryogenic properties of Cantor-type alloys make them candidates for:

Liquid hydrogen storage tanks for aerospace vehicles
LNG (liquefied natural gas) infrastructure components
Superconducting magnet structural supports
Space exploration equipment operating in extreme cold

5.1.2 Nuclear Applications

HEAs and HECs are being investigated for nuclear applications due to their radiation tolerance and high-temperature stability.

Radiation Tolerance of HEAs

HEAs show enhanced radiation tolerance compared to conventional alloys:

Reduced swelling: Sluggish diffusion suppresses void formation
Defect recombination: Chemical complexity promotes recombination
Phase stability: Entropy stabilization resists radiation-induced precipitation
Amorphization resistance: Random structure tolerates displacement damage

import numpy as np
import matplotlib.pyplot as plt

# Radiation damage comparison: HEA vs conventional alloys

def radiation_swelling_model(dpa, material_factor, temperature_factor):
    """
    Simplified radiation swelling model.

    Parameters:
    -----------
    dpa : array
        Displacements per atom
    material_factor : float
        Material-dependent swelling resistance (lower = better)
    temperature_factor : float
        Temperature-dependent factor

    Returns:
    --------
    array : Volumetric swelling (%)
    """
    # Incubation period followed by linear swelling
    incubation_dpa = 5 / material_factor
    swelling = np.where(dpa > incubation_dpa,
                        material_factor * temperature_factor * (dpa - incubation_dpa),
                        0)
    return swelling

dpa = np.linspace(0, 100, 100)

materials = {
    '316 SS': {'factor': 0.8, 'T_factor': 1.0},
    'Ferritic Steel': {'factor': 0.4, 'T_factor': 0.8},
    'Ni Superalloy': {'factor': 0.6, 'T_factor': 0.9},
    'CoCrFeMnNi HEA': {'factor': 0.2, 'T_factor': 0.7},
    'HfNbTaTiZr RHEA': {'factor': 0.15, 'T_factor': 0.6},
}

fig, axes = plt.subplots(1, 2, figsize=(14, 5))

# Swelling comparison
ax1 = axes[0]
for name, params in materials.items():
    swelling = radiation_swelling_model(dpa, params['factor'], params['T_factor'])
    ax1.plot(dpa, swelling, linewidth=2, label=name)

ax1.set_xlabel('Radiation Dose (dpa)', fontsize=12)
ax1.set_ylabel('Volumetric Swelling (%)', fontsize=12)
ax1.set_title('Radiation Swelling Resistance Comparison\n(Schematic at 500°C)', fontsize=14)
ax1.legend(loc='upper left')
ax1.grid(True, alpha=0.3)
ax1.set_xlim(0, 100)
ax1.set_ylim(0, 30)

# Defect density comparison
ax2 = axes[1]
defect_data = {
    '316 SS': [1.0, 0.8, 0.7, 0.6],
    'CoCrFeMnNi': [0.6, 0.4, 0.3, 0.25],
    'HfNbTaTiZr': [0.5, 0.3, 0.2, 0.15],
}
dpa_points = [10, 25, 50, 100]

x = np.arange(len(dpa_points))
width = 0.25

for i, (name, defects) in enumerate(defect_data.items()):
    ax2.bar(x + i*width, defects, width, label=name)

ax2.set_xticks(x + width)
ax2.set_xticklabels([f'{d} dpa' for d in dpa_points])
ax2.set_ylabel('Relative Defect Density', fontsize=12)
ax2.set_title('Residual Defect Density After Irradiation', fontsize=14)
ax2.legend()
ax2.grid(axis='y', alpha=0.3)

plt.tight_layout()
plt.show()

print("=== HEA Advantages for Nuclear Applications ===")
print("• Reduced void swelling (factor of 2-5 improvement)")
print("• Enhanced defect recombination (lower residual damage)")
print("• Phase stability under irradiation")
print("• Potential for accident-tolerant fuel cladding")

5.2 Energy Applications

5.2.1 Battery Applications

High-entropy oxides have shown promising performance as electrode materials in lithium-ion and next-generation batteries.

HEO Battery Electrodes

Rock-salt HEOs (e.g., (MgCoNiCuZn)O) function as conversion-type anodes:

\[ \text{HEO} + 2n\text{Li}^+ + 2n\text{e}^- \rightarrow n\text{Li}_2\text{O} + \text{HEA} \]

Benefits include:

High theoretical capacity (>600 mAh/g)
Excellent cycling stability (>1000 cycles demonstrated)
Entropy stabilization prevents phase segregation during cycling

5.2.2 Hydrogen Storage and Fuel Cells

Application	HEM Type	Mechanism	Performance
H₂ storage	TiZrNbHfTa HEA	Metal hydride formation	~2 wt% H₂, good kinetics
SOFC cathode	Perovskite HEO	O²⁻ transport, ORR catalysis	Enhanced ionic conductivity
SOFC interconnect	FeCoCrNiMn HEA	Electrical conduction, oxidation resistance	Stable ASR, CTE match
PEM catalyst	PtPdRhIrRu HEA NPs	HER/ORR catalysis	Reduced Pt loading

5.3 Catalytic Applications

High-entropy materials offer unique advantages as catalysts due to their diverse active sites, tunable adsorption energies, and synergistic effects.

flowchart LR subgraph Reactions HER[Hydrogen Evolution
2H⁺ + 2e⁻ → H₂] OER[Oxygen Evolution
2H₂O → O₂ + 4H⁺ + 4e⁻] ORR[Oxygen Reduction
O₂ + 4H⁺ + 4e⁻ → 2H₂O] CO2RR[CO₂ Reduction
CO₂ → CO, CH₄, etc.] end subgraph HEM Catalysts HEAA[HEA Nanoparticles] HEOC[Spinel HE Oxides] HECF[HE Perovskites] end HEAA --> HER HEAA --> ORR HEOC --> OER HECF --> CO2RR style HER fill:#d4edda style OER fill:#fff3cd style ORR fill:#f8d7da style CO2RR fill:#e2d5f1

5.3.1 Electrocatalysis

import numpy as np
import matplotlib.pyplot as plt

# Volcano plot concept for HEA catalysts

def volcano_activity(delta_G_H, exchange_current_0=1e-3, beta=0.5):
    """
    Calculate HER activity based on hydrogen adsorption free energy.

    Volcano relationship: activity is maximized when ΔG_H ≈ 0

    Parameters:
    -----------
    delta_G_H : float or array
        Hydrogen adsorption free energy (eV)
    exchange_current_0 : float
        Pre-exponential factor
    beta : float
        Symmetry factor

    Returns:
    --------
    float or array : Relative exchange current density
    """
    # Simplified Sabatier-type volcano
    k_B_T = 0.026  # eV at 300 K
    activity = exchange_current_0 * np.exp(-np.abs(delta_G_H) / k_B_T)
    return activity

# Catalysts and their approximate ΔG_H values
catalysts = {
    'Pt': -0.09,
    'Pd': -0.15,
    'Ni': -0.25,
    'Co': -0.30,
    'Fe': -0.45,
    'Mo': 0.10,
    'W': 0.22,
    'Pt-Ru-Ir-Pd-Au HEA': -0.05,
    'FeCoNiCuPt HEA': -0.08,
    'MoWVNbTa HEA': 0.03,
}

fig, axes = plt.subplots(1, 2, figsize=(14, 6))

# Volcano plot
ax1 = axes[0]
delta_G_range = np.linspace(-0.6, 0.4, 100)
activity_range = volcano_activity(delta_G_range)

ax1.plot(delta_G_range, np.log10(activity_range), 'k-', linewidth=2, alpha=0.5)

colors = {'HEA': '#e74c3c', 'Pure': '#3498db'}
for name, dG in catalysts.items():
    is_hea = 'HEA' in name
    activity = volcano_activity(dG)
    ax1.scatter(dG, np.log10(activity), s=150,
                c=colors['HEA'] if is_hea else colors['Pure'],
                edgecolors='black', linewidth=1.5, zorder=5)
    offset = (0.02, 0.3) if not is_hea else (0.02, -0.5)
    ax1.annotate(name.split(' HEA')[0] if is_hea else name,
                 (dG, np.log10(activity)),
                 textcoords="offset points", xytext=offset, fontsize=9)

ax1.axvline(x=0, color='gray', linestyle='--', alpha=0.5)
ax1.set_xlabel('ΔG_H (eV)', fontsize=12)
ax1.set_ylabel('log₁₀(Exchange Current Density)', fontsize=12)
ax1.set_title('HER Volcano Plot: HEA vs Pure Metal Catalysts', fontsize=14)
ax1.set_xlim(-0.6, 0.4)
ax1.grid(True, alpha=0.3)

# Add legend manually
from matplotlib.lines import Line2D
legend_elements = [
    Line2D([0], [0], marker='o', color='w', markerfacecolor='#3498db',
           markersize=10, label='Pure metals'),
    Line2D([0], [0], marker='o', color='w', markerfacecolor='#e74c3c',
           markersize=10, label='HEA catalysts'),
]
ax1.legend(handles=legend_elements, loc='lower right')

# Multi-element synergy illustration
ax2 = axes[1]
elements = ['Pt', 'Pd', 'Ru', 'Ir', 'Au']
individual_activities = [100, 85, 70, 95, 20]
he_activity = 120  # Synergistic enhancement

x_pos = np.arange(len(elements) + 1)
labels = elements + ['HEA\n(synergy)']
values = individual_activities + [he_activity]
colors_bar = ['#3498db']*5 + ['#e74c3c']

bars = ax2.bar(x_pos, values, color=colors_bar, edgecolor='black', linewidth=1)
ax2.axhline(y=np.mean(individual_activities), color='orange', linestyle='--',
            linewidth=2, label=f'Average: {np.mean(individual_activities):.0f}')

ax2.set_xticks(x_pos)
ax2.set_xticklabels(labels)
ax2.set_ylabel('Relative HER Activity (%)', fontsize=12)
ax2.set_title('Cocktail Effect in HEA Electrocatalysis', fontsize=14)
ax2.legend()
ax2.grid(axis='y', alpha=0.3)

for bar, val in zip(bars, values):
    ax2.text(bar.get_x() + bar.get_width()/2, val + 2,
             f'{val}', ha='center', fontsize=10)

plt.tight_layout()
plt.show()

print("=== HEA Catalysis Advantages ===")
print("• Diverse active site environments")
print("• Tunable adsorption energies through composition")
print("• Synergistic electronic effects (cocktail effect)")
print("• High stability against sintering and segregation")

5.3.2 Thermocatalysis

Beyond electrocatalysis, HEMs are being explored for traditional thermal catalysis:

Ammonia synthesis: HEA nanoparticles for N₂ activation
Methane reforming: Spinel HEOs for syngas production
Oxidation reactions: Perovskite HEOs for VOC destruction
Hydrogenation: Precious metal HEA for selective reduction

5.4 Protective Coatings

5.4.1 Thermal Barrier Coatings (TBCs)

HEO Thermal Barrier Coatings

High-entropy fluorite and pyrochlore oxides offer advantages over traditional YSZ:

Lower thermal conductivity: Enhanced phonon scattering
Higher phase stability: Entropy stabilization at high T
CMAS resistance: Some compositions resist silicate attack
Thermal cycling: Matched CTE with bond coat

import numpy as np
import matplotlib.pyplot as plt

# Thermal barrier coating performance comparison

coatings = {
    '8YSZ': {
        'kappa': 2.3,  # W/m·K
        'CTE': 10.5,   # 10^-6/K
        'T_max': 1200, # °C
        'CMAS': 'Poor'
    },
    'Gd₂Zr₂O₇': {
        'kappa': 1.6,
        'CTE': 10.0,
        'T_max': 1300,
        'CMAS': 'Moderate'
    },
    'La₂Zr₂O₇': {
        'kappa': 1.8,
        'CTE': 9.5,
        'T_max': 1250,
        'CMAS': 'Good'
    },
    '(La,Gd,Y,Nd,Sm)₂Zr₂O₇\nHE Pyrochlore': {
        'kappa': 1.1,
        'CTE': 10.2,
        'T_max': 1400,
        'CMAS': 'Excellent'
    },
    '(Hf,Zr,Ce,Y,La)O₂\nHE Fluorite': {
        'kappa': 1.3,
        'CTE': 11.0,
        'T_max': 1350,
        'CMAS': 'Good'
    },
}

fig, axes = plt.subplots(1, 3, figsize=(15, 5))

# Thermal conductivity
ax1 = axes[0]
names = list(coatings.keys())
kappa_values = [coatings[n]['kappa'] for n in names]
colors = ['#3498db', '#3498db', '#3498db', '#e74c3c', '#e74c3c']

bars1 = ax1.barh(names, kappa_values, color=colors, edgecolor='black', linewidth=1)
ax1.set_xlabel('Thermal Conductivity (W/m·K)', fontsize=11)
ax1.set_title('Thermal Conductivity\n(Lower is Better)', fontsize=12)
ax1.set_xlim(0, 3)
ax1.grid(axis='x', alpha=0.3)

for bar, val in zip(bars1, kappa_values):
    ax1.text(val + 0.05, bar.get_y() + bar.get_height()/2,
             f'{val:.1f}', va='center', fontsize=10)

# Maximum temperature
ax2 = axes[1]
T_values = [coatings[n]['T_max'] for n in names]

bars2 = ax2.barh(names, T_values, color=colors, edgecolor='black', linewidth=1)
ax2.set_xlabel('Maximum Operating Temperature (°C)', fontsize=11)
ax2.set_title('Temperature Capability\n(Higher is Better)', fontsize=12)
ax2.set_xlim(1100, 1500)
ax2.grid(axis='x', alpha=0.3)

for bar, val in zip(bars2, T_values):
    ax2.text(val + 5, bar.get_y() + bar.get_height()/2,
             f'{val}', va='center', fontsize=10)

# Radar chart for overall performance
ax3 = axes[2]
ax3.axis('off')
ax3.text(0.5, 0.95, 'CMAS Resistance Rating', transform=ax3.transAxes,
         fontsize=12, fontweight='bold', ha='center')

cmas_map = {'Poor': 1, 'Moderate': 2, 'Good': 3, 'Excellent': 4}
y_pos = 0.8
for name, props in coatings.items():
    rating = cmas_map[props['CMAS']]
    color = '#e74c3c' if 'HE' in name else '#3498db'
    ax3.text(0.1, y_pos, name.split('\n')[0][:20], transform=ax3.transAxes, fontsize=9)
    ax3.barh([y_pos], [rating], height=0.1, color=color, alpha=0.7)
    ax3.text(0.55 + rating*0.1, y_pos, props['CMAS'], transform=ax3.transAxes,
             fontsize=9, va='center')
    y_pos -= 0.15

ax3.set_xlim(0, 1)
ax3.set_ylim(0, 1)

plt.tight_layout()
plt.show()

print("=== HE TBC Advantages ===")
print("• 30-50% lower thermal conductivity than YSZ")
print("• Extended temperature capability (+100-200°C)")
print("• Enhanced phase stability under thermal cycling")
print("• Improved CMAS attack resistance in many compositions")

5.4.2 Wear and Corrosion Resistant Coatings

HEA coatings deposited by thermal spray, laser cladding, or PVD provide excellent wear and corrosion protection:

Coating Type	Deposition Method	Hardness (GPa)	Application
AlCoCrFeNi	Laser cladding	5-7	Industrial tooling
AlTiVCrNb	Magnetron sputtering	8-12	Cutting tools
(AlCrTiVZr)N	Cathodic arc	25-35	Hard coatings
CoCrFeMnNi	Cold spray	3-4	Corrosion protection

5.5 Biomedical Applications

HEAs are being explored for biomedical implants due to their tunable properties and potential for enhanced biocompatibility.

Biocompatible HEAs

HEAs for biomedical applications typically contain biocompatible elements:

Primary elements: Ti, Zr, Nb, Ta, Hf (excellent biocompatibility)
Avoid: Ni, Co, Cr (potential allergenic/toxic effects)
β-Ti type: TiZrNbTaMo with low elastic modulus (~50-80 GPa)

Design Criteria for Bio-HEAs

Low elastic modulus: Match bone (10-30 GPa) to prevent stress shielding
Corrosion resistance: Stable passive oxide film in body fluids
Non-toxic ion release: All constituent elements must be biocompatible
Osseointegration: Surface properties promoting bone bonding
Mechanical strength: Sufficient for load-bearing applications

5.6 Future Directions and Challenges

5.6.1 Machine Learning-Accelerated Discovery

flowchart TB ML[ML-Accelerated HEM Discovery] --> Data[Data Collection] ML --> Model[Model Development] ML --> Screen[High-Throughput Screening] ML --> Validate[Experimental Validation] Data --> D1[Literature mining] Data --> D2[High-throughput experiments] Data --> D3[DFT calculations] Model --> M1[Property prediction] Model --> M2[Phase prediction] Model --> M3[Composition optimization] Screen --> S1[Vast composition space] Screen --> S2[Multi-objective optimization] Validate --> V1[Targeted synthesis] Validate --> V2[Characterization] Validate --> V3[Feedback to model] style ML fill:#e7f3ff style Model fill:#d4edda style Screen fill:#fff3cd

import numpy as np
import matplotlib.pyplot as plt

# Demonstration of ML-accelerated HEA discovery concept

def compositional_space_size(n_elements, n_compositions_per_element=20):
    """
    Estimate size of compositional space for HEAs.

    Parameters:
    -----------
    n_elements : int
        Number of available elements to choose from
    n_compositions_per_element : int
        Number of composition increments per element

    Returns:
    --------
    dict : Space size estimates
    """
    from math import comb

    results = {}

    # 5-element equiatomic combinations
    n_5_equi = comb(n_elements, 5)

    # 5-element with composition variation (simplified)
    n_5_comp = comb(n_elements, 5) * (n_compositions_per_element ** 4)

    # Including 4-7 element systems
    total = sum(comb(n_elements, k) * (n_compositions_per_element ** (k-1))
                for k in range(4, 8) if k <= n_elements)

    return {
        '5-element equiatomic': n_5_equi,
        '5-element compositional': n_5_comp,
        'Total (4-7 elements)': total
    }

# Periodic table has ~40 elements commonly used in HEAs
n_elements = 40

spaces = compositional_space_size(n_elements)

print("=== HEA Compositional Space Size ===")
print(f"Available elements: {n_elements}")
for name, size in spaces.items():
    print(f"  {name}: {size:,.0f} compositions")

# ML screening efficiency
fig, axes = plt.subplots(1, 2, figsize=(14, 5))

# Traditional vs ML discovery rate
ax1 = axes[0]
years = np.arange(2004, 2030)

# Traditional: exponential growth in publications, linear in unique compositions
traditional_pubs = 10 * np.exp(0.25 * (years - 2004))
traditional_compositions = 50 + 200 * (years - 2004)

# ML-accelerated (starting 2020): much faster discovery
ml_start = 2020
ml_compositions = np.where(years >= ml_start,
                            traditional_compositions[years == ml_start][0] +
                            2000 * (years - ml_start),
                            traditional_compositions)

ax1.plot(years[years < 2025], traditional_compositions[years < 2025],
         'b-', linewidth=2, label='Traditional (experimental)')
ax1.plot(years[years >= 2020], ml_compositions[years >= 2020],
         'r--', linewidth=2, label='ML-accelerated (projected)')

ax1.axvline(x=2020, color='gray', linestyle=':', alpha=0.7)
ax1.set_xlabel('Year', fontsize=12)
ax1.set_ylabel('Unique HEA Compositions Characterized', fontsize=12)
ax1.set_title('HEA Discovery Rate: Traditional vs ML-Accelerated', fontsize=14)
ax1.legend()
ax1.grid(True, alpha=0.3)
ax1.set_xlim(2004, 2028)

# Property prediction accuracy improvement
ax2 = axes[1]
training_data = [100, 500, 1000, 5000, 10000, 50000]
accuracy_phase = [0.65, 0.78, 0.85, 0.92, 0.95, 0.97]
accuracy_hardness = [0.50, 0.65, 0.75, 0.85, 0.90, 0.93]
accuracy_yield = [0.45, 0.60, 0.70, 0.82, 0.88, 0.91]

ax2.semilogx(training_data, np.array(accuracy_phase)*100, 'o-',
             linewidth=2, markersize=8, label='Phase prediction')
ax2.semilogx(training_data, np.array(accuracy_hardness)*100, 's-',
             linewidth=2, markersize=8, label='Hardness prediction')
ax2.semilogx(training_data, np.array(accuracy_yield)*100, '^-',
             linewidth=2, markersize=8, label='Yield strength prediction')

ax2.axhline(y=90, color='green', linestyle='--', alpha=0.7, label='90% accuracy target')

ax2.set_xlabel('Training Dataset Size', fontsize=12)
ax2.set_ylabel('Prediction Accuracy (%)', fontsize=12)
ax2.set_title('ML Model Accuracy vs Training Data', fontsize=14)
ax2.legend()
ax2.grid(True, alpha=0.3)
ax2.set_ylim(40, 100)

plt.tight_layout()
plt.show()

5.6.2 Sustainability Considerations

As HEMs move toward commercialization, sustainability becomes increasingly important:

Sustainable HEM Development

Critical element reduction: Minimize use of Co, W, Ta, rare earths where possible
Recyclability: Design for end-of-life recovery and reprocessing
Energy-efficient processing: Optimize synthesis to reduce energy consumption
Life cycle assessment: Evaluate total environmental impact
Earth-abundant alternatives: Fe, Mn, Al, Si-based compositions

5.6.3 Remaining Challenges

Challenge	Current Status	Path Forward
Scale-up	Lab-scale predominantly	Industrial melting, powder production
Cost	Higher than conventional alloys	Process optimization, recycling
Standardization	No established standards	Industry working groups, testing protocols
Database development	Fragmented data	Centralized databases, FAIR principles
Oxidation at high T	Many RHEAs suffer poor oxidation	Protective coatings, Al/Cr additions
RT ductility (RHEAs)	Often brittle at room temperature	Composition optimization, processing

5.7 Summary

Key Concepts

Structural applications include aerospace (high-T turbine, cryogenic tanks), nuclear (radiation-tolerant cladding), and automotive (engine components)
Energy applications span Li-ion battery electrodes (HEO anodes), hydrogen storage (HEA hydrides), and fuel cell components
Catalytic applications leverage diverse active sites for HER, OER, ORR, and CO₂ reduction with synergistic effects
Protective coatings using HE oxides offer lower thermal conductivity and better CMAS resistance for TBCs; HEA coatings provide wear and corrosion protection
Biomedical applications focus on Ti-Zr-Nb-Ta-based HEAs with low modulus and excellent biocompatibility
ML-accelerated discovery is essential for exploring the vast compositional space (~10¹⁰+ compositions)
Sustainability considerations include critical element reduction, recyclability, and life cycle assessment
Key challenges remain in scale-up, cost, standardization, and property optimization for specific applications

5.8 Exercises

Conceptual Questions

Why are refractory HEAs promising for next-generation turbine engines? What are the main barriers to their adoption?
Explain how the entropy stabilization effect benefits HEO battery electrodes during cycling.
How does the "cocktail effect" manifest in HEA electrocatalysts for the HER?
What makes high-entropy pyrochlore oxides better TBCs than traditional YSZ?
Why must biomedical HEAs avoid Ni and Co? What elements should be preferred?

Quantitative Problems

Calculate the number of unique 5-element equiatomic HEA compositions possible from a palette of 30 elements. If each takes 1 day to synthesize and characterize, how many years would exhaustive exploration require?
A HE-TBC has thermal conductivity of 1.2 W/m·K compared to 2.3 W/m·K for YSZ. For a 200 μm coating with a 1200°C surface and 1000°C substrate, calculate the temperature drop across each coating.
An HEA electrocatalyst shows overpotential of 50 mV for HER at 10 mA/cm². If conventional Pt shows 30 mV, but the HEA uses 80% less Pt, calculate the Pt-normalized activity.

Computational Exercises

Design a simple ML workflow (using sklearn) to predict HEA phase from composition features. Use the screening parameters (VEC, delta, omega) as input features.
Create a Pareto optimization visualization showing the trade-off between thermal conductivity (minimize) and CTE match to Ni superalloy (target: 13×10⁻⁶/K) for a set of HE-TBC compositions.