Chapter 1: Fundamentals and Core Effects - Introduction to High-Entropy Materials

Learning Objectives

Understand the historical development and definition of high-entropy materials
Calculate configurational entropy for multi-component systems
Explain the four core effects: high entropy, lattice distortion, sluggish diffusion, and cocktail effects
Identify common crystal structures (FCC, BCC, HCP) in high-entropy alloys
Apply phase formation rules using VEC and atomic size difference parameters

1.1 Introduction: What are High-Entropy Materials?

For over a century, materials science has been dominated by alloys based on one or two principal elements. Steel is primarily iron, aluminum alloys are mostly aluminum, and nickel superalloys contain predominantly nickel. This conventional approach, while successful, explores only a tiny fraction of the vast compositional space available in multi-component systems.

High-entropy materials (HEMs) represent a revolutionary departure from this paradigm. These materials contain multiple principal elements (typically five or more) in near-equimolar ratios, creating systems with exceptionally high configurational entropy. This simple change in composition philosophy has unlocked materials with remarkable combinations of properties previously thought impossible.

High-Entropy Materials (HEMs)

Materials composed of multiple principal elements (typically 5 or more) in concentrations between 5 and 35 at.%, resulting in high configurational entropy. The term encompasses high-entropy alloys (HEAs), high-entropy ceramics (HECs), high-entropy oxides (HEOs), and other multi-principal element materials.

1.1.1 Historical Development

The field of high-entropy materials was independently pioneered in 2004 by two research groups:

Jien-Wei Yeh (Taiwan): Coined the term "high-entropy alloys" and introduced the concept of configurational entropy stabilization
Brian Cantor (UK): Developed the equiatomic CoCrFeMnNi alloy, now known as the "Cantor alloy"

The Cantor Alloy

The equiatomic CoCrFeMnNi alloy discovered by Cantor's group forms a single-phase FCC solid solution despite containing five elements. This remarkable stability challenged conventional alloy design principles and sparked worldwide research interest. The alloy exhibits exceptional ductility and fracture toughness, particularly at cryogenic temperatures.

timeline title High-Entropy Materials Timeline 2004 : First HEA papers by Yeh and Cantor 2010 : First refractory HEAs (Senkov) 2015 : First high-entropy ceramics 2018 : First high-entropy oxides for energy storage 2020 : Machine learning for HEA design 2024 : Commercial HEA applications emerging

1.1.2 Definition Criteria

High-entropy materials can be defined using several criteria:

Criterion	Definition	Example
Composition-based	5+ principal elements, each 5-35 at.%	CoCrFeMnNi (equiatomic)
Entropy-based	\(\Delta S_{conf} \geq 1.5R\)	Any 5+ element equiatomic system
Effect-based	Properties dominated by high-entropy effects	Single-phase solid solutions with unique properties

1.2 Configurational Entropy

1.2.1 Thermodynamic Foundation

The stability of any material system is governed by the Gibbs free energy:

\[ \Delta G_{mix} = \Delta H_{mix} - T \Delta S_{mix} \]

where \(\Delta H_{mix}\) is the enthalpy of mixing, \(T\) is temperature, and \(\Delta S_{mix}\) is the entropy of mixing. For solid solutions, the entropy of mixing consists primarily of configurational entropy arising from the random arrangement of atoms on lattice sites.

Configurational Entropy Formula

For an ideal n-component random solid solution, the configurational entropy is given by:

\[ \Delta S_{conf} = -R \sum_{i=1}^{n} x_i \ln x_i \]

where \(R = 8.314\) J/(mol·K) is the gas constant and \(x_i\) is the mole fraction of element \(i\).

1.2.2 Entropy Classifications

Materials can be classified by their configurational entropy:

Category	Entropy Range	Typical Composition
Low entropy	\(\Delta S_{conf} < 1.0R\)	Conventional alloys (1-2 principal elements)
Medium entropy	\(1.0R \leq \Delta S_{conf} < 1.5R\)	3-4 principal elements
High entropy	\(\Delta S_{conf} \geq 1.5R\)	5+ principal elements

Why 1.5R?

The threshold of \(1.5R\) corresponds to the entropy of an equiatomic 5-component system: \(\Delta S_{conf} = -5 \times (0.2 \ln 0.2) \times R = 1.61R\). This value is significant because it can overcome typical mixing enthalpies (\(\Delta H_{mix} \approx\) 5-15 kJ/mol) at moderate temperatures, promoting single-phase solid solution formation.

1.2.3 Python: Calculating Configurational Entropy

import numpy as np
import matplotlib.pyplot as plt

def configurational_entropy(compositions):
    """
    Calculate configurational entropy for a multi-component system.

    Parameters:
    -----------
    compositions : array-like
        Mole fractions of each component (must sum to 1)

    Returns:
    --------
    float : Configurational entropy in units of R (gas constant)
    """
    x = np.array(compositions)

    # Validate input
    if not np.isclose(np.sum(x), 1.0, rtol=1e-5):
        raise ValueError("Compositions must sum to 1")

    # Filter out zero values to avoid log(0)
    x_nonzero = x[x > 0]

    # Calculate entropy: -sum(x_i * ln(x_i))
    entropy = -np.sum(x_nonzero * np.log(x_nonzero))

    return entropy

# Example: Equiatomic systems with different numbers of components
n_elements = range(2, 11)
entropies = []

for n in n_elements:
    equiatomic = np.ones(n) / n  # Equiatomic composition
    S = configurational_entropy(equiatomic)
    entropies.append(S)
    print(f"{n} elements: S_conf = {S:.3f}R = {S * 8.314:.2f} J/(mol·K)")

# Visualization
fig, ax = plt.subplots(figsize=(10, 6))
ax.bar(n_elements, entropies, color='steelblue', edgecolor='navy', alpha=0.8)
ax.axhline(y=1.0, color='orange', linestyle='--', linewidth=2, label='Medium entropy threshold (1.0R)')
ax.axhline(y=1.5, color='red', linestyle='--', linewidth=2, label='High entropy threshold (1.5R)')

ax.set_xlabel('Number of Elements', fontsize=12)
ax.set_ylabel('Configurational Entropy (R)', fontsize=12)
ax.set_title('Configurational Entropy of Equiatomic Systems', fontsize=14)
ax.legend(loc='lower right')
ax.set_xticks(list(n_elements))
ax.set_ylim(0, 2.5)
ax.grid(axis='y', alpha=0.3)

plt.tight_layout()
plt.show()

# Output:
# 2 elements: S_conf = 0.693R = 5.76 J/(mol·K)
# 3 elements: S_conf = 1.099R = 9.13 J/(mol·K)
# 4 elements: S_conf = 1.386R = 11.53 J/(mol·K)
# 5 elements: S_conf = 1.609R = 13.38 J/(mol·K)
# ...

1.3 The Four Core Effects

Yeh proposed that the unique properties of high-entropy materials arise from four core effects that distinguish them from conventional alloys:

flowchart TB HEM[High-Entropy Materials] --> HE[High-Entropy Effect] HEM --> LD[Lattice Distortion Effect] HEM --> SD[Sluggish Diffusion Effect] HEM --> CE[Cocktail Effect] HE --> |Thermodynamic| HE1[Phase stability
Single-phase formation] LD --> |Structural| LD1[Solid solution hardening
Property anomalies] SD --> |Kinetic| SD1[Creep resistance
Thermal stability] CE --> |Synergistic| CE1[Unexpected properties
Property optimization] style HEM fill:#e7f3ff style HE fill:#d4edda style LD fill:#fff3cd style SD fill:#f8d7da style CE fill:#e2d5f1

1.3.1 High-Entropy Effect

High-Entropy Effect

The contribution of high configurational entropy to stabilize simple solid solution phases over intermetallic compounds at elevated temperatures. This is a thermodynamic effect that becomes more significant as temperature increases.

The high-entropy effect can be understood through Gibbs free energy competition. For a solid solution to be stable over intermetallic compounds:

\[ \Delta G_{SS} < \Delta G_{IM} \]

Since intermetallic compounds typically have more negative \(\Delta H_{mix}\) but lower entropy (ordered structures), high configurational entropy favors solid solutions at high temperatures through the \(T\Delta S\) term.

Temperature Dependence

The high-entropy effect is temperature-dependent. At high temperatures (\(T > 0.6-0.7 T_m\)), the entropy term dominates, stabilizing solid solutions. At lower temperatures, kinetic barriers may preserve metastable solid solutions, or decomposition into intermetallic compounds may occur during slow cooling.

1.3.2 Lattice Distortion Effect

Lattice Distortion Effect

The severe local lattice strain resulting from the random distribution of atoms with different sizes on the same crystallographic sites. This creates a distorted lattice that affects mechanical, thermal, and electrical properties.

In conventional dilute solid solutions, solute atoms create localized strain fields. In HEMs, every atom experiences strain from its dissimilar neighbors, resulting in:

Solid solution strengthening: Enhanced hardness and yield strength
Phonon scattering: Reduced thermal conductivity
Electron scattering: Increased electrical resistivity
Lattice friction: Higher Peierls stress for dislocation motion

import numpy as np
import matplotlib.pyplot as plt

def atomic_size_difference(compositions, radii):
    """
    Calculate atomic size difference parameter (delta).

    Parameters:
    -----------
    compositions : array-like
        Mole fractions of each component
    radii : array-like
        Atomic radii of each element in Angstroms

    Returns:
    --------
    float : Atomic size difference parameter (dimensionless, in %)
    """
    x = np.array(compositions)
    r = np.array(radii)

    # Average atomic radius
    r_avg = np.sum(x * r)

    # Delta parameter (Eq. from Zhang et al.)
    delta = 100 * np.sqrt(np.sum(x * (1 - r/r_avg)**2))

    return delta

# Example: Cantor alloy CoCrFeMnNi
elements = ['Co', 'Cr', 'Fe', 'Mn', 'Ni']
radii = [1.252, 1.249, 1.241, 1.350, 1.246]  # Goldschmidt radii in Angstroms
compositions = [0.2, 0.2, 0.2, 0.2, 0.2]  # Equiatomic

delta = atomic_size_difference(compositions, radii)
print(f"Cantor alloy (CoCrFeMnNi):")
print(f"  Atomic size difference (delta) = {delta:.2f}%")

# Example: Refractory HEA NbMoTaW
elements_ref = ['Nb', 'Mo', 'Ta', 'W']
radii_ref = [1.429, 1.363, 1.430, 1.367]
compositions_ref = [0.25, 0.25, 0.25, 0.25]

delta_ref = atomic_size_difference(compositions_ref, radii_ref)
print(f"\nRefractory HEA (NbMoTaW):")
print(f"  Atomic size difference (delta) = {delta_ref:.2f}%")

# Visualize lattice distortion
fig, axes = plt.subplots(1, 2, figsize=(14, 6))

# Ideal lattice
ax1 = axes[0]
n = 10
x_ideal, y_ideal = np.meshgrid(np.arange(n), np.arange(n))
ax1.scatter(x_ideal.flatten(), y_ideal.flatten(), s=200, c='steelblue',
            edgecolors='navy', linewidth=1.5)
ax1.set_title('Ideal Lattice (Single Element)', fontsize=14)
ax1.set_xlim(-0.5, n-0.5)
ax1.set_ylim(-0.5, n-0.5)
ax1.set_aspect('equal')
ax1.axis('off')

# Distorted HEA lattice
ax2 = axes[1]
np.random.seed(42)
colors = plt.cm.Set1(np.random.randint(0, 5, n*n))
sizes = np.random.uniform(150, 300, n*n)  # Variable sizes
x_distorted = x_ideal.flatten() + np.random.normal(0, 0.1, n*n)
y_distorted = y_ideal.flatten() + np.random.normal(0, 0.1, n*n)
ax2.scatter(x_distorted, y_distorted, s=sizes, c=colors,
            edgecolors='black', linewidth=0.5)
ax2.set_title('Distorted HEA Lattice (5 Elements)', fontsize=14)
ax2.set_xlim(-0.5, n-0.5)
ax2.set_ylim(-0.5, n-0.5)
ax2.set_aspect('equal')
ax2.axis('off')

plt.tight_layout()
plt.show()

1.3.3 Sluggish Diffusion Effect

Sluggish Diffusion Effect

The observation that atomic diffusion in HEMs is slower compared to conventional alloys. This is attributed to the fluctuating lattice potential energy landscape created by the random atomic environment around each lattice site.

Each diffusing atom in an HEM encounters a unique local environment at every lattice site, creating a wide distribution of activation energies for atomic jumps. This leads to:

Enhanced phase stability: Slower kinetics for phase decomposition
Improved creep resistance: Reduced diffusion-controlled deformation
Better oxidation resistance: Slower outward cation diffusion
Coarsening resistance: Slower precipitate growth

Controversy on Sluggish Diffusion

Recent studies have challenged the universality of sluggish diffusion. While some HEAs indeed show slower diffusion than expected, others exhibit normal or even enhanced diffusion rates. The effect appears to be composition-dependent and may not be a universal feature of all HEMs. Current research suggests the effect is most pronounced when there are large differences in melting points among constituent elements.

1.3.4 Cocktail Effect

Cocktail Effect

The synergistic combination of properties that results from mixing multiple principal elements, producing characteristics that cannot be predicted by simple rule-of-mixtures averaging. The whole becomes greater (or different) than the sum of its parts.

The cocktail effect encompasses several phenomena:

Synergistic strengthening: Multiple strengthening mechanisms operating simultaneously
Property optimization: Balancing opposing properties (e.g., strength vs. ductility)
Emergent properties: New functionalities not present in constituent elements
Tunable properties: Wide compositional space for property optimization

import numpy as np
import matplotlib.pyplot as plt

# Demonstrate cocktail effect: properties vs. rule-of-mixtures prediction

# Element properties (hypothetical normalized values)
elements = ['A', 'B', 'C', 'D', 'E']
hardness = np.array([200, 150, 300, 250, 180])  # HV
density = np.array([7.8, 8.9, 7.2, 8.4, 8.1])   # g/cm³

# Equiatomic composition
x = np.ones(5) / 5

# Rule of mixtures prediction
hardness_rom = np.sum(x * hardness)
density_rom = np.sum(x * density)

# Actual HEA properties (hypothetical - showing cocktail effect)
hardness_actual = hardness_rom * 1.4  # 40% enhancement
density_actual = density_rom * 0.98   # Slight reduction

print("=== Cocktail Effect Demonstration ===")
print(f"\nRule of Mixtures Prediction:")
print(f"  Hardness: {hardness_rom:.1f} HV")
print(f"  Density: {density_rom:.2f} g/cm³")
print(f"\nActual HEA Properties:")
print(f"  Hardness: {hardness_actual:.1f} HV (+{(hardness_actual/hardness_rom-1)*100:.0f}%)")
print(f"  Density: {density_actual:.2f} g/cm³")

# Visualization
fig, axes = plt.subplots(1, 2, figsize=(12, 5))

# Hardness comparison
ax1 = axes[0]
x_pos = np.arange(len(elements) + 2)
labels = elements + ['ROM', 'HEA']
values = list(hardness) + [hardness_rom, hardness_actual]
colors = ['#3498db']*5 + ['#95a5a6', '#e74c3c']
bars = ax1.bar(x_pos, values, color=colors, edgecolor='black', linewidth=1)
ax1.set_xticks(x_pos)
ax1.set_xticklabels(labels)
ax1.set_ylabel('Hardness (HV)', fontsize=12)
ax1.set_title('Cocktail Effect: Hardness Enhancement', fontsize=14)
ax1.axhline(y=hardness_rom, color='gray', linestyle='--', alpha=0.7)

# Add value labels
for bar, val in zip(bars, values):
    ax1.text(bar.get_x() + bar.get_width()/2, bar.get_height() + 5,
             f'{val:.0f}', ha='center', va='bottom', fontsize=10)

# Specific strength comparison
ax2 = axes[1]
specific_strength_rom = hardness_rom / density_rom
specific_strength_actual = hardness_actual / density_actual
categories = ['Rule of Mixtures', 'Actual HEA']
values_ss = [specific_strength_rom, specific_strength_actual]
colors_ss = ['#95a5a6', '#27ae60']
bars_ss = ax2.bar(categories, values_ss, color=colors_ss, edgecolor='black', linewidth=1)
ax2.set_ylabel('Specific Strength (HV·cm³/g)', fontsize=12)
ax2.set_title('Cocktail Effect: Specific Strength', fontsize=14)

for bar, val in zip(bars_ss, values_ss):
    ax2.text(bar.get_x() + bar.get_width()/2, bar.get_height() + 0.5,
             f'{val:.1f}', ha='center', va='bottom', fontsize=12)

plt.tight_layout()
plt.show()

1.4 Crystal Structures in HEAs

Despite containing many elements, high-entropy alloys typically form simple solid solution crystal structures rather than complex intermetallic phases:

Structure	Description	Typical Elements	Example HEAs
FCC	Face-Centered Cubic	Co, Ni, Cu, Al, Mn	CoCrFeMnNi, Al₀.₃CoCrFeNi
BCC	Body-Centered Cubic	W, Mo, Ta, Nb, V, Cr	WMoTaNb, AlCoCrFeNi
HCP	Hexagonal Close-Packed	Ti, Zr, Hf, Co, Re	GdHoLaTbY, CoReRuOs
FCC+BCC	Dual-phase mixture	Mixed element groups	AlₓCoCrFeNi (x > 0.5)

Refractory High-Entropy Alloys

Refractory HEAs based on Group 4, 5, and 6 transition metals (Ti, Zr, Hf, V, Nb, Ta, Cr, Mo, W) typically form BCC structures. These alloys are promising for high-temperature structural applications beyond the capability of nickel superalloys (\(T > 1000°C\)).

1.5 Phase Formation Rules

Predicting whether an HEA will form a single-phase solid solution or multiple phases (including intermetallics) is crucial for alloy design. Several empirical parameters have been developed:

1.5.1 Valence Electron Concentration (VEC)

Valence Electron Concentration

The weighted average of valence electrons per atom:

\[ VEC = \sum_{i=1}^{n} x_i \cdot VEC_i \]

where \(x_i\) is the mole fraction and \(VEC_i\) is the valence electron count of element \(i\).

The VEC parameter correlates with crystal structure formation:

\(VEC \geq 8.0\): FCC phase predominates
\(VEC < 6.87\): BCC phase predominates
\(6.87 \leq VEC < 8.0\): Mixed FCC + BCC phases

1.5.2 Atomic Size Difference Parameter

The atomic size difference (\(\delta\)) quantifies lattice distortion:

\[ \delta = 100\sqrt{\sum_{i=1}^{n} x_i \left(1 - \frac{r_i}{\bar{r}}\right)^2} \]

where \(\bar{r} = \sum x_i r_i\) is the average atomic radius.

\(\delta < 6.6\%\): Single-phase solid solution likely
\(\delta > 6.6\%\): Multiphase or intermetallic formation likely

1.5.3 Omega Parameter

The omega parameter (\(\Omega\)) combines entropy and enthalpy effects:

\[ \Omega = \frac{T_m \Delta S_{mix}}{|\Delta H_{mix}|} \]

where \(T_m\) is the average melting temperature.

\(\Omega > 1.1\): Solid solution formation favored
\(\Omega < 1.1\): Intermetallic compound formation likely

import numpy as np
import matplotlib.pyplot as plt

def calculate_phase_parameters(elements, compositions, vec_values, radii, melting_temps, delta_h_mix):
    """
    Calculate phase formation parameters for HEA design.

    Parameters:
    -----------
    elements : list
        Element symbols
    compositions : array-like
        Mole fractions
    vec_values : array-like
        Valence electron concentrations
    radii : array-like
        Atomic radii in Angstroms
    melting_temps : array-like
        Melting temperatures in Kelvin
    delta_h_mix : float
        Enthalpy of mixing in kJ/mol

    Returns:
    --------
    dict : Phase formation parameters
    """
    x = np.array(compositions)
    vec = np.array(vec_values)
    r = np.array(radii)
    T_m = np.array(melting_temps)

    # VEC
    VEC = np.sum(x * vec)

    # Atomic size difference (delta)
    r_avg = np.sum(x * r)
    delta = 100 * np.sqrt(np.sum(x * (1 - r/r_avg)**2))

    # Configurational entropy
    x_nonzero = x[x > 0]
    S_conf = -8.314 * np.sum(x_nonzero * np.log(x_nonzero))  # J/(mol·K)

    # Average melting temperature
    T_m_avg = np.sum(x * T_m)

    # Omega parameter
    if abs(delta_h_mix) > 0:
        Omega = T_m_avg * S_conf / (1000 * abs(delta_h_mix))  # Convert kJ to J
    else:
        Omega = float('inf')

    return {
        'VEC': VEC,
        'delta': delta,
        'S_conf': S_conf,
        'T_m': T_m_avg,
        'Omega': Omega
    }

# Example: Cantor alloy CoCrFeMnNi
elements = ['Co', 'Cr', 'Fe', 'Mn', 'Ni']
compositions = [0.2, 0.2, 0.2, 0.2, 0.2]
vec_values = [9, 6, 8, 7, 10]
radii = [1.252, 1.249, 1.241, 1.350, 1.246]
melting_temps = [1768, 2180, 1811, 1519, 1728]
delta_h_mix = -4.16  # kJ/mol (from literature)

params = calculate_phase_parameters(elements, compositions, vec_values,
                                     radii, melting_temps, delta_h_mix)

print("=== Cantor Alloy (CoCrFeMnNi) Phase Parameters ===")
print(f"VEC: {params['VEC']:.2f}")
print(f"  -> Predicted structure: {'FCC' if params['VEC'] >= 8 else 'BCC' if params['VEC'] < 6.87 else 'FCC+BCC'}")
print(f"Atomic size difference (delta): {params['delta']:.2f}%")
print(f"  -> {'Single phase likely' if params['delta'] < 6.6 else 'Multi-phase likely'}")
print(f"Configurational entropy: {params['S_conf']:.2f} J/(mol·K) = {params['S_conf']/8.314:.2f}R")
print(f"Average melting temperature: {params['T_m']:.0f} K")
print(f"Omega parameter: {params['Omega']:.2f}")
print(f"  -> {'Solid solution favored' if params['Omega'] > 1.1 else 'Intermetallic likely'}")

# Visualization: Phase formation map
fig, ax = plt.subplots(figsize=(10, 8))

# Generate data for multiple HEAs
hea_data = [
    {'name': 'CoCrFeMnNi', 'delta': 3.27, 'omega': 4.3, 'phase': 'FCC'},
    {'name': 'AlCoCrFeNi', 'delta': 5.61, 'omega': 2.8, 'phase': 'FCC+BCC'},
    {'name': 'WMoTaNb', 'delta': 2.28, 'omega': 6.1, 'phase': 'BCC'},
    {'name': 'AlCrMoNbTi', 'delta': 5.84, 'omega': 1.8, 'phase': 'BCC'},
    {'name': 'CuNiCoZnAl', 'delta': 6.92, 'omega': 0.9, 'phase': 'Multi'},
]

colors = {'FCC': '#3498db', 'BCC': '#e74c3c', 'FCC+BCC': '#9b59b6', 'Multi': '#95a5a6'}
markers = {'FCC': 'o', 'BCC': 's', 'FCC+BCC': 'D', 'Multi': 'X'}

for hea in hea_data:
    ax.scatter(hea['delta'], hea['omega'], c=colors[hea['phase']],
               marker=markers[hea['phase']], s=200, edgecolors='black',
               linewidth=1.5, label=f"{hea['name']} ({hea['phase']})")

# Add decision boundaries
ax.axvline(x=6.6, color='gray', linestyle='--', alpha=0.7, label='delta = 6.6%')
ax.axhline(y=1.1, color='gray', linestyle='-.', alpha=0.7, label='Omega = 1.1')

# Shade regions
ax.fill_between([0, 6.6], [1.1, 1.1], [10, 10], alpha=0.1, color='green',
                label='Solid solution region')
ax.fill_between([6.6, 10], [0, 0], [1.1, 1.1], alpha=0.1, color='red',
                label='Intermetallic region')

ax.set_xlabel('Atomic Size Difference delta (%)', fontsize=12)
ax.set_ylabel('Omega Parameter', fontsize=12)
ax.set_title('HEA Phase Formation Map', fontsize=14)
ax.set_xlim(0, 10)
ax.set_ylim(0, 8)
ax.legend(loc='upper right', fontsize=9)
ax.grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

1.6 Summary

Key Concepts

High-entropy materials contain multiple principal elements (typically 5+) in near-equimolar ratios, creating systems with high configurational entropy
The field was established in 2004 by Yeh and Cantor, with the CoCrFeMnNi "Cantor alloy" as the archetypal example
Configurational entropy is calculated as \(\Delta S_{conf} = -R \sum x_i \ln x_i\), with \(\geq 1.5R\) defining high-entropy systems
The four core effects distinguish HEMs from conventional alloys:
- High-entropy effect: Thermodynamic stabilization of solid solutions
- Lattice distortion effect: Local strain from atomic size mismatch
- Sluggish diffusion effect: Slower kinetics due to fluctuating potential landscape
- Cocktail effect: Synergistic property combinations
HEAs typically form simple FCC, BCC, or HCP structures rather than complex intermetallics
Phase formation rules using VEC, delta, and Omega parameters help predict structure and phase stability

1.7 Exercises

Conceptual Questions

Explain why the high-entropy effect becomes more significant at elevated temperatures.
How does the lattice distortion effect in HEAs differ from solid solution strengthening in dilute alloys?
Why might the sluggish diffusion effect be beneficial for high-temperature structural applications?
Give an example of the cocktail effect leading to a property that exceeds rule-of-mixtures predictions.
Why do most HEAs form simple crystal structures (FCC, BCC) rather than complex intermetallics?

Quantitative Problems

Calculate the configurational entropy for the following alloys and classify them:
- (a) Binary alloy: Cu₅₀Ni₅₀
- (b) Medium entropy: CoCrNi
- (c) High entropy: Al₀.₅CoCrFeNi
For the refractory HEA VNbMoTaW:
- (a) Calculate the VEC (V: 5, Nb: 5, Mo: 6, Ta: 5, W: 6)
- (b) Predict the crystal structure
- (c) Calculate delta using radii (V: 1.316, Nb: 1.429, Mo: 1.363, Ta: 1.430, W: 1.367 Angstroms)
At what temperature does the entropy contribution \(T\Delta S_{conf}\) equal 10 kJ/mol for an equiatomic 6-component system?
Design an HEA composition that would be expected to form a single-phase BCC structure with VEC between 5 and 6.5.

Computational Exercises

Write a Python function to screen compositions for solid solution formation using the combined criteria:
- delta < 6.6%
- Omega > 1.1
- -15 < \(\Delta H_{mix}\) < 5 kJ/mol
Create a phase diagram showing VEC vs. atomic size difference for 20 known HEAs, color-coded by observed phase(s).