Chapter 5: Ternary Phase Diagrams and CALPHAD Method

Ternary Phase Diagrams of Fe-Cr-Ni and Al-Cu-Mg Systems and Principles of Phase Diagram Calculation Using CALPHAD Method

📖 Reading Time: 20-25 minutes 📊 Difficulty: Beginner 💻 Code Examples: 0 📝 Exercises: 0

This chapter covers Ternary Phase Diagrams and CALPHAD Method. You will learn essential concepts and techniques.

Learning Objectives

In this chapter, you will learn how to read ternary phase diagrams , which are essential for practical materials, and the principles of the CALPHAD method (CALculation of PHAse Diagrams) , which forms the foundation of computational materials science. Although ternary systems are more complex than binary systems, many industrial materials such as stainless steels (Fe-Cr-Ni) and high-strength aluminum alloys (Al-Cu-Mg) are ternary or higher-order multicomponent systems.

Skills You Will Acquire in This Chapter

💡 Importance of Ternary Phase Diagrams and CALPHAD Method

Ternary phase diagrams represent the equilibrium state of three-component alloys in three-dimensional space (two composition axes + temperature axis). However, since 3D diagrams are difficult to read, they are typically visualized as 2D sections such as isothermal sections, liquidus projections, and vertical sections. The CALPHAD method is a technique for calculating phase diagrams using thermodynamic databases and is indispensable for predicting phase diagrams in regions where experiments are difficult and for designing new alloys.

1. Composition Representation Using Gibbs Triangle

The composition of a ternary system A-B-C is represented by the Gibbs triangle. Each vertex of the equilateral triangle corresponds to a pure component (A, B, C), and each edge represents a binary system (A-B, B-C, C-A).

1.1 Principles of Triangular Coordinate System

How to Read Composition

The composition of point P (\(x_A, x_B, x_C\)) is read by the following procedure:

  1. The length of the perpendicular from point P to edge BC is proportional to \(x_A\)
  2. The length of the perpendicular from point P to edge CA is proportional to \(x_B\)
  3. The length of the perpendicular from point P to edge AB is proportional to \(x_C\)
  4. \(x_A + x_B + x_C = 1\)(or 100%)
# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0


            <h4>Code Example 1: Composition Representation in Gibbs Triangle</h4>
            <pre><code class="language-python">import numpy as np
import matplotlib.pyplot as plt
from matplotlib.patches import Polygon

def ternary_to_cartesian(a, b, c):
    """Convert ternary coordinates to Cartesian coordinates"""
    x = 0.5 * (2*b + c) / (a + b + c)
    y = (np.sqrt(3)/2) * c / (a + b + c)
    return x, y

# Draw Gibbs triangle
fig, ax = plt.subplots(figsize=(10, 9))

# Triangle vertices (A, B, C)
vertices = np.array([[0, 0], [1, 0], [0.5, np.sqrt(3)/2]])
triangle = Polygon(vertices, fill=False, edgecolor='black', linewidth=2)
ax.add_patch(triangle)

# Vertex labels
ax.text(-0.05, -0.05, 'A (Fe)', fontsize=14, fontweight='bold')
ax.text(1.05, -0.05, 'B (Cr)', fontsize=14, fontweight='bold')
ax.text(0.5, np.sqrt(3)/2 + 0.05, 'C (Ni)', fontsize=14, fontweight='bold', ha='center')

# Draw iso-composition lines (grid)
for i in range(1, 10):
    t = i / 10
    # Iso-composition lines for component A (parallel to BC edge)
    x1, y1 = ternary_to_cartesian(t, 1-t, 0)
    x2, y2 = ternary_to_cartesian(t, 0, 1-t)
    ax.plot([x1, x2], [y1, y2], 'gray', linewidth=0.5, alpha=0.5)

    # Iso-composition lines for component B (parallel to CA edge)
    x1, y1 = ternary_to_cartesian(1-t, t, 0)
    x2, y2 = ternary_to_cartesian(0, t, 1-t)
    ax.plot([x1, x2], [y1, y2], 'gray', linewidth=0.5, alpha=0.5)

    # Iso-composition lines for component C (parallel to AB edge)
    x1, y1 = ternary_to_cartesian(1-t, 0, t)
    x2, y2 = ternary_to_cartesian(0, 1-t, t)
    ax.plot([x1, x2], [y1, y2], 'gray', linewidth=0.5, alpha=0.5)

# Plot sample composition point (Fe-18Cr-8Ni: SUS304 stainless steel)
a_sample, b_sample, c_sample = 0.74, 0.18, 0.08  # Mole fraction
x_sample, y_sample = ternary_to_cartesian(a_sample, b_sample, c_sample)
ax.plot(x_sample, y_sample, 'ro', markersize=10, label='SUS304 (Fe-18Cr-8Ni)')
ax.text(x_sample + 0.03, y_sample, 'SUS304', fontsize=11, color='red')

# Other important composition points
compositions = {
    'SUS316': (0.68, 0.17, 0.12),  # Fe-17Cr-12Ni
    'SUS430': (0.83, 0.17, 0.00),  # Fe-17Cr (Ferritic)
}

for name, (a, b, c) in compositions.items():
    x, y = ternary_to_cartesian(a, b, c)
    ax.plot(x, y, 'bs', markersize=8)
    ax.text(x + 0.03, y, name, fontsize=10, color='blue')

ax.set_xlim(-0.1, 1.1)
ax.set_ylim(-0.1, 1.0)
ax.set_aspect('equal')
ax.axis('off')
ax.legend(loc='upper right')
ax.set_title('Gibbs Triangle: Composition Representation of Fe-Cr-Ni Ternary System', fontsize=15, fontweight='bold')

plt.tight_layout()
plt.savefig('gibbs_triangle.png', dpi=150, bbox_inches='tight')
plt.show()

print("📌 Composition Confirmation:")
print(f"SUS304: Fe={a_sample*100:.1f}%, Cr={b_sample*100:.1f}%, Ni={c_sample*100:.1f}%")
print(f"Total: {(a_sample + b_sample + c_sample)*100:.1f}%")

💡 Practical Example: Composition Representation of Stainless Steels

Stainless steels are representative examples of Fe-Cr-Ni ternary systems. SUS304 (Fe-18Cr-8Ni) is austenitic stainless steel, and SUS430 (Fe-17Cr) is ferritic stainless steel. By confirming which phase region of the phase diagram these compositions are in on the Gibbs triangle, the crystal structure at room temperature (austenite FCC or ferrite BCC) can be predicted.

2. Isothermal Sections

An isothermal section is a cross-sectional diagram showing the phase equilibrium of a ternary system at a specific temperature. Each phase region and phase boundary is drawn on the Gibbs triangle.

2.1 How to Read Isothermal Sections

💡 Tie-lines and Lever Rule

At composition point P within a two-phase region, two phases (α and β) on the tie-line coexist. The fraction of each phase can be calculated using the lever rule, as in binary systems:

\[ f_\alpha = \frac{|\text{P-β}|}{|\text{α-β}|}, \quad f_\beta = \frac{|\text{P-α}|}{|\text{α-β}|} \]

However, the distance is the distance in composition space on the Gibbs triangle.

3. Liquidus Projection

A liquidus projection is a diagram showing the temperature at which the liquid phase begins to solidify (liquidus temperature) as contour lines on the Gibbs triangle. It is useful for understanding cooling paths and solidification processes.

3.1 Components of Liquidus Projection

Tracking Cooling Paths

On the liquidus projection, the cooling path of an alloy can be tracked as follows:

  1. Start cooling from composition point P
  2. When the liquidus temperature is reached, primary crystals (α, β, or γ) begin to crystallize
  3. As cooling progresses, the composition of the liquid changes along the primary crystallization line
  4. Descend the primary crystallization line and reach the eutectic valley
  5. Descend the eutectic valley and completely solidify at the ternary eutectic point

4. Vertical Sections (Pseudo-Binary Sections)

A vertical section is a temperature-composition diagram along a specific line on the Gibbs triangle (e.g., a line from edge A-B to vertex C). It has a similar appearance to binary phase diagrams.

4.1 Applications of Vertical Sections

💡 Application Example of Vertical Sections

In welding of stainless steels, the phase state during solidification is important. By using vertical sections, it is possible to predict the solidification path at a specific Cr/Ni ratio and design compositions that avoid the formation of harmful phases (such as σ phase) that cause weld cracking.

5. Ternary Eutectic Points and Invariant Reactions

In ternary systems, there are invariant reactions such as ternary eutectic reactions \( L \rightarrow \alpha + \beta + \gamma \). According to Gibbs’ phase rule, these reactions occur at specific temperatures and compositions.

5.1 Invariant Reactions in Ternary Systems

Reaction TypeReaction EquationCharacteristics
Ternary Eutectic\( L \rightarrow \alpha + \beta + \gamma \)Liquid decomposes into three solid phases
Ternary Peritectic\( L + \alpha + \beta \rightarrow \gamma \)Liquid and two solid phases react to form a new solid phase
Ternary Monotectic\( L_1 \rightarrow L_2 + \alpha + \beta \)Liquid separates into two liquids and a solid phase
Quasi-Peritectic\( L + \alpha \rightarrow \beta + \gamma \)Liquid and solid phase react to form two solid phases
# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0


            <h4>Code Example 5: Determination of Ternary Eutectic Point</h4>
            <pre><code class="language-python">import numpy as np
import matplotlib.pyplot as plt
from matplotlib.patches import Polygon
from mpl_toolkits.mplot3d import Axes3D

def ternary_to_cartesian(a, b, c):
    """Convert ternary coordinates to Cartesian coordinates"""
    total = a + b + c
    x = 0.5 * (2*b + c) / total
    y = (np.sqrt(3)/2) * c / total
    return x, y

# Ternary Eutecticreaction visualization
fig = plt.figure(figsize=(14, 6))

# Left: Ternary eutectic point on Gibbs triangle
ax1 = fig.add_subplot(121)

# Triangle vertices
vertices = np.array([[0, 0], [1, 0], [0.5, np.sqrt(3)/2]])
triangle = Polygon(vertices, fill=False, edgecolor='black', linewidth=2)
ax1.add_patch(triangle)

ax1.text(-0.05, -0.05, 'A', fontsize=14, fontweight='bold')
ax1.text(1.05, -0.05, 'B', fontsize=14, fontweight='bold')
ax1.text(0.5, np.sqrt(3)/2 + 0.05, 'C', fontsize=14, fontweight='bold', ha='center')

# Ternary eutectic point
eutectic_comp = (0.40, 0.35, 0.25)  # A, B, C
e_x, e_y = ternary_to_cartesian(*eutectic_comp)
ax1.plot(e_x, e_y, 'r*', markersize=20, label='Ternary eutectic point E')

# Compositions of three solid phases in equilibrium
alpha_comp = (0.85, 0.10, 0.05)
beta_comp = (0.15, 0.75, 0.10)
gamma_comp = (0.20, 0.15, 0.65)

alpha_x, alpha_y = ternary_to_cartesian(*alpha_comp)
beta_x, beta_y = ternary_to_cartesian(*beta_comp)
gamma_x, gamma_y = ternary_to_cartesian(*gamma_comp)

ax1.plot(alpha_x, alpha_y, 'go', markersize=10, label='α phase')
ax1.plot(beta_x, beta_y, 'bo', markersize=10, label='β phase')
ax1.plot(gamma_x, gamma_y, 'mo', markersize=10, label='γ phase')

# Tie-triangle
tie_triangle = Polygon(
    [ternary_to_cartesian(*alpha_comp),
     ternary_to_cartesian(*beta_comp),
     ternary_to_cartesian(*gamma_comp)],
    fill=False, edgecolor='red', linewidth=2, linestyle='--', label='Tie-triangle'
)
ax1.add_patch(tie_triangle)

# Composition labels
ax1.text(alpha_x + 0.05, alpha_y, 'α', fontsize=11, color='green', fontweight='bold')
ax1.text(beta_x + 0.05, beta_y, 'β', fontsize=11, color='blue', fontweight='bold')
ax1.text(gamma_x + 0.05, gamma_y, 'γ', fontsize=11, color='purple', fontweight='bold')
ax1.text(e_x + 0.03, e_y + 0.05, 'E', fontsize=11, color='red', fontweight='bold')

ax1.set_xlim(-0.1, 1.1)
ax1.set_ylim(-0.1, 1.0)
ax1.set_aspect('equal')
ax1.axis('off')
ax1.legend(loc='upper right', fontsize=9)
ax1.set_title('(a) Ternary eutectic point and Tie-triangle', fontsize=13, fontweight='bold')

# Right: Cooling curves
ax2 = fig.add_subplot(122)

time = np.linspace(0, 100, 500)

# Cooling curve (ternary eutectic composition)
temp_eutectic = 900 - 5*time
temp_eutectic[temp_eutectic < 550] = 550  # Arrest at eutectic temperature
temp_eutectic[time > 60] = 550 - 3*(time[time > 60] - 60)

# Cooling curve (non-eutectic composition)
temp_noneutectic = 950 - 5*time
temp_noneutectic[(temp_noneutectic < 600) & (temp_noneutectic > 550)] = \
    600 - 0.5*(time[(temp_noneutectic < 600) & (temp_noneutectic > 550)] - 50)
temp_noneutectic[temp_noneutectic < 550] = 550
temp_noneutectic[time > 70] = 550 - 3*(time[time > 70] - 70)

ax2.plot(time, temp_eutectic, 'r-', linewidth=2, label='Eutectic composition (point E)')
ax2.plot(time, temp_noneutectic, 'b-', linewidth=2, label='Non-eutectic composition')

# Eutectic temperature line
ax2.axhline(550, color='gray', linestyle='--', linewidth=1, alpha=0.7)
ax2.text(10, 560, 'Ternary eutectic temperature T_E', fontsize=10, color='gray')

ax2.set_xlabel('Time (arbitrary units)', fontsize=12)
ax2.set_ylabel('Temperature (°C)', fontsize=12)
ax2.set_title('(b) Cooling Curves: Arrest at Eutectic Temperature', fontsize=13, fontweight='bold')
ax2.set_xlim(0, 100)
ax2.set_ylim(400, 1000)
ax2.grid(True, alpha=0.3)
ax2.legend(loc='upper right', fontsize=10)

plt.tight_layout()
plt.savefig('ternary_eutectic.png', dpi=150, bbox_inches='tight')
plt.show()

print("📌 Characteristics of ternary eutectic reaction:")
print(f"• Eutectic composition: A={eutectic_comp[0]*100:.0f}%, B={eutectic_comp[1]*100:.0f}%, C={eutectic_comp[2]*100:.0f}%")
print("• Reaction: L → α + β + γ")
print("• Cooling curve: Significant arrest at eutectic temperature (release of latent heat)")
print("• Tie-triangle: 3triangle connecting equilibrium compositions of three solid phases")

Gibbs’ Phase Rule and Ternary Systems

Applying Gibbs’ phase rule \( F = C - P + 2 \) to ternary systems:

In ternary eutectic reactions, since liquid L and three solid phases α, β, γ coexist (P=4), they occur only at specific temperatures and compositions.

6. Principles of CALPHAD Method

The CALPHAD method (CALculation of PHAse Diagrams) is a technique for calculating phase diagrams using thermodynamic databases. It combines experimental data and theoretical models to predict phase equilibria in complex multicomponent systems.

6.1 Basic Concepts of CALPHAD Method

Advantages of CALPHAD Method

  1. Extrapolation capability : Predict phase diagrams in regions without experimental data (high temperature, extreme compositions)
  2. Extension to multicomponent systems : Extrapolate from binary system data to ternary, quaternary, and higher-order systems
  3. Time and cost reduction : Significantly reduce the number of experimental trials
  4. Integrated approach : Can calculate not only phase diagrams but also heat capacity, activity, chemical potential, etc.

6.2 Gibbs Free Energy Model for Phases

The Gibbs free energy of a solution phase in binary system A-B is expressed in the following form:

\[ G_m = x_A {}^0G_A + x_B {}^0G_B + RT(x_A \ln x_A + x_B \ln x_B) + {}^{\text{ex}}G_m \]

The excess Gibbs free energy is approximated by the Redlich-Kister polynomial :

\[ {}^{\text{ex}}G_m = x_A x_B \sum_{i=0}^{n} {}^iL_{A,B} (x_A - x_B)^i \]

💡 Model Extension to Ternary Systems

In ternary system A-B-C, in addition to binary parameters (A-B, B-C, C-A), a ternary interaction parameter is introduced:

\[ {}^{\text{ex}}G_m^{\text{ABC}} = {}^{\text{ex}}G_m^{\text{AB}} + {}^{\text{ex}}G_m^{\text{BC}} + {}^{\text{ex}}G_m^{\text{CA}} + x_A x_B x_C L_{\text{ABC}} \]

Here, \( L_{\text{ABC}} \) is the ternary interaction parameter. In many cases, extrapolation from binary data provides sufficient accuracy, so \( L_{\text{ABC}} = 0 \) is approximated.

7. CALPHAD Method Workflow

Phase diagram calculation using the CALPHAD method is performed in the following steps:

5 Steps of CALPHAD Workflow

  1. Literature review : Collect existing experimental data (phase diagrams, heat capacity, activity, etc.)
  2. Model selection : Select appropriate thermodynamic models for each phase (liquid, solid solution, compound)
  3. Parameter Optimization : Experimental dataOptimize model parameters to best fit
  4. Database construction : Store optimized parameters in TDB file (Thermo-Calc DataBase)
  5. Phase diagram calculation and verification : Calculate phase diagrams using the database and verify by comparison with experimental data

7.1 Structure of Thermodynamic Databases

In the CALPHAD method, thermodynamic data are stored in TDB files (Thermo-Calc DataBase format). Representative databases:

💡 Actual CALPHAD Calculation Software

The above is an extremely simplified example for educational purposes. In actual CALPHAD calculations, the following software is used:

In the next chapter, we will perform practical phase diagram calculations using pycalphad.

Exercises

Exercise 1: Reading Composition on Gibbs Triangle

Problem: In the Fe-Cr-Ni ternary system, plot point P (Fe: 70%, Cr: 20%, Ni: 10%) on the Gibbs triangle and calculate the composition distance to SUS304 (Fe: 74%, Cr: 18%, Ni: 8%).

Hint

Convert ternary coordinates to Cartesian coordinates, then calculate the Euclidean distance. The distance in composition space serves as an indicator of similarity in actual material properties.

Sample Answer

# Composition distance calculation on Gibbs triangle
def ternary_to_cartesian(a, b, c):
    x = 0.5 * (2*b + c) / (a + b + c)
    y = (np.sqrt(3)/2) * c / (a + b + c)
    return x, y

# Composition 1: Point P
P_comp = (0.70, 0.20, 0.10)
P_x, P_y = ternary_to_cartesian(*P_comp)

# Composition 2: SUS304
SUS304_comp = (0.74, 0.18, 0.08)
SUS304_x, SUS304_y = ternary_to_cartesian(*SUS304_comp)

# Euclidean distance
distance = np.sqrt((P_x - SUS304_x)**2 + (P_y - SUS304_y)**2)

print(f"Point P: ({P_x:.4f}, {P_y:.4f})")
print(f"SUS304: ({SUS304_x:.4f}, {SUS304_y:.4f})")
print(f"Composition distance: {distance:.4f}(normalized distance on triangle)")
print(f"Composition difference: ΔFe={abs(0.70-0.74)*100:.1f}%, ΔCr={abs(0.20-0.18)*100:.1f}%, ΔNi={abs(0.10-0.08)*100:.1f}%")

Exercise 2: Tie-line in Isothermal Section

Problem: In the isothermal section of the Fe-Cr-Ni system at 1200°C, assume an alloy with composition Fe: 60%, Cr: 25%, Ni: 15% is in the L+γ two-phase region. When the liquid composition is Fe: 50%, Cr: 30%, Ni: 20%, and the γ phase composition is Fe: 65%, Cr: 22%, Ni: 13%, find the fraction of each phase.

Hint

Apply the lever rule in ternary coordinates. The inverse ratio of distances from the alloy composition point to each phase gives the phase fraction.

Sample Answer

# Lever rule in ternary coordinates
alloy_comp = np.array([0.60, 0.25, 0.15])  # Fe, Cr, Ni
L_comp = np.array([0.50, 0.30, 0.20])
gamma_comp = np.array([0.65, 0.22, 0.13])

# Vector distance calculation
dist_alloy_to_L = np.linalg.norm(alloy_comp - L_comp)
dist_alloy_to_gamma = np.linalg.norm(alloy_comp - gamma_comp)
dist_L_to_gamma = np.linalg.norm(L_comp - gamma_comp)

# Lever rule
f_gamma = dist_alloy_to_L / dist_L_to_gamma
f_L = dist_alloy_to_gamma / dist_L_to_gamma

print(f"Fraction of liquid (L): {f_L*100:.1f}%")
print(f"γ phasefraction: {f_gamma*100:.1f}%")
print(f"Total: {(f_L + f_gamma)*100:.1f}%")

# Verification: Mass conservation
reconstructed_comp = f_L * L_comp + f_gamma * gamma_comp
print(f"\nVerification (Mass conservation law):")
print(f"Original composition: Fe={alloy_comp[0]*100:.1f}%, Cr={alloy_comp[1]*100:.1f}%, Ni={alloy_comp[2]*100:.1f}%")
print(f"Reconstructed composition: Fe={reconstructed_comp[0]*100:.1f}%, Cr={reconstructed_comp[1]*100:.1f}%, Ni={reconstructed_comp[2]*100:.1f}%")

Exercise 3: Fitting Redlich-Kister Equation

Problem: Activity data for a binary system A-B (at 1000 K, excess Gibbs free energies at x_B = 0.2, 0.4, 0.6, 0.8 are 5000, 8000, 8000, 5000 J/mol) have been obtained. Determine the Redlich-Kister parameters L0, L1 by the least squares method.

Hint

Redlich-Kisterequation \( {}^{\text{ex}}G_m = x_A x_B (L_0 + L_1(x_A - x_B)) \) to fit the experimental data. scipy.optimize.curve_fit is convenient.

Sample Answer

from scipy.optimize import curve_fit

# Experimental data
x_B_data = np.array([0.2, 0.4, 0.6, 0.8])
ex_G_data = np.array([5000, 8000, 8000, 5000])  # J/mol

# Redlich-Kister model (L0, L1)
def redlich_kister_model(x_B, L0, L1):
    x_A = 1 - x_B
    return x_A * x_B * (L0 + L1 * (x_A - x_B))

# Fitting
popt, pcov = curve_fit(redlich_kister_model, x_B_data, ex_G_data)
L0_fit, L1_fit = popt

print(f"Optimized parameters:")
print(f"  L0 = {L0_fit:.1f} J/mol")
print(f"  L1 = {L1_fit:.1f} J/mol")

# Visualization
x_B_fine = np.linspace(0.01, 0.99, 100)
ex_G_fit = redlich_kister_model(x_B_fine, L0_fit, L1_fit)

plt.figure(figsize=(9, 6))
plt.plot(x_B_data, ex_G_data, 'ro', markersize=10, label='Experimental data')
plt.plot(x_B_fine, ex_G_fit, 'b-', linewidth=2, label=f'Fit (L₀={L0_fit:.0f}, L₁={L1_fit:.0f})')
plt.xlabel('x_B', fontsize=13)
plt.ylabel('Excess Gibbs Free Energy (J/mol)', fontsize=13)
plt.title('Optimization of Redlich-Kister Parameters', fontsize=14, fontweight='bold')
plt.legend(fontsize=11)
plt.grid(True, alpha=0.3)
plt.tight_layout()
plt.show()

Exercise 4: Application of CALPHAD Method

Problem: Consider the composition of SUS316L (Fe-17Cr-12Ni-2.5Mo) in the Fe-Cr-Ni ternary system. Discuss how Mo addition changes the solidification temperature from the CALPHAD perspective. (Qualitative discussion is acceptable)

Hint

Mo is a high melting point element (2623°C) and acts as a solid solution strengthening element in steel. Consider the effect on liquid phase stability.

Sample Answer

Qualitative Discussion:

CALPHAD calculation (concept) : Using Thermo-Calc’s TCFE (steel) database, vertical sections of Fe-17Cr-12Ni-xMo (x = 0, 1, 2, 3 wt%) can be calculated to quantify changes in liquidus and solidus.

Summary

In this chapter, we learned how to read ternary phase diagrams and the principles of the CALPHAD method.

Review of Key Points

  1. Gibbs triangle : Represents ternary composition on a 2D equilateral triangle. Each vertex is a pure component, each edge is a binary system.
  2. Isothermal section : Shows phase equilibrium at specific temperature. Represents two-phase and three-phase regions with tie-lines and tie-triangles.
  3. Liquidus projection : Contour map of liquidus temperature. Primary crystallization lines and eutectic valleys are important. Useful for tracking cooling paths.
  4. Vertical section : Temperature-composition diagram with fixed composition ratio. Can be analyzed in the same format as binary phase diagrams.
  5. Ternary eutectic reaction : Invariant reaction L → α + β + γ. Tie-triangle represents equilibrium compositions of three phases.
  6. CALPHAD method : Phase diagram calculation by Gibbs free energy minimization. Utilizes thermodynamic database (TDB).
  7. Redlich-Kister equation : Models excess Gibbs free energy of solution phase. Expresses non-ideality with interaction parameter L.
  8. CALPHAD workflow : Literature review → Model selection → Parameter optimization → Database construction → Phase diagram calculation and verification.

💡 Next Steps

In the next chapter, we will learn practical phase diagram calculation using pycalphad. Using the open-source Python library pycalphad, we will calculate phase diagrams from actual thermodynamic databases (TDB) and quantitatively analyze the relationship between temperature, composition, and phase fraction. Let’s experience the power of the CALPHAD method through phase diagram calculations of practical materials such as Fe-C system, Al-Cu system, and Ni-based superalloys.

Learning Check

Check if you can answer the following questions:

← Chapter 4: Reading and Analysis of Binary Phase Diagrams

Disclaimer

← Previous Chapter Back to Series Index